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Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions: • The exam consists two parts: 5 short answers (6 points each) and your pick of 2 out 3 long answer problems (35 points each). • Where possible, show all work, partial credit will be given. • Personal notes on two sides of a 8X11 page are allowed. • Total time: 3 hours Good luck! Short Answers: S1. Consider a free particle moving in 1D. Shown are two different wave packets in position space whose wave functions are real. Which corresponds to a higher average energy? Explain your answer. S2. Consider an atom consisting of a muon (heavy electron with mass m μ 200m e ) bound to a proton. Ignore the spin of these particles. What are the bound state energy eigenvalues? What are the quantum numbers that are necessary to completely specify an energy eigenstate? Write out the values of these quantum numbers for the first excited state. S3. Two particles of spins 1 s r and 2 s r interact via a potential 2 1 s s a H r r = . a) Which of the following quantities are conserved: 2 2 1 2 2 2 1 2 1 s s s s s s s s r r r r r r r r , , , , , + = ? b) If s 1 = 5 and s 2 = 1, what are the possible values of the total angular momentum quantum number s? (a) (b) ψ ( x ) ψ ( x ) ψ = 0

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Page 1: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New Mexico

Fall 2004

Instructions:• The exam consists two parts: 5 short answers (6 points each) and your pickof 2 out 3 long answer problems (35 points each).• Where possible, show all work, partial credit will be given.• Personal notes on two sides of a 8X11 page are allowed.• Total time: 3 hours

Good luck!

Short Answers:S1. Consider a free particle moving in 1D. Shown are two different wave packets inposition space whose wave functions are real. Which corresponds to a higher averageenergy? Explain your answer.

S2. Consider an atom consisting of a muon (heavy electron with mass

mµ ≈ 200me)bound to a proton. Ignore the spin of these particles. What are the bound state energyeigenvalues? What are the quantum numbers that are necessary to completely specify anenergy eigenstate? Write out the values of these quantum numbers for the first excitedstate.

S3. Two particles of spins 1sr and

2sr interact via a potential

21ssaHrr ⋅=′ .

a) Which of the following quantities are conserved:2

2122

2121

ssssssssrrrrrrrr ,,,,, += ?

b) If

s1 = 5 and

s2 = 1, what are the possible values of the total angularmomentum quantum number s?

(a) (b)

ψ (x)

ψ (x)

ψ (x)

ψ (x)

ψ = 0

ψ (x)

Page 2: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

S4. The energy levels En of a symmetric potential well V(x) are denoted below.

E1E0

E2

E3

E4V(x)

x=-b x=-a x=bx=a

(a) How many bound states are there?(b) Sketch the wave functions for the first three levels (n=0,1,2). For each, denote theregions where the particle is classically forbidden.(c) Describe the evolution of the wave function if we start in an equal superposition ofeigenstates n=0 and n=1.

S5. A particle with well defined energy E is scattered from dimensional step potential ofheight

V0 . What is the probability of reflection when

E < V0 and

E > V0. E

Long Answers: Pick two out of three problems belowL1. A harmonically bound particle experiences a constant force F.(a) Argue that the interaction Hamiltonian associated with this force is

ˆ H int = −Fˆ x .

(b) Assuming the force is weak compared to the harmonic binding, what is the lowestnonvanishing perturbation to the energy of the bound states

(c) Show that this lowest nonvanishing perturbation is

ΔE = −Fx0( )2hω

, where

x0 =h

2mω

independent of level. Hint: Recall

ˆ x = x0 ˆ a + ˆ a †( ) , where

ˆ a †, ˆ a are the creation,annihilation operators.

(d) Solve the Schrödinger equation exactly and find the new energy eigenstates andeigenvalues including the harmonic potential and perturbing force.

(e) Show that your exact energy spectrum agrees with the perturbation result to theappropriate order in the perturbation’s small parameter.

V0

Page 3: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L2. Consider a spin-1 particle. The matrix representations of the components of spinangular momentum are

ˆ S x =12

0 1 01 0 10 1 0

⎢ ⎢ ⎢

⎥ ⎥ ⎥ ,

ˆ S y =1

i 2

0 1 0−1 0 10 −1 0

⎢ ⎢ ⎢

⎥ ⎥ ⎥ ,

ˆ S z =1 0 00 0 00 0 −1

⎢ ⎢ ⎢

⎥ ⎥ ⎥

expressed in the ordered basis,

+1 , 0 , −1{ }, where the ket is labeled by the eigenvalueof

ˆ S z in units of h.

(a) Show that

ψ A =12

+1 − 2 0 + −1( ),

ψ B =12

+1 + 2 0 + −1( ) , and

ψ c =12

+1 − −1( ) are eigenvectors of

ˆ S x . What are their eigenvalues?

(b) Suppose a beam of particles identically prepared in the state

Ψ =34

+1 +i

2 20 +

1+ i4

−1 is put through a Stern-Gerlach apparatus with the

gradient B-field along z. Three beams emerge.

Explain this phenomenon. What are the relative intensities of these beams?

(c) Now suppose the central beam is passed through a second Stern-Gerlach device,oriented with the gradient field along the x-axis.

How many beams will emerge and with what relative intensities?

S

N

Page 4: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L3. A deuteron is bound state of proton and neutron (mp ~ mn~m~939 MeV/c2).It can be modeled as one particle of reduced mass (µ = mp mn/(mp+mn) ) in acentral square well potential of width b and depth −V0 , as shown in the figure.For the deuteron there is only one bound state with a binding energy of EB = -2.2MeV, which can be assumed to have no orbital angular momentum.

(a) Find the general solution for the reduced radial wave function u(r) = r R(r)for the two regions r<b and r>b. (You do not need to normalize the wavefunction)

(b) Show that from the general solution you get a relation between EB, V0, b ofthe form:

tankb = −kq

,

where k, q are functions of m, EB and V0.

(c) What is the width of the potential (in fm) if its depths is V0 = 38.5 MeV?(Use fmMeVc 197=h and note that kb must be positive!)

(d) Can there be a second bound state of this potential?

V(r)

EB

b

r

- V0

Page 5: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics

Department of Physics and Astronomy

University of New Mexico

Fall 2005

Instructions:

• The exam consists of two parts: 5 short answers (6 points each) and your

choice of 2 out 3 long answer problems (35 points each).

• Where possible, show all work, partial credit will be given.

• Personal notes on two sides of a 8X11 page are allowed.

• Total time: 3 hours

Good luck!

Short answers:

S1. Consider a balanced two-port interferometer sketched below, with 50/50 beam

splitters, each arm having a path length L. The optical path length of one arm can be

made longer through a phase shift !. A single photon of frequency " is prepared and

injected into the input port.

• What is the probability of detecting a photon at the output-port (detector shown) as a

function of !?

• If the other arm of the interferometer is blocked with a completely absorbing barrier

(shown below), what is probability of detecting a photon at the denoted output-port vs. !?

!

Phase shifter

!

Detector mirror

beam-splitter

Page 6: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

S2. A particle of mass m moves freely in a two-dimensional, infinite height, rectangular

potential width a and height b.

What are the energy levels and energy eigenfunctions of the system? Under what

conditions are there degeneracies in the spectrum?

S3. A particle is trapped in the ground state harmonic well of frequency

!0

. Its state

vector is

u0

. At time t0 the curvature of the well is changed suddenly so that the fre-

quency is doubled. The energy eigenstates of the new well are denoted

vn

n = 0,1,...{ } .

• What is the probability of finding the particle in the first excited state of the new well?

• Suppose that the frequency is not changed suddenly, but with some arbitrary time

dependence. Describe the condition under which the probability of finding the particle in

the new ground state is approximately 1.

S4. A spin 3/2 nucleus is placed in a magnetic field B in the z-direction. The nuclear

magnetic dipole moment is described by the operator

r µ = gµN

r I , where

µN

is the nuclear

magneton (with g-factor) and

r I the nuclear spin magnetic moment operator in units of h.

The Hamiltonian describing the interaction of the spin with the field is

H = !r

µ "B .

• What are the possible eigenvalues and eigenvectors. What is the ground state of the

system?

• Roughly describe an experiment you might perform to measure the spectrum of

eigenvalues.

S5. The valence shell of He has the configuration 1s2. What is the total spin of the

ground state? What is the degeneracy of the ground state? Is there any spin-orbit

coupling in the ground state?

b

a �

V = !

x

y

V = !

V = 0

Page 7: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L1. Tunneling and decay

Consider a particle moving in 1D incident on thin potential barrier. We will model this

barrier as a repulsive delta-function potential at the origin,

V (x) = U0!(x).

(a) What are the units of U0?

(b) Use the time-independent Schrödinger equation to show that the derivative of the

wave function at the origin is discontinuous by,

d!

dx0+

"d!

dx0"

=2m

h2U0! (0).

(c) Now consider the scattering of a free particle with energy E. We denote the

probability amplitude for transmission t and reflection r.

Show that

r =!i"

k + i",

t =k

k + i! where

k = 2mE /h2

and

! =mU

0

h2

. Is the sum of the

probabilities for reflection and transmission equal to one?

(d) A simple model of #-particle decay is as follows. The confinement of the #-particle

in the nucleus is model by a square well of width L. Decay occurs via tunneling through

a finite delta-function barrier as above.

For an alpha particle near the ground state, estimate the decay rate to free space.

V (x) = U0!(x)

eikx

teikx

re!ikx

!

L

E

V (x) = U0!(x " L)

Page 8: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L2. Coherent states of the harmonic oscillator

Consider a harmonic oscillator in one dimension for a particle of mass m and oscillator

frequency ". A special class of wave packets known as “coherent states” can be

constructed from a superposition stationary states (defined by

ˆ H n = h! n + 1/2( ) n ),

! = e"!

2/ 2

!( )n

n!n=0

#

$ n ,

where # is a complex number. It has the property that it is an eigenstate of the

“annihilation operator” satisfying,

ˆ a ! = ! ! .

(a) Show that the mean position and momentum of the wave packet are

ˆ x =2h

m!Re(") ,

ˆ p = 2hm! Im(").

(b) Show that the uncertainty variances in position and momentum are,

!x2

=h

2m",

!p2

=hm"

2.

Is this a minimum uncertainty wave packet? Explain.

(c) Suppose at time t=0 the oscillator is prepared in this wave packet,

! (0) = " .

Show that a later time, the particle is still in a coherent state (up to an overall irrelevant

phase) but with

! "!e# i$t , i.e.

! (t) = "e#i$t

Page 9: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L3. The Stark effect

Consider a hydrogen atom in a uniform static electric field F (we use F here to avoid

confusion with the energy E). For a fixed nucleus, and field in z-direction, the

perturbation on the electron is described by the Hamiltonian

H1

= eFz = eFrcos! ,

where z=rcos! is the z-coordinate of position of the electron relative to the nucleus (also

expressed in spherical coordinates). We ignore spin and any relativistic effects in this

problem.

(a) Show that the perturbation has no effect on the energy of the ground state, 1s, to first

order in F. Explain physically.

(b) Show that to second order, the shift on the ground state is

!E1s

(2)= "2a

0F2

nlm z 1,0,02

1"1

n2

m="l

l

#l=0

n"1

#n=2

$

# ,

where

a0 is the Bohr radius and

nlm are the unperturbed stationary states of hydrogen

with principle quantum number n, angular momentum magnitude l, and projection m.

(c) Given the facts

Y0,0(!,") =

1

4# and

Y1,0(!,") =

3

4#cos! (spherical harmonics),

show that the perturbation simplifies to

!E1s

(2)= "2a

0F2

In

2

1"1

n2

n=2

#

$ ,

where

In

=1

3dr r R

n1(r)R

10(r)

0

!

" ,

with

Rnl(r) the reduced radial wave function for unperturbed hydrogen.

Page 10: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy

University of New Mexico Fall 2006

Instructions: • The exam consists of 10 short answers (10 points each). • Where possible, show all work, partial credit will be given. • Personal notes on two sides of an 8X11 page are allowed. • Total time: 3 hours Good luck! 1. A Hydrogen atom is in a state with principle quantum number n = 2. (4pts) What are the possible values of angular momentum in this state? (6pts) Consider the state with angular momentum quantum numbers l = 1, m = 1. Using the Virial Theorem, find the average speed of the electron.

Page 11: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

2. The raising and lowering operators for the quantum harmonic oscillator satisfy † 1 1

1

a n n n

a n n n

= + +

= −

for energy eigenstates n with energy En. Determine the first-order shift in the n = 2

energy level due to the perturbation ( )2†H V a aΔ = + , where V is a constant.

Page 12: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

3. For atoms of low to moderate Z, the LS coupling scheme is appropriate. Consider an atom with two optically active electrons with quantum numbers l1 = 2, s1 = ½; l2 = 3, s2 = ½. Find the possible values of l, s, and the resulting values of the total angular momenta j. Be sure to specify which values of j go with each l and s combination.

Page 13: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

4. In a photoelectric experiment, monochromatic light of wavelength 400 nm shines on a metal anode with a work function of 2.5 eV. What electric potential between the anode and cathode would be required to stop all of the photoelectrons from reaching the cathode? If one were now to use monochromatic light of half the wavelength and half the intensity with the same metal anode, what will be the new stopping potential?

Page 14: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

5. Imagine a system with just two linearly independent states: 1 0

1 , 20 1⎛ ⎞ ⎛ ⎞

≡ ≡⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

so that

a general state can be written as 2 21 2 ,S a b with a b 1= + + =

⎞⎟

. Suppose the

Hamiltonian for this system has the form: . If the system starts out (at t = 0)

in state

h gg h

⎛= ⎜⎝ ⎠

H

1 , what is its state at time t? Give a physical example of such a system.

Page 15: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

6. The energy levels En of a symmetric potential well are denoted below:

(1pt) How many bound states are there? (4pts) Sketch the wavefunctions for the n = 0, 1, 2, and 4 levels. For each, denote the

regions where the particle is classically forbidden. (5pts) Describe the evolution of the wavefunction if we start in an equal superposition of

the n = 0 and n = 1 eigenstates.

Page 16: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

7. Given a particle with the Hamiltonian ( )2ˆ

2pH Vm

= + x , find ˆd pdt

. Interpret this

result.

Page 17: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

8. A particle of mass m scatters off of a 3-D potential given by:

( ) 0 0

00V r R

V rr R

− <⎧= ⎨ ≥⎩

Find the s-wave scattering phase shift for the scattered wave.

Page 18: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

9. Consider a 2-D Hilbert space (describing e.g. a spin ½ particle). Let us denote abstractly two different orthonormal basis sets for this space: { },+ − and { },↑ ↓ .

The inner products of these states are given by: 1 1,2 2

↑ + = ↑ − = ↓ + = − ↓ − = .

Let 1 23 3

iψ = + + − be the state of the particle. Write the state ψ as a

superposition of the basis vectors { },↑ ↓ . What is the probability of finding the

photon in the ↓ state?

Page 19: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

10. A spinless particle of mass m resides in two dimensions and experiences the potential

( ) ( 2 21,2

V x y k x y= + ) , with k a positive constant. What are the conserved quantities for

this potential? Give the energies and degeneracies of the first two energy levels.

Page 20: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy

University of New Mexico Spring 2007

Instructions: • The exam consists of 10 problems, 10 points each. • Partial credit will be given if merited. • Personal notes on two sides of an 8 × 11 page are allowed. • Total time is 3 hours. Problem 1: A particle in a potential well is in a superposition of bound states, described by the wave function

! (x) =2

3u0(x) +

1+ i

12u2(x) +

e" i# / 4

6u3(x),

where

un(x) are the orthonormal energy eigenfunctions satisfying,

Hun(x) = E

nun(x) ,

(n=0 ground state, n>0, nth excited state). (i) Show that the wave function is normalized. (ii) If energy is measured, what values are possible and with what probability? (iii) What is the expectation value of energy in terms of the eigenvalues, En? Problem 2: A particle of mass m is in a one-dimensional well described by,

(i) Sketch the ground state and first excited state wave function in the well. (ii) What are the energy eigenvalues?

V (x) =

1

2kx

2, x > 0

!, x < 0

"

# $

% $

Page 21: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Problem 3: Consider a double-well consisting of two-finite square wells, separated by a thick barrier, sketched below. (i) Assuming each well supports 2 bound states in isolation (i.e., with an infinitely thick barrier), sketch the energy levels and eigenfunctions for the double well. (ii) If we now have a periodic train of wells, describe the nature of the energy spectrum. Problem 4: Show that the ground state of a one-dimensional harmonic oscillator is a minimum uncertainty wave packet, i.e.,

!x!p = h /2 . Hint: Recall that the position and momentum operators can be expressed in terms of creation and annihilation operators,

ˆ x = h /(2m!) ˆ a + ˆ a †( ) ,

ˆ p = !i hm" /2 ˆ a ! ˆ a †( ).

Problem 5: To coarsest approximation, a homonuclear diatomic molecule is described by a rigid rotator, with equal mass nuclei M, attached to a rigid rod of length a. (i) Taking the center of mass to be fixed, show that the energy eigenvalues are,

El

= h2l(l +1) /Ma

2 ; l=0,1,2… (Hint: write the appropriate Hamiltonian first based on your knowledge of the classical problem). (ii) What are eigenfunctions and degeneracy of the lth level? Problem 6: A “hydrogenic atom” consists of a single electron orbiting a nucleus with Z protons (Z=1 is hydrogen, Z=2 is singly ionized helium, Z=3 is doubly ionized lithium, etc.). (i) What are the energy levels as a function of the principle quantum number n, for a given Z? (ii) What is the degeneracy of the level n (include electron spin, but ignore nuclear spin)? (iii) Given that the Lyman-alpha line (transition from n=2 to n=1) in hydrogen lies in the far ultraviolet (

! = 121.6 nm), what part of the spectrum does the Lyman-alpha lie in for Z=2 and Z=3?

L

L

M a

Page 22: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Problem 7: Consider a hydrogen atom in a uniform electric field E in the z-direction. The perturbation Hamiltonian is

ˆ H 1

= !eEˆ z , where

ˆ z is the z-coordinate operator of the electron relative to the nucleus. (i) Show that the shift in the ground state vanishes to first order in the perturbation (ii) Write an expression for the second order shift of the ground state. Problem 8: The dynamics of a spin-1/2 electron placed in a magnetic field B in the z-direction is governed by the Hamiltonian

ˆ H = µBB ˆ !

z, where µB is the Bohr magneton

and

ˆ ! z is the z-component Pauli spin-operator. A time t=0 the spin is prepared as spin-

up along x, which can be expressed as a superposition of spin-up and down-up along z,

! (t = 0) = "x

= "z

+ #z( ) / 2 .

What is the probability of finding the electron in

!x

at a later time t? Problem 9: Consider two spin-1/2 particles, labeled A and B. The familiar complete set of commuting operators

C1

=r ˆ S

2= (

r ˆ s

A+

r ˆ s

B), ˆ S

Z= ˆ s

Az+ ˆ s

Az,

r ˆ s

A

2,

r ˆ s

B

2{ } ,

leads to the “coupled basis” of triplets and singlet. An alternative set that leads to the so-called “Bell basis” is

C2

=r ˆ s A

2,

r ˆ s B

2, ˆ !

Axˆ ! Bx

, ˆ ! Az

ˆ ! Bz{ } .

Show that these two sets are not compatible. Note:

r s =

h

2

r ! .

Problem 10: Show that the Bell-basis, defined by the four kets,

! ±( )= "

z

A,"

z

B± #

z

A,#

z

B( ) / 2 , and

! ±( )= "

z

A,#

z

B± #

z

A,"

z

B( ) / 2 , are simultaneous eigenvectors of the set C2 defined in problem 9. What are the eigenvalues of all four operators? Hint: Recall, that for a given spin,

ˆ ! z

= "z

"z# $

z$z

,

ˆ ! x

= "z

#z

+ #z

"z

.

Page 23: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy

University of New Mexico Fall 2008

Instructions: • The exam consists of 10 problems, 10 points each. • Partial credit will be given if merited. • Personal notes on two sides of an 8 × 11 page are allowed. • Total time is 3 hours. Problem 1: A highly collimated, nearly monoenergetic beam of electrons, energy E, is incident on mask with two very thin long slits, separated by distance d. The electrons are detected downstream art a distance L, very far away, L>>d. (i) What is the probability density (unnormalized) of detecting an electron as a function of the position of the detector y, and any other relevant parameters? (ii) We know that measuring the system can affect the wave function, and the probabilities of outcomes of subsequent measurements. If a photon is scattered off the electrons just after they pass the slits, estimate the minimum wavelength of that light such that the interference pattern disappears. Problem 2: A very cold gas of identical bosons will condense into a Bose-Einstein-Condensate (BEC) when the average spacing between the particles is equal to the deBroglie wavelength. (i) Given a gas at density n in three dimensions, find an approximate formula for the critical temperature at which the gas condenses to BEC in terms of n, the mass of the particles m, Planck’s constant, and Boltzmann’s constant. (ii) A composite particle like an atom can be boson and condensed to BEC. The isotope of lithium (Z=3) 7Li is a boson while the isotope 6Li is fermion. Explain how this can be.

L

d

Detector y

Photon source λ

Page 24: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Problem 3: Consider a particle of mass m moving in a one dimensional well with an infinitely high wall at the origin, and a wall of potential height V0 at x=a. What is the minimum height V0 such that the well will support at least one bound state. Problem 4: Consider a simple harmonic oscillator for a particle of mass m and frequency ω. Find the uncertainty product

ΔxΔp for the nth energy eigenstate (with n=0 the ground state). For what n is this the minimum possible value? Problem 5: Consider a diatomic molecule of atoms with masses m1 and m2. We can model the binding potential between the atoms by a spherically symmetric “Lennard-Jones” potential

V (r) = C12 / r12 −C6 / r

6 , where C12, C6 are constants, and r is the relative distance between the atoms. (i) Ignoring any internal structure of the atoms, write a Hamiltonian that describes their relative motion in terms of the Lennard-Jones potential, the relative-coordinate angular momentum, and any other relevant parameters. (ii) Close to the ground state, the potential looks like a harmonic well. In that regime, and for zero angular momentum, what are the approximate energy levels of the molecule? You may express your answer in terms of the Lennard-Jones potential and its derivatives. Problem 6: Consider a spin-1 particle, spin operator S. In the basis of eigenstates of S2 and Sz, ordered as

S = 1,ms = 1 , S = 1,ms = 0 , S = 1,ms = −1{ } , the matrix representations of the three components of spin-angular momentum are

Sx =h

2

0 1 01 0 10 1 0

⎢ ⎢ ⎢

⎥ ⎥ ⎥ , Sy =

h

2

0 −i 0i 0 −i0 i 0

⎢ ⎢ ⎢

⎥ ⎥ ⎥ , Sz = h

1 0 00 0 00 0 −1

⎢ ⎢ ⎢

⎥ ⎥ ⎥ .

(i) Show that these matrices satisfy the expected commutation relations for angular momentum (you need only pick one commutator to show). (ii) What are the eigenvalues of Sx?

V0 �

a

Page 25: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Problem 7: The hyperfine interaction between electron and nuclear spin is described by a Hamiltonian of the form

H = AI ⋅S , where I is the nuclear spin angular momentum, and S the electron spin operator. (i) Ignoring any other effects, what are the good quantum numbers that specify the eigenstates of this Hamiltonian. (ii) What are the eigenvalues of H in terms of these quantum numbers, and what are the degeneracies of the energy levels? Problem 8: Two non-interacting spin ½ identical particles are placed in a one-dimensional harmonic well. (i) Write the two-particle ground state including motional and spin degrees of freedom. (ii) A magnetic field B is applied. What is the energy shift to first order in B due to the interaction with the magnetic moment of the spins? Problem 9: A particle of mass m is placed in a three dimensional, isotropic harmonic well, described by potential,

V (x, y,z) =12mω 2 x 2 + y 2 + z 2( ) .

(i) What is the energy and degeneracy of the first excited state?

A perturbation potential of the form

V1(x, y) = Axy is now added. (ii) Calculate the shifts of the sublevels of the first excited state to lowest nonvanishing order in perturbation theory? Problem 10: Two electrons a and b are in a spin-singlet state

Ψ =12

↑a↓

b− ↓

a↑

b( ) ,

where

↑ , ↓ are spin-up and down along the z-axis. The two electrons fly apart and are analyzed in separated Stern-Gerlach apparatuses. (i) If electron-a is measured along the z-axis and found to be spin-up, what is the probability for electron-b to be found spin-up along z? (ii) If electron-a is measured along the x-axis and found to be spin-up, what is the probability for electron-b to be found spin-up along x? (iii) If electron-a is measured along the z-axis and found to be spin-up, what is the probability for electron-b to be found spin-up along x?

S

N

Ψ =12

↑a↓

b− ↓

a↑

b( )

S

N

Page 26: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim August 18, 2009 p.1

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New MexicoFall 2009

Instructions:

• the exam consists of 10 problems, 10 points each;

• partial credit will be given if merited;

• personal notes on two sides of 8× 11 page are allowed;

• total time is 3 hours.

Table of Constants and Conversion factors

Quantity Symbol ≈ valuespeed of light c 3.00× 108 m/sPlanck h 6.63× 10−34 J·selectron charge e 1.60× 10−19 Ce mass me 0.511 Mev/c2 = 9.11× 10−31 kgp mass mp 938 Mev/c2 = 1.67× 10−27 kgfine structure α 1/137Boltzmann kB 1.38× 10−23 J/KBohr radius a0 = h/(mecα) 0.53× 10−10 mconversion constant hc 200 eV·nmconversion constant kBT@ 300K 1/40 eVconversion constant 1eV 1.60× 10−19 J

1. At what speed is the deBroglie wavelength of an alpha particle equal to that ofa 10 KeV photon.

2. Argue using the deBroglie wavelength whether or not QM is needed to describethe physical system:a) atomic electron (note that for the ground state of hydrogen 〈v〉 = cα)b) hydrogen gas with number density 1021/cm3 at T=300K

3. Three essential features of the non-relativistic Schroedinger equation are that itis linear, it depends on the first time derivative and it explicitly contains thecomplex number i in the time derivative term. State the physical consequencesof each of these features.

4. A particle is confined to a 1D box 0 < x < a and at t = 0 has the wave function

ψ = Ax(a− x).

(a) Determine the normalization constant A.(b) Find the expectation value of the energy for this state.(c) How does your result in (b) compare qualitatively (>,=, <) with the groundstate energy and explain your reasoning.

Page 27: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim August 18, 2009 p.2

5. A particle moving in 1D with kinetic energy E0 is incident on an infinitely widepotential barrier with V0 > E0. Find the penetration depth of the particle intothe barrier.

6. A photon is in the state

|ψ〉 =1

5(3|x〉+ i4|y〉

where |x〉, |y〉 refer to photon states polarized along the x and y axes respectively.(a) What is the probability of the photon being right circularly polarized?(b) Suppose the photon passes through a plate that introduces a relative phaseshift of π between the states |x〉, |y〉. What is the probability of the photon beingright-circularly polarized now?

Page 28: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim August 18, 2009 p.3

7. Consider two electrons with equal and opposite momenta directed along the x-axis and produced in the entangled spin state

1√2(| ⇑z,⇓z〉 − | ⇓z,⇑z〉).

The electron with momentum ~p is the first in the product ket, and the −~pis the second. This is sketched in the figure below, where the boxes labeled“SG +z” represent Stern-Gerlach devices that deflect electrons in the | ⇑z〉state in the +z direction and electrons in the | ⇓z〉 state in the −z direc-tion thereby measuring the electron spin components along the z direction.

a) What are the possible outcomes for the two SG measurements? What are theprobabilities for the outcomes given this entangled state.b) What is the total spin of this state? (Think of the symmetry of this state andhow you would write down the other possible two-electron product states.)c) The experiment is repeated with SG devices oriented in the y direction therebymeasuring the spin components along the y axis. What is the probability foreach of the possible outcomes now? (No calculations needed if you use thesymmetry property of this state.)

8. Calculate the first order correction to the energy of a simple 1D harmonic oscil-lator with displacement in the x direction due to the perturbation bx4.

Page 29: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim August 18, 2009 p.4

9. a) For the 1/r potential (simple hydrogen atom) the states are labeled by quan-tum numbers n,`,m`,s,ms. The states with magnetic quantum number m` butthe same ` are degenerate. Why? Same question for the spin quantum numberss,ms.b) A typical energy level (not to scale) diagram is shown below, where each boxrepresents states of the same n,` and is labeled by the degeneracy including spin(number of states in the box). Complete the diagram up to n=3.

2

l

n

1

2

3

0 1

2

3

10. Add the spin-orbit coupling to the simple hydrogen atom. For the n = 2 level,label all the states according to the “good” quantum numbers. Show that thetotal number of states sums to 2n2 = 8 (2 for spin).

Page 30: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim August 19, 2010 p.1

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New MexicoFall 2010

Instructions:

• the exam consists of 10 problems, 10 points each;

• partial credit will be given if merited;

• personal notes on two sides of 8× 11 page are allowed;

• total time is 3 hours.

1. A polished silicon surface is seen by a free neutron as an (infinitely sharp) im-penetrable barrier. Under the influence of gravity, unlike a classical particle thatrests on a hard horizontal surface, the neutron will hover above the surface.(a) Sketch the wave function of the ground state of such a neutron.(b) Derive an approximate expression for the expectation value of the height ofthe neutron above the silicon using the uncertainty principle (in terms of theacceleration of gravity g, the mass of the neutron mn, and other constants).

2. Consider a particle of mass m in a one-dimensional potential well V (x) = −V0

for |x| < a and equal to zero otherwise. Show that at least one bound state willalways exist.

3. Consider a HamiltonianH = H1 + H2.

(a) When can the energy eigenvalue of the total Hamiltonian H be written asthe sum of eigen-energies Ei for the Hi ( i = 1, 2),

E = E1 + E2?

(b) For a diatomic molecule, what is the rigid rotor approximation and showexplicitly that E can be written as the sum of vibrational and rotational energies.

4. Find the reflection coefficient for 1 dimensional scattering of a particle of massm off of a delta function potential V = −aδ(x).

5. Consider a complex potential V = V0 − iΓ where V0 and Γ are real constantswith dimensions of energy. Use the one dimensional Schrodinger equation toobtain an equation for the probability density and show that the total probabilityP =

∫ψ?ψdx is not conserved; Find dP/dt.

6. An electron is in the spin-up state with respect to a uniform magnetic field. Thefield direction is suddenly rotated at t=0 by 60◦. What is the probability tomeasure the electron in the spin-up state with respect to the new field directionimmediately after the field is rotated?

Page 31: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim August 19, 2010 p.2

7. The spin of an electron interacts with a uniform, static magnetic field with

H0 = ω0Sz.

At t = 0 the spin state is

|ψ(0)〉 =1√2

(| ↑〉+ | ↓〉)),

with the notation | ↑〉 = |12,+1

2〉, | ↓〉 = |1

2,−1

2〉.

Find 〈Sx〉 as a function of time t.

8. For the two-dimensional harmonic oscillator, the unperturbed Hamiltonian isgiven by

H0 =1

2m

(p2

x + p2y

)+

1

2mω2

(x2 + y2

).

Use perturbation theory to determine the first-order energy shifts to the groundstate and the (degenerate) first-excited states due to the perturbation

H1 = 2bxy.

9. Consider a particle of mass m in an infinite spherical well of radius a. Determinethe energy eigenstates and eigenvalues for ` = 0. The energy eigenstates shouldbe written in terms of a normalization constant (A) that you leave undetermined.

10. Consider the two-electron states of Helium in perturbation theory, where wetake the e-e Coulomb interaction to be the perturbation.a) What is the unperturbed energy and degeneracy of the multiplet of firstexcited states?

Explicitly construct these states in terms of single electron hydrogen atom(spatial) energy eigenstates |ni, `i,mi〉i and spinors |±〉i ≡ |1

2,±1

2〉i (where

i = 1, 2 labels the electron) for states with total orbital angular momentumquantum number ` and total spin quantum number s for the cases:b) ` = 0, s = 1; andc) ` = 1, s = 0.

Page 32: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
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Page 34: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
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Page 37: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 38: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 39: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 40: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 41: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 42: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 43: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 44: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 45: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 46: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 47: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New Mexico

Fall 2013

Instructions:�The exam consists of 10 short-answer problems (10 points each).�Where possible, show all work; partial credit will be given if merited.�Personal notes on two sides of an 8�11 page are allowed.�Total time: 3 hours.

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �Useful Formulae and Constants:Z 1

�1dx exp(�ax2)x2n = (2n� 1)!!

p�

2na2n+12

0;0;0

(r; �; �) =1p�a30

exp

�� r

a0

�; a0 =

4��0~2

me2

m = 9:1�10�31kg, ~ = 1:05�10�34Js, �0 = 8:8�10�12C2

Nm2; e = 1:6�10�19C

hSx; Sy

i= i~Sz

x =

r~

2m!0

�b+ by

�p = �i

r~m!02

�b� by

1

Page 48: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P1. A particle with mass m moving in two-dimensions is con�ned to anin�nite rectangular-well potential having length a and width b: What are theenergy eigenfunctions and associated eigenvalues of the system? Under whatconditions are there degeneracies in the spectrum?

P2. A harmonic oscillator with frequency !0 is in the ground state. At timet0 the frequency is abruptly reduced by a factor of two. For t > t0; what is theprobability that the new oscillator, with frequency !0=2; will be in its groundstate?

P3. Calculate the root-mean-square (RMS) speed of the electron in a hy-drogen atom in the ground state (in m/s).

P4. An electron beam is prepared so that each particle is in the spin state

j i = 1p3j"i+ i

q23 j#i ; where the states j"i and j#i are eigenstates of Sz: Along

an axis z0 a measurement of Sz0 is made, and Sz0 = +}=2 is always found.(Sz0 = �}=2 is never found.) What is the angle � between the z0 axis and thez axis?

P5. A particle�s wave function is

(x; y; z) = C(xy + 2yz)e��r

where C and � are constants. What are the possible outcomes if L2 is measured?What are the possible outcomes if Lz is measured? You may need the followingspherical harmonics:

2

Page 49: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P6. A free electron is initially prepared in a Gaussian wave packet,

(x) =1

(2��2)1=4exp(� x2

4�2)

with standard deviation � = 10 �m in the x direction. Estimate the time (inseconds) for the uncertainty in the electron�s position in x to increase to 20 �m:

P7. An electron with mass M is constrained to move on a ring of radius ain the xy plane. A uniform magnetic �eld B passes through the ring in the zdirection. The Schroedinger equation is

1

2M

�}ia

@

@�� eBa

2

�2 (�) = E (�):

What are the energy eigenvalues E? Sketch a graph of the ground state energyas a function of B:

P8. A harmonic oscillator with frequency !0 and mass m is prepared in thestate

j i = e�12 j�j

21Xn=0

�npn!jni

where � is a complex number, and jni is an eigenstate of the number operatorn = byb; where by and b are the raising and lowering operators. Show that j iis an eigenstate of b; and �nd the associated eigenvalue. Find hx(t)i :

P9. Consider a particle with mass m in a three-dimensional spherical"square" well centered at the origin, described by the potential energy func-tion,

U(r) =

��U0; r < a0; r � a

;

sketched in the �gure below. The radial Schroedinger equation is

� ~2

2m

d2

dr2�+

�~2` (`+ 1)2mr2

+ U(r)

�� = E�

where ` is the angular momentum quantum number, and �(r) = r(r). Whatis the minimum depth U0 so that there is at least one bound state?

3

Page 50: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P10. At t = 0; the wave function for a particle with mass m moving in thex direction through a channel with cross sectional area A is

(x; y; z) ' 1pAL

�sin

�x

a+ i sin

2�x

a

�in the neighborhood of x = 0: Here L and a are constants. Use the probability�ux associated with (x; y; z) to �nd the time rate of change in the probabilityto �nd the particle to the left of the origin x = 0, at t = 0:

4

Page 51: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics

Department of Physics and Astronomy

University of New Mexico

Fall 2014

Instructions:

The exam consists of 10 problems (10 points each).

Where possible, show all work; partial credit will be given if merited.

Useful formulae are provided below; crib sheets are not allowed.

Useful Formulae:

1

2E n

; †0

ˆ2

x b bm

; †0ˆ2

mp i b b

2

20

1/4

2

0 02

0

1 1( ) / ;

2 !

x

x

n nn

x e H x x xx mn

;

2 2

( ) 1n

n x x

n n

dH x e e

dx

ˆ ˆ ˆ,x y zS S i S

0 1 0 1 0, ,

1 0 0 0 1x y z

i

i

, ˆ ˆ( , ) expD n i n S

States of total spin J for two spin-1 particles:

1 2

1 2

2, 2 1,m 1

12, 1 1,0 0,1

2

12, 0 1, 1 2 0,0 1,1

6

12, 1 1,0 0, 1

2

2, 2 1,m 1

11, 1 1,0 0,1

2

11, 0 1, 1 1,1

2

11, 1 1,0 0, 1

2

10, 0 1, 1 0,0 1,1

3

J M m

J M

J M

J M

J M m

J M

J M

J M

J M

___________________________________________________________________________

Page 52: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P1. At time 0t a particle with mass m is incident from the left ( 0)x on a step potential,

0( ) , 0

( ) 0, 0

V x V x

V x x

If the particle has energy0

4

3E V , what is the probability that it will be on the right (x>0) after a

long time ( t )?

P2. Consider a particle with mass m in a one-dimensional quadratic potential given by

21( )

2U x kx , 0k , in the state

1/4

22( ) exp( ),

ax ax x

Evaluate the expectation value of the energy of the particle in this state, and find the value of the

parameter a for which this energy is a minimum. Useful integral:

2 2 1/2exp 1n

nn

n

ddxx x

d

P3. Consider a two-state system for which the Hamiltonian is

ˆ 1 2 2 1 1 1 2 2H V ,

where V and are real. At time 0t the system is in the state 1 . What is the probability that

the system will be in the same state at a later time t ?

P4. A particle with mass m is constrained to move along the perimeter of a circle of radius

in the xy plane. Write down a Hamiltonian H and find the eigenvalues and associated

eigenfunctions in terms of the azimuthal angle . Find another observable that distinguishes

between the degenerate states of , and obtain the eigenfunctions and eigenvalues of the

operator associated with this observable.

H

x

y

z

Page 53: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P5. Consider a particle with mass m moving in one dimension, its dynamics described by the

harmonic oscillator Hamiltonian, 2

2 2

0 0

ˆ 1ˆ ˆ2 2

pH m x

m . At time t = 0 the state is prepared so that

the expectation values of position and momentum are, respectively, 0x x and

0p p .

Derive expressions for ˆ( )x t and ˆ( )p t in terms of 0x and

0p .

P6. A particle with mass m moves about the origin in a three-dimensional spherically

symmetric potential ( )V r . Its dynamics is determined by the Hamiltonian

22

2

ˆ1ˆ ˆ ( )2

r

LH P V r

m r

, where, in spherical coordinates,

1ˆrP r

i r r

and

22 2

2 2

1 1ˆ sinsin sin

L

. Consider finding the eigenfunctions of H using

the method of separation of variables. Taking the ansatz that an eigenfunction

( , , ) ( ) ( ) ( )r R r can be written as a product of three functions, one for each

coordinate, show that the Schroedinger equation can be separated into three independent

eigenvalue problems, one for each of the functions , , and R , respectively. [Note: is the

azimuthal angle, determining longitude.]

P7. A photon traveling in the positive z direction is in the state

1

310

R L

where R and L refer to right and left circularly polarized states, respectively. What is the

probability for the photon to be polarized along the x axis?

P8. Two spin-1 bosons are confined to a harmonic oscillator potential in the x direction and are

subjected to a uniform magnetic field, 0ˆB B z . Ignoring particle-particle interactions and orbital

magnetic coupling, the Hamiltonian is

2 2

2 21 21 2 1 2

1 1

2 2 2 2

p pH kx kx S S B

m m

,

where the subscripts 1 and 2 refer to particles 1 and 2, respectively. By appropriately combining

harmonic oscillator eigenfunctions and two-particle spinnors (consult the formula table on page

1), write down explicit expressions for the two-body wave functions and their associated

energies for the ground state and the first excited state(s). Include both space and spin degrees of

freedom. Assume that0 0B .

Page 54: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P9. An electron with mass m moving in one-dimension is confined by an infinite square-well

potential to the region / 2 / 2L x L . The eigenfunctions for the time-independent

Schroedinger equation may be written in two different ways,

2( ) cos ,

2( ) sin ,

n

n

n xx

L L

n xx

L L

depending on whether the integer n is even or odd. Suppose that the electron is perturbed with an

electric field E , giving rise to a perturbing potential ( )V x eEx . To lowest non-vanishing order

in E , what is the shift in the ground state energy? Assume that the effect of E can be adequately

described with a truncated basis set consisting of only the two lowest-energy eigenfunctions.

[Useful Integral: 32

sin cos / 29

d

]

P10. An electron is prepared in the spin-up state with respect to the z axis. Consider a second

axis 'z that is titled from the z axis by an angle of 0105 . What is the probability that a

measurement of the spin along the 'z axis will give a value of / 2 ?

Page 55: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics

Department of Physics and Astronomy

University of New Mexico

Summer 2015

Instructions:

The exam consists of 10 problems (10 points each).

Where possible, show all work; partial credit will be given if merited.

Useful formulae are provided below; crib sheets are not allowed.

Total time: 3 hours

Useful Information:

0

1

2E n

; †

0

ˆ2

x b bm

; †0ˆ2

mp i b b

2

20

1/4

2

0 02

0

1 1( ) / ;

2 !

x

x

n nn

x e H x x xx mn

;

2 2

( ) 1n

n x x

n n

dH x e e

dx

191eV 1.6 10 J

1 19 2 2 34

04 9 10 Nm /C ; 2 6.626 10 Js

20

19

0

0 0

0.529 A; 1eV 1.6 10 J; 27.2eV4

ea

a

2

B

e

e

m

2 2

, , 22n m

mcE

n

Page 56: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

2

2 21; 197eV-nm;m 0.511MeV/c ; m 938MeV/c

137e p

ec

c

2 2

2 1

2 1 !!exp( )

2

n

n n

ndx x ax

a

22

1

2 2

exp exp exp4

exp

kdx ax ikx

a a

dx x a a ka

1

0

( 1) 1exp ; 1 ; ; 1 !

2

s

s

sx dx ax s s s n n

a

0 1 0 1 0

; ;1 0 0 0 1

x y x

i

i

ˆ ˆ ˆ,x y zS S i S

,

______________________________________________________________________________

P1. An electron with spin initially in an eigenstate of ˆxS with an eigenvalue / 2 is placed in a

uniform magnetic field 0ˆB B z . Show that the expectation value ˆ ( )xS t is a periodic function

of time, and calculate its period (in seconds) for the case that 0B = 1.0 Tesla.

P2. The Hamiltonian for a harmonic oscillator is given by

0

1ˆ ˆˆ2

H b b

where the operators b and †b satisfy the commutation relation †ˆ ˆ, 1b b

. Consider an

eigenstate n of H with eigenvalue 0

1

2n

, where n is a positive integer. Prove that the

two states, ( ) ( )ˆC b n , and

( ) ( )ˆC b n , are also eigenstates of H , where ( )C and

( )C are normalization constants. Find their respective eigenvalues and normalization constants.

ˆˆ ( , ) expD n i n S

Page 57: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P3. If the photon had a non-vanishing mass , electrons and protons would attract one another

via a screened (Yukawa) potential of the form

,

where the screening parameter . Such an effect would cause a shift in the hydrogen

emission spectrum. To lowest non-vanishing order in , what would be the energy (eV) shift in

the Lyman series 2p-to-1s spectral line if the photon mass were as large as the 20.320eV/c

neutrino mass? (Note: The energy associated with a spectral line is the energy difference

between initial state and final atomic states. As such, a shift in the position of a spectral line will

only occur if the initial and final state energies are each perturbed by different amounts.)

P4. A nonrelativistic beam of particles with kinetic energy E=1.0 eV is normally incident on a

fall-away 3.0 eV potential step. A two-dimensional slice of the potential step with energy on the

vertical axis is sketched for reference below. If the particle number density in the incoming beam

is 3 -31.0 10 m , what will be the number density in the transmitted beam?

P5. A 5 eV electron moves freely along the x axis. In momentum-space the electron wave

function is a Gaussian, peaked at momentum 0k . In real-space the wave function evolves in

time according to

2

02

'( , ) exp exp

4 ''2

x tx t A i k x t

it

,

where A and are constants and 0 0

2 2' / , '' /k k k k

k k

are the first and

second derivatives of the dispersion relation ( )k . If the uncertainty in position initially is

1.0mmx , after how much time (in seconds) will 10.0 mmx ?

*m

2

0

( )4

reU r e

r

* /m c

Page 58: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P6. The nuclei of many large atoms are found to be aspherical, their rotational motion described

by the Hamiltonian for an axially symmetric rotator,

2 2 2

1 22 2

x y zL L L

HI I

How much smaller than 1I does 2I have to be for the energy levels of the 1 multiplet to

begin to overlap the energy levels of the 2 multiplet?

P7. A free particle with mass m is subjected to a spatially-uniform sinusoidal driving force,

0( ) sinF t F t ,

with frequency and amplitude 0F directed along the x-axis. Obtain an expression for the

expectation value of the position ˆ( )x t as a function of time in terms of the expectation values

of the initial position ˆ(0)x and the initial momentum ˆ(0)p .

P8. The Hamiltonian for an anharmonic oscillator in one dimension is given by

2

4ˆˆ ˆ2

pH gx

m

where g is a positive constant. The ground state energy E can be estimated using the variational

method. For a trial wave function, 1/4

22( ) expx x

, it is found that

2

2

3ˆ2 16

gH

m

Show that this implies,

4/3 4/3 1/3

2/3

3

4

gE

m

.

Page 59: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P9. A deuteron is a bound state of a proton and a neutron, 2939 MeV/cN Pm m . The

attraction between the two particles can be modeled by a spherical square-well potential of width

b and depth 0V , as illustrated in the figure. The reduced radial wavefunction ( ) ( )u r rR r for

the ground state is determined from the time-independent Schroedinger equation for motion

relative to the center of mass,

2 2

2( )

2

d uV r u Eu

dr

Assuming a bound state exists, sketch the shape of the ground state wave function. For a width

1.63 fmb ( 151fm 10 m ) calculate the threshold depth 0V (in MeV) required to have at least

one bound state.

P10. The wave function for a particle moving freely in three dimensions is given by

2 2 2

3/4 22

1( , , ) exp

42

x y zx y z

where is a constant. Calculate the distribution of expected outcomes of a measurement of the

particle’s energy E. What is the mean of this distribution?

Page 60: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New Mexico

Spring 2005

Instructions:• The exam consists of two parts: 5 short answers (6 points each) and yourchoice of 2 out 3 long answer problems (35 points each).• Where possible, show all work, partial credit will be given.• Personal notes on two sides of a 8X11 page are allowed.• Total time: 3 hours

Good luck!

Short answers:S1. Consider a free particle described by a Gaussian wavepacket of width

Δx0 . Estimatethe time it would take for the width to double. Explain your reasoning.

S2. Estimate the speed of the electron around the nucleus in the ground state ofhydrogen. Express its ratio with the speed of light in terms of the fine structure constant.How good of approximation is it to use nonrelativistic theory?

S3. A particle of mass M moves on a ring of radius a. Show that the energies are

quantized as

Em = m 2 h2

2Ma 2, where

m = 0,±1,±2,K Explain the degeneracies.

S4. A one dimensional simple harmonic oscillator is prepared at t=0 in the state

ψ =121 + i 3

42 , where

n is the eigenstate with energy

(n +1/2)hω . What is the state

at a later time? What is the probability distribution of possible n? What is the meanexcitation number

n as a function of time?

S5. Consider a particle of mass m moving in 3 dimensions. Which of the followingobservables can be measured simultaneously:

ˆ A = ˆ x p y − ˆ y p x ,

ˆ B = ˆ y p z − ˆ z p y ,

ˆ C = ˆ x 2 + ˆ y 2 + ˆ z 2 = ˆ r 2 ,where

( ˆ x , ˆ y , ˆ z ) and

( ˆ p x , ˆ p y , ˆ p z ) are the three components of position and momentum,

respectively, and

ˆ r is the radius from the origin.

Page 61: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L1. A one dimensional model of Hydrogen A very schematic one dimensional model of a hydrogen atom is given by treating theCoulomb attraction of the electron to the proton by a delta function. With nucleus at theorigin, the potential for the electron is

V(x) = −e2δ (x) (in cgs units), where e is themagnitude of electron and proton charges.

(a) Using the boundary conditions that the wave function is everywhere continuous, but

its derivative at the delta function is discontinuous,

dψdx 0−

−dψdx 0+

=2me2

h2ψ (0) , show that

there is one bound state, with eigenfunction

ψ (x) = κe−κ x , with

κ =me2

h2, corresponding

to bound-state energy

Eb = −me4

2h2.

(b) How does the binding energy compare with the binding energy in a real 3D hydrogenatom in its ground state? How does the rms width of the wave function compare with the

Bohr radius a0 =

h2

me2? (Given:

dx x 2eax = eax (2+ 2ax + a2x 2) /a3∫ )

(c) Now consider the atom at the origin, inside a box of width 2L, having infinite walls,and located symmetrically with respect to the origin.Let V(x) = 0 inside the box, except at the origin, where V(x) = −e2δ (x)

∞ ∞

x=+Lx=–L x=02e (x)δ−

V=0V=0

Show that the equation determining the binding energy of the ground state is

κa0 = tanh(κL) , with energy eigenvalue E = −

h2κ 2

2m.

Sketch the wave function in this case.

(d) With the proton at the center of the box, determine the smallest value of L / a0 , thatpermits the electron to remain bound to the nucleus (Hint: solve graphically).

Page 62: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L2. Diatomic Molecule: Anharmonic OscillatorA model for the binding potential of two atoms into a molecule by a central force is theso-called “Lennard-Jones” potential,

V(r) =Ar12

−Br 6

where r is the distance between atoms and A and B are positive coefficients

The two-atom energy eigenstates separates into center-of-mass and relative coordinates.The relative coordinate further separates into radial and angular coordinates, as,

ψ n,l ,m (r,θ,φ) =unl (r)r

Ylm (θ,φ) ,

where

Ylm (θ,φ ) is a spherical harmonic. The (reduced) radial wave function satisfies,

−h2

2md 2

dr 2+

h2l(l +1)2mr2

+V (r)⎛ ⎝ ⎜

⎞ ⎠ ⎟ unl (r) = Eunl (r) , where m is the reduced mass.

(a) What is the meaning of each of the terms in this Schrödinger equation?(b) Show that classically, for l=0, the equilibrium separation between the atoms is

r0 = 2A /B( )1/ 6 , and the vibrational frequency about equilibrium is

ω = ′ ′ V 0 /m where

′ ′ V 0 =d 2Vdr 2

(r0) .

(c) If you approximate the motion near the ground state as a harmonic oscillator, what isthe ground state energy for l=0?(d) Now included the first anharmonic correction to the l=0 potential. To lowest non-vanishing order in perturbation theory, show that the shift in the ground-state energy is,

δEground = −ξ 2 h

2mω⎛ ⎝ ⎜

⎞ ⎠ ⎟

3 n ˆ a + ˆ a †( )30

2

nhωn≠0∑ ,

where

ξ =16d 3Vdr 3 r0

,

n are the eigenstates of the harmonic oscillator, and

ˆ a † / ˆ a are the

usual raising/lowing operators.

V(r)

r (scaled units)

r0

Page 63: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

L3. The hyperfine structure of hydrogen

The ground electronic state of hydrogen has “hyperfine structure” – a splitting due of thespectral lines due to the interaction of the spins of the electron and proton.

(a) Including electron and nuclear spins, but neglecting their interaction, write thequantum numbers associated with the four degenerate sublevels of the hydrogen groundstate.

Now lets add the hyperfine interaction. The interaction Hamiltonian has the form,

ˆ H int = Aδ (3)(r) r ˆ s ⋅r ˆ i

where A is a constant depending on the magnetons, rs is the electron’s spin-1/2 angular

momentum and ri is the proton’s spin-1/2 angular momentum.

(b) Let rf =

ri + rs be the total spin angular momentum. Show that ground state manifold

are defined with good quantum numbers

n = 1, l = 0,s = 1/2, i = 1/2, f ,mf . What are thepossible values of f and mf ?

(c) The perturbed 1s ground state now has hyperfine splitting. The energy level diagramis sketched below. Label all the quantum numbers for the four sublevels shown.

(d) Show that the energy level splitting is

ΔEHF = Aψ1s(0)2 where

ψ1s(0) is the value ofthe 1s electron orbital at the position of nucleus.

This splitting gives rises to the famous 21cm radio frequency radiation used inastrophysical observations.

ΔEHF

1s

Page 64: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy

University of New Mexico Spring 2006

Instructions: • The exam consists of 10 short answers (10 points each). • Where possible, show all work, partial credit will be given. • Personal notes on two sides of an 8X11 page are allowed. • Total time: 3 hours Good luck!

1. A particle of mass m moves in a potential: ( ) 212

V x kx cx= + . Find the eigenvalue of

the nth state (n= 0,1,2,…) to lowest nonvanishing order in perturbation theory, treating the linear term cx as a perturbation.

Page 65: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

2. A particle of mass m is in the ground state of an infinite square well potential,

( )0, 0

,otherwisex a

V x≤ ≤⎧

= ⎨∞⎩. Suddenly, the well expands to three times its original size – the

right wall moving from a to 3a – leaving the wave function momentarily undisturbed. The energy of the particle is now measured. Write an integral expression for the probability of finding the particle in the ground state of the new potential.

Page 66: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

3. Given the eigenfunction below, make a sketch of the potential (on the same graph), indicating on the graph the energy level associated with this eigenfunction. Describe the energy of this eigenfunction relative to the lowest allowed energy.

Page 67: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

4. Quarks carry spin ½. Three quarks bind together to make a baryon (such as the proton or neutron); two quarks (or more precisely a quark and an antiquark) bind together to make a meson (such as the pion or kaon). Assume the quarks are in the ground state (so the orbital angular momentum is zero). What magnitudes of the spin vector are possible for baryons and mesons?

Page 68: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

5. In the representation in which S2 and Sz are both diagonal, a spin- ½ wave function is: 11310 i

ψ ⎛ ⎞= ⎜ ⎟

⎝ ⎠.

What is the probability that a measurement of Sx on ψ yields –ħ/2?

Page 69: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

6. A particle with zero spin in a spherically symmetric potential is in a state represented by the wavefunction: ( ) ( ), , 2 rx y z C xy yz e αψ −= + , where C and α are constants.

a) What is the eigenvalue of L2 for this particle? b) If Lz were measured, what are the possible outcomes?

You may need the following spherical harmonics:

( )

00

1 01 1

2 2 2 1 0 22 2 2

14

3 3sin cos8 4

15 15 5sin sin cos 3cos 132 8 16

i

i i

Y

Y e Y

Y e Y e Y

ϕ

ϕ ϕ

π

θ θπ

θ θ θπ π π

± ±

± ± ± ±

=

= =

= = =

m

m m m θ −

Page 70: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

7. Operators A and obey the following equations: B ˆab abA aψ ψ= and ˆ

ab abB bψ ψ= , where the abψ form a complete set in Hilbert space and a and b are real numbers. Show

that the operator is Hermitian. ( ˆ ˆAB)

Page 71: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

8. A particle in a finite square well potential:

( ) 0

0,,

0,

x aV x V a x a

x a

< −⎧⎪= − − ≤ ≤⎨⎪ >⎩

,

with only three bound states, initially is in a state such that the probability to measure the energy is given by: P(E1) = 1/3, P(E2) = 1/3 and P(E3) = 1/3. The parity is then measured in such a way that the particle stays bound, and found to be -1. If some time later, the energy is measured, what value is found?

Page 72: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

9. In a photoelectric experiment, electrons are emitted from a surface illuminated by light of wavelength 4000Å, and the stopping potential for these electrons is found to be Φ0 = 0.5V. What is the longest wavelength of light that can illuminate this surface and still produce a photoelectric current?

Page 73: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

10. A free electron is prepared in a Gaussian wave packet with standard deviation σx. Estimate the time for the uncertainty in the electron’s position to increase to 2 σx.

Page 74: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy

University of New Mexico Spring 2007

Instructions: • The exam consists of 10 problems, 10 points each. • Partial credit will be given if merited. • Personal notes on two sides of an 8 × 11 page are allowed. • Total time is 3 hours. Problem 1: A nearly monoenergic particle is moving in one dimension where the potential energy make a sharp jump as shown. What is the probability for the particle to transmit across the step in according to classical mechanics and according to quantum mechanics? Problem 2: A particle of mass m is trapped in one dimension in a box with hard walls (infinite energy to escape) of width a. At time t=0, the particle is in the ground state. The width of the box is suddenly expanded to 2a. What is the probability of finding the particle in the ground state of the new well (you may leave your answer as an integral)? What is the probability of finding the particle in the first excited state of the new well (give a number). Problem 3: A particle of mass m moves in two dimensions constrained to a ring of radius a. The particle is otherwise “free” (no other external forces act on the particle). What are the conserved physical quantities? What are energy eigenvalues and degeneracies of these levels? How are the degeneracies related to the conserved quantities?

V0

E

a 2a

a

Page 75: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Problem 4: A wave packet in a 1D harmonic oscillator is described by the state,

! = e"#

2/ 2

# n

n!n

n

$ ,

where α is a complex number and

n is an eigenstate of the number operator,

ˆ N ! ˆ a † ˆ a ,

where

ˆ a †, ˆ a are the usual creation and annihilation operators.

(i) Show that this state is normalized? (ii) Is this an energy eigenstate? (iii) Show that

ˆ a ! = " ! .

Problem 5: Consider a particle with angular momentum J=1. Show that,

!a

=1

20 +

1

21 + "1( ) ,

!b

= "1

20 +

1

21 + "1( ) ,

!c

=1

21 " "1( )

are eigenstates of the x-component of angular momentum,

ˆ J x, with the eigenvalues you

expect. Here

m m = !1,0,1{ } denote the three eigenstates of

ˆ J z,

ˆ J z

m = m m . (Hint, express

ˆ J x in terms of raising and lowering operators).

Problem 6: Consider again, a particle with angular momentum J=1, whose dynamics is governed by a Hamiltonian

ˆ H = ! ˆ J z

2 . At time t=0, we prepare the particle in the state

! (0) = !a

, as given in Problem 5. Show that at a time

t = ! /(2" ),

! (t = " /(2# ) =e$i" / 4 !

a$ ei" / 4 !

b

2.

Problem 7: Consider two distinguishable noninteracting particles of mass m, each trapped in a separate harmonic well in 1D. The Hamilton is

H =p1

2

2m+p2

2

2m+1

2m!

2x1

+ a( )2

+1

2m!

2x2" a( )

2 .

(i) Show that this Hamiltonian separable in relative coordinate and center of mass. (ii) Express the total energy eigenvalues of the two-body system in terms energy

eigenvalues associated with the motion of center of mass and relative coordinate.

Page 76: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Problem 8: Neglecting the spin of the electron or proton, what is the degeneracy of the first excited state of a hydrogen atom? What are the quantum numbers that uniquely specify each of the substates? Now add electron spin and spin-orbit coupling. What are the good quantum numbers? Sketch an energy level diagram for each of these levels of description. Calculation of the energy level splitting is not necessary. Problem 9: Consider the Hamiltonian describing two bound states of an atom. Written in bra-ket notation, the unperturbed Hamiltonian in this subspace is,

ˆ H 0

= E1!

1!

1+ E

2!

2!

2.

Interaction with a weak external field is described by the Hamiltonian

ˆ H 1

= Eint

!1!

2+ !

2!

1( ) ,

where

Eint

E2! E

1

<< 1. To lowest nonvanishing order in perturbation theory, what are the

shifts in the energies of the two bound states? Problem 10: Given the total Hamiltonian

ˆ H = ˆ H 0

+ ˆ H 1 in Problem 9, solve for the exact

energy eigenvalues. Perform a Taylor series expansion on your result to find the perturbation shift in eigenvalues to lowest order in

E int /(E2 ! E1).

Page 77: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Quantum Prelim January 18, 2008 p.1

Preliminary Examination: QuantumDepartment of Physics and Astronomy

University of New MexicoWinter 2008

Instructions:

• the exam consists of 10 problems, 10 points each;

• partial credit will be given if merited;

• personal notes on two sides of 8× 11 page are allowed;

• total time is 3 hours.

1. A spin-1/2 particle is in the state,

|ψ〉 =1√3

{| ↑ z〉 + i

√2| ↓ z〉

},

where in this notation | ↑ z〉 labels the state that has spin-up with respect tothe z-axis. Find the probability for a particle in this state to pass through anStern-Gerlach device oriented along +x with spin-up followed by a Stern-Gerlachdevice oriented along +z with spin-down. Note that | ↑ x〉 = 1√

2{| ↑ z〉 + | ↓ z〉}.

2. A particle is incident from the left (x < 0) on step potential,

V (x) = V0, x > 0V (x) = 0, x < 0

a) For E < V0, sketch the wave function for all x.

b) Let’s call the wave function for x < 0 Ψ−. Then for E < V0 Ψ− must be ofthe form,

Ψ− = eikx + eiδe−ikx,

where δ is a real constant. Why?

c) What is δ in the limit V0 >> E?

3. Consider a particle in a 1-D box 0 < x < a which at t = 0 is in the state,

ψ(x) =

√3

a

(2x/a

), 0 < x < a/2

ψ(x) =

√3

a

(2 − 2x/a

), a/2 < x < a,

Determine the probabilities P1, P2 to measure the ground state energy E1 andthe first excited state energy E2 at t=0. (use

∫ π/20 u sin(u)du = 1 )

Page 78: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Quantum Prelim January 18, 2008 p.2

4. Consider a system with orthonormal basis states |1〉 and |2〉. The Hamiltonianin this basis is given by,

H =

(E0 −iAiA E0

)

where E0 and A are real, positive constants. Find the energy eigenvalues andcorresponding normalized energy eigenstates. Be clear as to which state goeswith which eigenvalue.

5. Consider a spinless particle that is constrained to move on a circle of radius Rbut free to move around the circle. In a cylindrical coordinate system with thecircle in the x-y plane, the wave function depends only on the azimuthal angle φ.

a) Write the Hamiltonian for this system.b) What are the energy eigenvalues and normalized eigenstates? Identify all goodquantum numbers and any degeneracies.c) What is the uncertainty in φ for states of definite energy and angular momen-tum?

6. Consider a particle subject to the harmonic oscillator Hamiltonian,

H =p2

2m+mω2x2

2.

For an initial state of the particle given by a superposition of energy eigenstates,

|Ψ(x, 0)〉 =1√2(|n〉 + |n+ 1〉).

Calculate 〈x〉 as a function of time.

7. Consider a diatomic molecule as a rigid rotor with moment of inertia I and mag-netic moment ~µ = −µ0

~L, µ0 being a postive constant. The molecule is in auniform magnetic field ~B along the z-axis.

(a) What is the Hamiltonian for this system?

(b) If at t = 0 the wave function is,

Ψ(0) =Y11 + Y10√

2,

calculate 〈Lx〉 as a function of time, using

Lx =L+ + L−

2

Page 79: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Quantum Prelim January 18, 2008 p.3

8. A particle moving in a central potential V(r) has a Hamiltonian,

H =1

2m{Pr

2+L2

r2} + V (r),

where

Pr =h

i

1

r

∂rr.

a) Show that

Ψ(~r) =u(r)

rY`m(θ, φ)

satisfies the time independent Schrodinger equation and find the eigenvalue equa-tion for u(r).

b) For an infinite spherical well of radius a, find the ground state and first ` = 0excited state wave functions (up to a normalization constant) and the correspond-ing energies.

9. Consider a particle in a 1D box of length a with 0 < x < a at the middle of thebox. Calculate the first-order correction to the energies for all energy eigenstatesdue to the purturbation,

Hint = αδ(x− a/2)

where δ() is the Dirac delta function and α is a constant.

10. For the 2P3/2 hydrogen multiplet (states |n, `, j,mj〉 with n = 2, L = 1 andJ = 3/2 ) calculate the correction due to the Zeeman effect in the weak-fieldapproximation,

HB =eB

2mc(Lz + 2Sz).

Note: States of definite total angular momentum |j,mj〉 are related to the productstates of orbital angular momentum times spin |`,m`〉|s,ms〉 according to:

|32,±3

2〉 = |1,±1〉|1

2,±1

2〉

|32,+1

2〉 =

√13|1, 1〉|1

2,−1

2〉 +

√23|1, 0〉|1

2,+1

2〉

|32,−1

2〉 =

√13|1,−1〉|1

2,+1

2〉 +

√23|1, 0〉|1

2,−1

2〉

Page 80: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 81: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 82: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 83: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 84: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 85: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 86: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 87: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 88: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 89: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 90: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:
Page 91: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim version January 7, 2010 p.1

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New MexicoSpring 2010

Instructions:

• the exam consists of 10 problems, 10 points each;

• partial credit will be given if merited;

• personal notes on two sides of 8 × 11 page are allowed;

• total time is 3 hours.

Table of Constants and Conversion factors

Quantity Symbol ≈ valuespeed of light c 3.00 × 108 m/sPlanck h 6.63 × 10−34 J·selectron charge e 1.60 × 10−19 Ce mass me 0.511 Mev/c2 = 9.11 × 10−31 kgp mass mp 938 Mev/c2 = 1.67 × 10−27 kgamu u 931.5 Mev/c2 = 1.66 × 10−27 kgfine structure α 1/137Boltzmann kB 1.38 × 10−23 J/KBohr radius a0 = h/(mecα) 0.53 × 10−10 mconversion constant hc 200 eV·nm = 200 MeV·fmconversion constant kBT@ 300K 1/40 eVconversion constant 1eV 1.60 × 10−19 J

1. The earliest conception of the neutron was as a bound state of electron andproton. Suppose some force existed that could confine the electron to the insideof the proton. Estimate the energy of the electron in the ground state.

2. The carbon monoxide molecule absorbs radiation at a wavelength of 2.6 millime-ters, corresponding to the rotational excitation of the molecule from ` = 0 to` = 1. Calculate the molecular bond length.

3. Derive the continuity equation

∂ρ

∂t+∂J

∂x= 0

for the Schrodinger equation in 1 dimension. What is the physical significanceof ρ?

Page 92: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim version January 7, 2010 p.2

4. A particle is confined to a 1D box 0 < x < a and at t = 0 has the wave function:

ψ(x) = 2

√3

a

(x

a

), x <

a

2

ψ(x) = 2

√3

a

(1 − x

a

), x >

a

2

Find the expectation value of the energy for this state.

5. An electron with energy E0 = V0/3 is incident from the left on a step-downpotential: V (x) = 0 for x < 0 and V (x) = −V0 for x > 0. What is thetransmission probability T?

6. Consider the ground state of a particle in 1 dimension bound in the potentialsketched below (v(x) is infinite for x < 0, = −V0 for 0 < x < a and zero forx > a).(a) Make a sketch of the ground state wave function.(b) What is the minimum depth V0 such that at least one bound state exists?

Page 93: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim version January 7, 2010 p.3

7. Two spin-1/2 particles interact via the Hamiltonian

H =2A

h2

(S1

xS2x + S1

y S2y

)

(a) Explain why the z-component of the total spin Sz = S1z + S2

z is conserved.(b) In the basis of total spin |s,m〉 (i.e. eigenstates of S2, Sz) the Hamiltonian isdiagonal. Find the energy of the ground state.

8. Consider an electron bound in a harmonic oscillator potential with a linear per-turbation.

H =p2

2m+

1

2mω2x2 + ε

x

x0

where x0 =√h/mω and ε << hω. What is the correction to the ground state

energy to second order due to this perturbation? Recall that

x =x0√2

(a+ a†

)9. Consider a particle in a harmonic oscillator potential. For an initial state of the

particle given by a superposition of energy eigenstates,

|Ψ(x, 0)〉 =1√2(|n〉 + |n+ 1〉).

Calculate 〈x〉 as a function of time. Recall that the eigenstate energies are En =hω(n+ 1/2) and use the expression for x from the previous problem.

10. Consider the spin-orbit coupling of Hydrogen in the n=3 excited state. Labelthe degenerate multiplets as 3Lj, where L = S, P,D corresponding to ` = 0, 1, 2,and j is the total angular momentum quantum number. Give the degeneracies ofeach multiplet.

Page 94: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New Mexico

Spring 2011

Instructions:�The exam consists of 10 problems (10 points each).�Where possible, show all work; partial credit will be given if merited.�Personal notes on two sides of an 8�11 page are allowed.�Total time: 3 hours.

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �Useful Information:

1. Pauli sigma matrices: �x =�0 11 0

�; �y =

�0 �ii 0

�; �z =

�1 00 �1

�2.R �=20

dx sin(2x) sin(x) = �2=33. For a particle with mass m moving under the in�uence of an attractive

one-dimensional square-well potential, V (x) = �V0 for jxj � a and V (x) = 0otherwise, the energy eigenvalues for the bound states are determined from thetranscendental equationp

�� y2y

=

�tan y (even parity)� cot y (odd parity)

;

where � = 2ma2V0=~2 and y = qa with q =q

2m~2 (V0 � E):

4. Some hydrogen wave functions:

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

1

Page 95: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P1. A particle with mass m moving to the right with kinetic energy Eencounters a step potential of height V; as shown in the �gure. What is theprobability that the particle will be re�ected if E = (4=3)V ?

P2. Consider a beam of electrons, to be �red a great distance, L = 104 km,along the x axis. In momentum-space the electron wavefunction is a stronglypeaked Gaussian at ~k0 with energy ~! = (~k0)2

2m = 10 eV. The real-spacewavefunction evolves in time according to

(x; t) = A exp [i (k0x� !t)] exp"� (x� !0t)2

4��2 + i

2!00t�# ;

where A and � are constants and !0 = (@!=@k)jk=k0denotes di¤erentiation of! with respect to k .If the initial uncertainty in the position of an electron is �x = 1:0 mm,

approximately what will be �x on arrival?

P3. The Hamiltonian for an anharmonic oscillator in one dimension is givenby

H =p2

2m+ gx4

where g is positive. Use the uncertainty principle to estimate the ground stateenergy. Express your answer as a function of g; ~; and the mass m:

2

Page 96: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P4. A particle with mass m is initially in the ground state of a one-dimensional in�nite square well with width a: If the square well is suddenlystretched to a new width 2a; what is the probability that a measurement of theenergy will �nd the particle to be in the ground state of the new square well?

P5. The motion of a particle with mass m moving in one dimension isdescribed by the Hamiltonian

H =p2

2m+mgx:

Obtain an expression for the expectation value of the position hx(t)i in termsof the initial expectation values hp(0)i and hx(0)i :

P6. The nucleus of a gold atom is found to be aspherical, for its rotationalmotion is described by the Hamiltonian for the axially symmetric rotator,

H =L2x + L

2y

2I1+L2z2I2

where the moment of inertia I1 > I2: Sketch the splittings in the rotationalenergy spectrum for ` = 0; 1; and 2:

3

Page 97: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P7. The Hamiltonian for a harmonic oscillator is given by

H = ~!0�aya+

1

2

�;

where the operators a and ay satisfy the commutation relation�a; ay

�= 1:

Consider an eigenstate jni of H with eigenvalue�n+ 1

2

�~!0: Prove that

the two states����(�)E = C(�)a jni and

����(+)E = C(+)ay jni are also eigenstates

of H; where C(�) and C(+) are normalization constants. Find their respectiveeigenvalues and normalization constants.

P8. In quasi one-dimensional conductors, the Coulomb repulsion betweenconduction electrons is screened out at large distances. All that remains issmall phonon-mediated interaction. This interaction is attractive, and is oftenmodeled by a square-well potential.

Consider two electrons in a carbon nanotube, each in the same spin state,their center of mass free to move in the x direction. Suppose that they areattracted to one another by the square-well interaction V (x1 � x2) = �V0 forjx1 � x2j � a and V (x1 � x2) = 0 otherwise, where the argument x1 � x2 isthe di¤erence between the displacements x1 and x2 of electron 1 and electron2 respectively. What is the minimum depth V0 required for a bound state forthe two electron system? You may want to make use of the square-well solutionlisted in the table.

4

Page 98: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P9. An electron initially in an eigenstate of Sx with eigenvalue ~=2 is placedin a uniform magnetic �eld ~B = B0 z: At some later time t a measurement isto be made of Sy: What is the probability that a value ~=2 will be found?

P10. If the photon actually had a small mass m�; electrons and protonswould attract one another via a screened (Yukawa) potential of the form

U(r) =�e24��0r

e� r;

where the screening parameter = m�c=~. This would show up as causing ashift in the hydrogen spectrum.To �rst order in , calculate the shift in the energy E0;0;0 of the hydrogen

ground state. What is the �rst order shift in the energy En;`;m for arbitraryquantum numbers n; `; and m?

5

Page 99: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New Mexico

Spring 2012

Instructions:�The exam consists of 10 problems (10 points each).�Where possible, show all work; partial credit will be given if merited.�Personal notes on two sides of an 8�11 page are allowed.�Total time: 3 hours.

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �Useful Information:

1. Pauli sigma matrices: �x =�0 11 0

�; �y =

�0 �ii 0

�; �z =

�1 00 �1

�2. Some hydrogen wave functions:

3. Hydrogen expectation values:1r3

�n;`= 1

a30n3

1`(`+1=2)(`+1)

4. Commutation relations: [A;BC] = B [A;C] + [A;B]C

5. Lorentzian Fourier Transform:R1�1 dx exp(i!x) �=�

�2+x2 = exp(� j�!j)� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

1

Page 100: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P1. An electron with mass m and spin Sz = ~2 moves in the z direction,

parallel to a magnetic �eld ~B = Bz. For z < 0; the magnetic �eld is uniform,with B = B1, and for z > 0 it is again uniform, with B = B2. The Hamiltonian,

H =p2

2m+

�2�B1

~ Sz; z < 02�B2

~ Sz; z > 0;

is time-independent, and e¤ectively discontinous at z = 0; assuming that thedistance over which B changes from B1 to B2 is su¢ ciently short. The electronis initially located to the left of the origin, and moves to the right, in the zdirection, with momentum ~k: If its spin does not �ip, what is the probabilityP that the electron will cross the origin and continue moving to the right? Forwhat conditions will P = 0?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

2

Page 101: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P2. The spin-orbit coupling for the hydrogen atom is described by theinteraction Hamiltonian

Hso =�1:77� 10�10eV

��a30r3

� ~S � ~L~2

To �rst order, how does the spin-orbit interaction perturb the energy levels ofthe the 2p state? Express your answers in eV. The Bohr radius is denoted bya0.

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

P3. Consider a particle with massm con�ned to a linear potential V (r) = krin three dimensions. For ` = 0; the radial Schroedinger equation is�

� ~2

2m

d2

dr2+ kr

�u(r) = Eu(r)

where u(r) = r (r) satis�es boundary conditions u(0) = u(1) = 0: Using atransformation to dimensionless variables, show that the allowed energies mustbe proportional to

" =

�~2k2

2m

�1=3:

3

Page 102: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P4. A tritium atom consists of one electron bound to a triton [H3 = (nnp)]:The triton is unstable and decays into a He3 nucleus by converting a neutron toa proton while casting o¤ an electron and an antineutrino [(nnp)! (npp)+e�+��e]. As compared to the period of the orbiting electron, this nuclear decayoccurs practically instantaneously ( t < 10�13 sec). If the original electronremains bound to the nucleus, a positively charged He3 ion is formed. What isthe probability that the ion will be in its ground state? It might help to you torecall the ground state hydrogen wave function, and the integralZ 1

0

exp(��x)xndx = n!

an+1:

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

P5. The expectation value h~r � ~pi may be shown to be a constant for anystationary state. From the equation of motion, d

dt h~r � ~pi =i~ h[H;~r � ~p]i ; show

that �p2

m

�=D~r � ~rV (~r)

Efor a particle moving in a potential V (~r). For the ground state of hydrogen,

hHi =Dp2

2m �e2

r

E= �13:6 eV. Use this, together with the relation above, to

determine the value ofDp2

2m

E.

4

Page 103: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P6. Consider a particle con�ned by an in�nite square well 0 < x < a in onedimension. The initial wave function

(x; 0) =1pa

�sin

�x

a+ i sin

2�x

a

�is a complex superposition of the ground state and the �rst excited state. Cal-culate the probability �ux. In which direction will the particle most likely bemoving at x = a=2; to the right, or to the left?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

P7. The Hamiltonian for a harmonic oscillator is given by

H = ~!0�aya+

1

2

�:

Recall that the operators a and ay satisfy the commutation relation�a; ay

�=

1; that the position operator x =�ay + a

�q ~2m!0

; and that the momentum

operator p = i�ay � a

�q~m!02 : Suppose that the oscillator is prepared in an

initial statej (0)i = exp

�gay�j0i exp

��g2=2

�;

where g is real. What is the physical signi�cance of this state? Calculate hx(t)i :

5

Page 104: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P8. Consider two noninteracting electrons moving in a one dimensionalin�nite square well of width a in a uniform magnetic �eld ~B = Bz. The Hamil-tonian describing this system is

H =p212m

+p222m

+2�

~

�~S1 + ~S2

�� ~B

for 0 < x < a: The subscripts 1 and 2 refer to the two electrons, respectively.

The particle-in-a-box eigenfunctions are n(x) =q

2a sin(

n�xa ): Write down an

expression for the two-particle ground state wavefunction when B = 0: Be sureto include a description of both space and spin. Write down an expression forthe �rst excited state, and describe any degeneracies. Which of these becomesthe new ground state at su¢ ciently high B?

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

P9. A particle with spin 12 is at rest in a magnetic �eld

~B = B0 (x cos�+ y sin�)

Its Hamiltonian is given byH = � ~B � ~S

Here x and y are unit vectors, and ~S is the spin operator. At time t=0 theparticle is in state

��ms =12

�; where ms~ is the eigenvalue of the operator Sz:

Calculate the probability that, at some later time t; the system will be found inthe state

��ms = � 12

�:

6

Page 105: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P10. Under the action of a time-dependent Hamiltonian, H = H0 +H1(t);to �rst order in H1; the state j (t2)i is connected to the state j (t1)i throughthe integral expression,

j�(t2)i ' j�(t1)i+1

i~

Z t2

t1

dt eiH0t=~H1(t)e�iH0t=~ j�(t1)i ;

where j�(t)i = eiH0t=~ j (t)i is the state vector in the interaction representation.Consider a two-state system for which

H0 = "1 j1i h1j+ "2 j2i h2j :

The system is subjected to a perturbation

H1(t) =V0

1 + 2t2(j1i h2j+ j2i h1j)

that turns on and o¤ with a rate as t goes from �1 to 1: If j (�1)i = j1i ;to �rst order in V0; what is the probability that j (1)i = j2i? Show that theadiabatic theorem is satis�ed when H1(t) is slowly varying.

7

Page 106: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New Mexico

Spring 2013

Instructions:�The exam consists of 10 problems (10 points each).�Where possible, show all work; partial credit will be given if merited.�Personal notes on two sides of an 8�11 page are allowed.�Total time: 3 hours.

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �Useful Formulae and Constants:Z 1

�1dx exp(�ax2)x2n = (2n+ 1)!!

p�

2na2n+12

0;0;0

(r; �; �) =1p�a30

exp

�� r

a0

�;

a0 =4��0~2

me2hSx; Sy

i= i~Sz

x =

r~

2m!0

�b+ by

�p = �i

r~m!02

�b� by

n(x) =

r2

Lsin�n�xL

�;

En =n2�2~2

2mL2

1

Page 107: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P1. Consider the time-independent Schroedinger equation for a one dimen-sional forced harmonic oscillator,

�~22m

@2

@x2+

�1

2m!2x2 + Fx

� = E :

By making an appropriate coordinate transformation, determine how the energyeigenvalues depend on the constant force F:

2

Page 108: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P2. Consider the Hamiltonian for the motion of the spin of an electron ina constant magnetic �eld B,

H = �2�~B Sz:

By solving Heisenberg�s equations of motion, show that the x component of thespin operator evolves in time, such that

Sx(t) = Sx cos (!t) + Sy sin (!t)

where ! = 2�~B:

3

Page 109: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P3. Consider a two-level atom, with ground state j1i ; and �rst excited statej2i . In the presence of an electric �eld E0, the Hamiltonian is given by

H =�

2(j2i h2j � j1i h1j) + p0E0 (j1i h2j+ j2i h1j)

where p0 is the transition-dipole matrix element. What are the eigenstates andeigenvalues of H?

4

Page 110: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P4. The time-independent Schroedinger equation for the hydrogen atom,written explicitly in spherical coordinates, is as follows:

�~22m

�1

r2@

@r

�r2@

@r

�+

1

r2 sin �

@

@�

�sin �

@

@�

�+

1

r2 sin2 �

@2

@�2

� � e2

4��0r = E

Apply the method of separation of variables to reduce this partial di¤erentialequation to three ordinary di¤erential equations. Assume that the wave function (r; �; �) = R(r)P (�)F (�) can be written in a product form, introduce appro-priate separation constants, and obtain (but do not solve) three equations, onefor each of the functions R(r); P (�); and F (�), in their respective variables r; �;and �:

5

Page 111: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P5. Consider a particle with mass m con�ned to a harmonic oscillatorpotential in one dimension,

U(x) =1

2m!20x

2:

Use the variational method to �nd a lower bound for the ground state energyE0. Take the trial wavefunction to be (x) =

���

�1=4exp(��

2 x2), with � to be

determined, and show that the bound,

E0 �~!02;

you obtain variationally agrees with the exact ground state energy. What doesthis imply for the trial wavefunction?

6

Page 112: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P6. Consider the Coulomb potential

U0(r) = �e2

4��0r

describing the attraction between the electron and the proton in the hydrogenatom: Due to the �nite radius a ' 2 � 10�15 m of the proton, one expectsmodi�cations to this potential at short range. In one proposal, the proton isto be modeled as a point charge surrounded by a spherical "skin" consisting ofa double-layer at the radius a: The e¤ect of the double-layer is to introduce adiscontinuity �, such that the actual potential is given by

U(r) =

�U0(r); r > a

U0(r) + �; r � a

To �rst order, what is the shift in the ground state energy for � ' 1 eV ? Notethat exp

�� ra0

�' 1 for r << a0:

7

Page 113: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P7. Consider a particle of mass m con�ned to a one-dimensional box withwidth L: The particle is initially in the ground state. The box is suddenlydoubled in size, from L to 2L: Show that the probability to �nd the particle inthe �rst excited state of the new box is 1/2.

8

Page 114: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P8. An atom subjected to an electric �eld E will acquire an induced dipolemoment p = �E, where � is the atomic polarizability, thereby lowering itselectronic energy by

�" = �12�E2:

Consider a model of an atom consisting single electron with mass m andcharge e; con�ned by a three-dimensional harmonic oscillator potential withfrequency !0. Assume that the atom is subjected to an electric �eld in thex-direction, so that the total potential energy is U(~r) = 1

2m!20r2 + eEx: By

considering the perturbation of the ground state, �nd an expression for theatomic polarizability �:

9

Page 115: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P9. A free particle with mass m traveling to the right with energy E isre�ected from a square barrier of height V; as shown in the �gure. What is theprobability that the particle will penetrate the barrier to a depth greater thana given value `?

10

Page 116: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P10. Two noninteracting electrons are con�ned to a square well potentialU(x) in one dimension. For x > L; or x < 0; the potential is in�nite. For0 < x < L; the potential is independent of position, but dependent on spin,such that

U(x) = ��2~2

mL2

2~S1 � ~S2~2

+1

2

!Find the ground state wavefunction and its associated energy for the two-particlesystem (include both space and spin)?

11

Page 117: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics

Department of Physics and Astronomy

University of New Mexico

Spring 2014

Instructions:

The exam consists of 10 short-answer problems (10 points each).

Where possible, show all work; partial credit will be given if merited.

Personal notes on two sides of an 811 page are allowed. For some problems it may be

helpful to use the formulae provided below.

Useful Formulae:

1

2E n

; †

0

ˆ2

x b bm

; †0ˆ2

mp i b b

2

20

1/4

2

0 02

0

1 1( ) / ;

2 !

x

x

n nn

x e H x x xx mn

;

2 2

( ) 1n

n x x

n n

dH x e e

dx

ˆ ˆ ˆ,x y zS S i S

0 1 0 1 0, ,

1 0 0 0 1x y z

i

i

___________________________________________________________________________

P1. A particle with mass m is incident from the left ( 0)x on a step potential,

0( ) , 0

( ) 0, 0

V x V x

V x x

If the particle has energy 0E V , the wave function for 0x can be written in the form

( ) ikx i ikxx e e e .

Derive an expression for the phase in terms of 0, , and E m V .

Page 118: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P2. Consider a particle in a one-dimensional box / 2 / 2a x a in the state

2

5

5( ) , / 2 / 2

2

ax x a x a

a

Evaluate the expectation value of the energy of the particle in this state, and show that this is

greater than the ground state energy.

P3. Consider a two-state system for which the Hamiltonian is

ˆ 1 2 2 1H iA ,

where A is real. At time 0t the system is in the state 1 . What is the probability that the

system will be in the same state at a later time t ?

P4. A thin rod with mass m and length is constrained to lie in the xy plane with one end fixed

at the origin. It is free to rotate through an angle about the z-axis, as shown in the figure. What

is the Hamiltonian for this system? If the rod is prepared in a state of definite angular momentum

ˆ,z what is its wave function ? What is the expectation value of its angular velocity?

[The moment of inertia is 21/ 3 m ].

Page 119: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P5. Consider a particle with mass m moving in one dimension, its dynamics described by the

harmonic oscillator Hamiltonian, 2

2ˆ 1ˆ ˆ2 2

pH kx

m . For an initial state given by a superposition of

energy eigenstates, 1

12

n n , determine the time evolution of the expectation value

21ˆ

2kx of the potential energy.

P6. A particle moves in a three-dimensional spherical “square-well” potential,

0,

( ), r>R

r RV r

Its dynamics is determined by the Hamiltonian 2

2

2

ˆ1ˆ ˆ ( )2

r

LH P V r

m r

, where

1ˆrP r

i r r

Taking the wave function to be of the form, ,

( )( ) ,m

u rr Y

r in spherical coordinates,

show that the radial Schroedinger equation is

22

2

1''( ) ( ) ( ) ( ) ( )

2 2u r V r u r u r Eu r

m mr

.

Explain, either mathematically, or intuitively, why should be zero in the ground state. Write

down an expression for ( )u r in the ground state, and determine its associated energy eigenvalue.

P7. A photon is in the state 1

25

x i y where x and y refer to photon states

polarized along the x and y axes, respectively. What is the probability for the photon to be right

circularly polarized?

P8. Two electrons are confined to a harmonic oscillator potential in the x direction and are

subjected to a uniform magnetic field 0ˆB B z . Ignore electron-electron interactions and ignore

orbital magnetic coupling. The Hamiltonian is

2 2

2 21 21 2 1 2

1 1 2

2 2 2 2

Bp pH kx kx S S B

m m

,

where the subscripts 1 and 2 refer to particles 1 and 2, respectively, and B is the Bohr

magneton. By appropriately combining harmonic oscillator eigenfunctions, write down explicit

expressions for the two-body wave functions and their associated energies for the ground state

and the first excited state(s). Include both space and spin degrees of freedom.

Page 120: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P9. A particle moving in two-dimensions is confined by an infinite square-well potential to the

region / 2 / 2, / 2 / 2L x L L y L . The eigenfunctions for the time-independent

Schroedinger equation may be written in four different ways;

,

,

,

,

2( , ) cos cos ;

2( , ) cos sin ;

2( , ) sin cos ;

2( , ) sin sin ;

m n

m n

m n

m n

m x n yx y

L L L

m x n yx y

L L L

m x n yx y

L L L

m x n yx y

L L L

the particular choice depends on whether m and n are even or odd. What is the energy and

degeneracy of the first excited state, and what are the eigenfunctions? If the square-well is

perturbed with a spatially-dependent potential 0 2

( , y)xy

V xL

, where 2 2 2

0 / 2mL , what

is the splitting of the first excited state, to lowest (non-vanishing) order?

[Useful Integral: 32

sin cos / 29

d

]

P10. An electron is prepared in the spin-up state with respect to the z axis. Consider a second

axis 'z that is titled from the z axis by an angle of 030 . What is the probability that a

measurement of the spin along the 'z axis will give a value of / 2 ?

Page 121: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

Preliminary Examination: Quantum Mechanics

Department of Physics and Astronomy

University of New Mexico

Spring 2015

Instructions:

The exam consists of 10 problems (10 points each).

Where possible, show all work; partial credit will be given if merited.

Useful formulae are provided below; crib sheets are not allowed.

Useful Formulae:

1

2E n

; †

0

ˆ2

x b bm

; †0ˆ2

mp i b b

2

20

1/4

2

0 02

0

1 1( ) / ;

2 !

x

x

n nn

x e H x x xx mn

;

2 2

( ) 1n

n x x

n n

dH x e e

dx

ˆ ˆ ˆ,x y zS S i S

, ˆ ˆ( , ) expD n i n S

191eV 1.6 10 J

2 1

; 197MeV fm137

ec

c

2 2

3

1exp( )

2dx x ax

a

22

12 2

exp exp / exp4

exp( ) exp

akdx ax ikx a

dx x a ikx a ka

Page 122: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P1. A system consists of two spins, 1s and

2s in a uniform magnetic field 0ˆB B z directed

along the z axis. Their evolution is governed by the Hamiltonian,

1 2 1 22H s s B s s

Which of the following quantities are constants of the motion: 2 2 2

1 2 1 2 1 2, , , , , , zs s s s s s s s s ? If

both are spin-1/2, what are the eigenvectors and eigenvalues of H?

P2. Consider a spin-1 particle. The matrix representation of the components of spin-1 angular

momentum are,

0 1 0 0 1 0 1 0 0

ˆ ˆ ˆ1 0 1 , 1 0 1 , 0 0 02 2 2

0 1 0 0 1 0 0 0 1

x y zS S Si

The matrices are expressed in the ordered basis 1 , 0 , 1 where the kets are labeled by the

eigenvalues of ˆzS in units of .

Show that 11 2 0 1

2A , 1

1 2 0 12

B , and

1

1 12

C are eigenvectors of ˆxS and find their eigenvalues. When a beam of spin-1

particles initially prepared in the state 1 is put through a Stern-Gerlach apparatus having the

magnetic field gradient in the x-direction, three beams emerge. What are the relative intensities

of the three beams?

Page 123: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P3. A deuteron is a bound state of a proton and a neutron, 2939 MeV/cN Pm m . The attraction

between the two particles can be modeled by a square-well potential of width b and depth0V ,

as illustrated in the figure.

The reduced radial wavefunction ( ) ( )u r rR r for the ground state is determined from the time-

independent Schroedinger equation for motion relative to the center of mass,

2 2

2( )

2

d uV r u Eu

dr

Assume a solution of the form ( ) sinu r A kr for the region 0<r<b, and ( ) expu r B qr

for b r , where A, B are normalization constants; find the relations for k and q in terms of E, and

show that E is determined by the transcendental equation tan /kb k q . If 0 38.5 MeVV ,

what is the minimum width b (in fm) below which there are no bound states?

(Note: 197 MeV-fmc )

P4. The wavefunction for a particle moving freely on the x-axis is given by

2

01/4 22

1( ) exp exp

42

xx ik x

where and 0k are constants. Calculate the distribution of expected outcomes of a measurement

of the particle’s energy E. What is the mean of this distribution?

P5. Consider a particle moving in three dimensions. Show which of the following observables

can be measured simultaneously:

2 2 2 2, , , , ,x z y y x z z y x zL yp zp L zp xp L xp yp z p r x y z

Here , ,x y z and , ,x y zp p p are the three Cartesian components of position and momentum

respectively, and r is the radius from the origin.

Page 124: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P6. A simple three-parameter potential that is often used to characterize the atomic separation

distance r in a diatomic molecule is Morse potential, given by

2

0( ) 1 expU r D a r r ,

and sketched in the graph. Here 0r is the equilibrium bond distance, D is the dissociation energy,

and a is a parameter that determines with width of the potential. For the 2N molecule in the

ground state, the equilibrium bond distance is 1.0977 angstroms, and the dissociation energy is

9.79 eV. For a zero-point vibrational energy of 0.146 eV, what is the value of a (in angstroms)?

The atomic weight of N is 14.0067 grams per mole ( 262.33 10 kg per atom) .

P7. The hyperfine splitting of the ground state in hydrogen is due to the interaction of electron

spin s and the proton spin i , and can be described by the interaction Hamiltonian,

int ( )H A r s i

where the constant A depends on the electron and proton magnetic moments and

( )r x y z is a delta-function in three dimensions. Using the hydrogen wave

functions provided in the table, obtain an expression for the hyperfine splitting in terms of A and

the Bohr radius 0a , to lowest order in

intH .

1 2 3 4 5Angstroms

5

10

15

20

Electron Volts

Page 125: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

P8. Consider a two-port interferometer with 50/50 beam splitters, each arm having a path length

L. The optical path for one arm can be made longer through a phase shift . A single photon of

frequency is prepared and injected into the input port. If one of the two mirrors is only

partially reflecting, with a reflection probability (reflectance) R 1 as indicated in the figure

below, what is the probability of detecting the photon at the denoted output-port versus and

R?

P9. A particle with mass m is trapped in the ground state of a harmonic oscillator well described

by the potential energy function 2

1

1( )

2U x kx . Suddenly the particle is subjected to a constant

force F, so that the new potential well is given by 2

2

1( )

2U x kx Fx . What is the probability

that the particle will be in the ground state of the new well?

P10. The eigenstates n of the harmonic oscillator Hamiltonian † 1/ 2H b b are labeled

by the quantum number n, with energy eigenvalue 1/ 2nE n . A coherent state,

2

/2

0 !

n

n

e nn

is a superposition of these eigenstates, weighted respectively by !

n

n

, where is any complex

number. Coherent states have the property that they are eigenstates of the lowering-operator, i.e.,

b . Consider the time evolution of a coherent state. If the initial state (0) ,

where ie show that 2( ) '

it

t e

where

'i t

e

. Use this result to show that the

expectation value of the position evolves according to 0( ) cosx t x t , where 0

2mx

.

Page 126: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim January 15 2016 p.1

Preliminary Examination: Quantum MechanicsDepartment of Physics and Astronomy

University of New MexicoJanuary 2016

Instructions:

• the exam consists of 10 problems, 10 points each;

• partial credit will be given if merited;

• total time is 3 hours.

Table of Constants and Conversion factors

Quantity Symbol ≈ valuespeed of light c 3.00× 108 m/sPlanck h 6.63× 10−34 J·selectron charge e 1.60× 10−19 Ce mass me 0.511 Mev/c2 = 9.11× 10−31 kgp mass mp 938 Mev/c2 = 1.67× 10−27 kgamu u 931.5 Mev/c2 = 1.66× 10−27 kgfine structure α 1/137Boltzmann kB 1.38× 10−23 J/KBohr radius a0 = h/(mecα) 0.53× 10−10 mconversion constant hc 200 eV·nm = 200 MeV·fmconversion constant kBT@ 300K 1/40 eVconversion constant 1eV 1.60× 10−19 J

FormulasPauli spin matricies:

σx =

(0 11 0

)σy =

(0 −ii 0

)σz =

(1 00 −1

)

Rotation of spinor about n direction by an angle φ:

R(φn) = cos

2

)I − i~σ · n sin

2

)

Angular momentum raising,lowering operators

J± |j,m〉 = h√j(j + 1)−m(m± 1) |j,m± 1〉

H.O. lowering operator

a =

√mω

2h

(x+

i

mωpx

)

Page 127: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim January 15 2016 p.2

1. The carbon monoxide molecule absorbs radiation at a wavelength of 2.6 millime-ters, corresponding to the rotational excitation of the molecule from ` = 0 to` = 1. Calculate the molecular bond length.

2. Consider a particle of mass m in a time independent but complex potentialV (x) = V0(x)− iΓ where Γ is independent of both space and time. Show that inthis case the probability density is not conserved and find its time dependence.

3. Consider an electron bound in a harmonic oscillator potential with a linear per-turbation.

H =p2

2m+

1

2mω2x2 + ε

x

x0

where x0 =√h/mω and ε << hω. What is the correction to the ground state

energy to lowest non-vanishing order in perturbation theory?

4. Consider a particle in a harmonic oscillator potential. For an initial state of theparticle given by a superposition of energy eigenstates,

|Ψ(x, 0)〉 =1√2

(|n〉+ |n+ 1〉).

Calculate 〈x〉 as a function of time.

5. A particle of mass m moves on a flat ring of radius a but is otherwise free to move.Taking the symmetry axis of the ring to be the z axis, write the Hamiltonian inposition space. Find the energy eigenstate position space wave functions andenergy eigenvalues. What are the conserved physical quantities? How are thedegeneracies related to the conserved quantities?

6. Consider the Ammonia molecule NH3. The three hydrogens lie in a plane andform an isosceles triangle with the nitrogen along an axis perpendicular to theplane. The position-space wave function for the nitrogen moving in the potentialof the three hydrogens has two linearly independent states: |1〉 and |2〉 corre-sponding to N above and below the plane of the hydrogens. (see Figure) TheHamiltonian in this basis has the form

[H] =

[E0 −A−A E0

]

(a) If the system is in state |1〉 at t = 0, what is the probability to find the systemin state |1〉 at time t?(b) What is the physical interpretation of the off-diagonal elements?

Page 128: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim January 15 2016 p.3

Figure 1: States |1〉 (left) and |2〉 (right) of the ammonia molecule.

7. A particle with low momentum hk is scattered by a hard sphere of radius a forwhich we can take the potential to be

V (r) =

{∞ r < a0 r > a

where r is the distance from the center of the sphere.(a) The radial wave function for the ` = 0 partial wave is of the form u(r) =rR(r) = c sin (kr + δ0) where δ0 is the zeroth partial wave phase shift and c is aconstant. Sketch u(r) for 0 < r < ∞, clearly indicating r = a. Determine thephase shift δ0.(b) What is the k → 0 total cross section limit? Compare to the quantum to theclassical result Recall the total cross section is

σ =4π

k2∑`

(2`+ 1) sin2 δ`

where δ` is the `th partial wave phase shift.

8. Consider two electrons produced in an entangled spin state with total spin S = 0(spin singlet state). What is the probability to measure one electron with spin-upalong the direction a and the other electron with spin-up along the direction b,where these directions are arbitrary?

Page 129: Preliminary Examination: Quantum Mechanics Department … · Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions:

QM Prelim January 15 2016 p.4

9. A hydrogen atom in the presence of a magnetic field will have an additionalinteraction (Zeeman effect)

HZeeman =µB

h~B ·

(~L+ 2 ~S

)where µB = 5.8× 10−5eV/T is the Bohr magneton.(a) What constitutes the weak field regime? Give an order of magnitude estimateof an upper limit on B, in Tesla, for the weak field approximation to be valid.(b) In the weak field approximation, find the correction to each of the states ofthe 2P 1

2doublet. The explicit form these states is:

|12,1

2〉 =

√2

3|1, 1〉 |1

2,−1

2〉 −

√1

3|1, 0〉 |1

2,1

2〉

|12,−1

2〉 =

√1

3|1, 0〉 |1

2,−1

2〉 −

√2

3|1,−1〉 |1

2,1

2〉

where the notation refers to |j,mj〉 (total angular momentum quantum numberj) on the left hand side and |1,m`〉 |12 ,ms〉 are the product of orbital and spinwave functions on the right hand side.

10. Two identical, non-interacting spin-1/2 particles of mass m are in the same onedimensional harmonic oscillator potential.

H =p212m

+1

2mω2x21 +

p222m

+1

2mω2x22

a) Determine the ground state(s) in terms of harmonic oscillator states |n〉i andspin |1/2,m〉i where i is the particle label. What is the energy and degeneracyof the ground state?b) Determine the first excited state(s). What is the energy and degeneracy ofthe first excited state?