1

Review of Phys 300

The Classical Point of View

A system is a collection of particles that interacts themselves via internal

forces and that may interact with the world outside via external fields.

Intrinsic properties of a classical system (e.g. rest mass and

charge) are independent of its physical environment and

therefore don’t depend on the particle’s location and don’t

evolve with time.

Extrinsic properties of a classical system (e.g. position and

momentum) evolve with time in response to the forces on the

particle.

According to the classical physics, all intrinsic and extrinsic properties of

a particle could be known to infinite precision and we could measure the

precise value of both position and momentum of a particle at the same

time (classical physics describes a determinate universe).

Classical physics predicts the trajectory (i.e., the values of its position and

momentum for all times after some initial time 0t ) of a particle,

0, ;r t p t t t trajectory , where the linear momentum is given by

vd

p t m r t m tdt

with m the mass of the particle. Trajectories are

called the state descriptors of Newtonian physics. The evolution of the

state represented by the trajectory is given by 2

2,

dm r t V r t

dt where

,V r t is the potential energy of the particle. To obtain the trajectory for

0t t , one only need to know ,V r t and the initial conditions ( the values

of r and p at the initial time 0t ).

As a result, classical physics (Newton’s laws) predicts the future of any

system that is known its initial conditions.

2

Near the end of the 19th century, theoretical physics was based on three

fundamentals:

Newton’s theory of mechanics

Maxwell’s theory of electromagnetic phenomena

Thermodynamics and kinetic theory of gasses

At the end of the 19th century and the beginning of the 20th century, a crisis

appeared in physics. A series of experimental observations couldn’t be

explained by concepts of classical physics.

Blackbody radiation

Photoelectric effect

Compton effect

Particle diffraction

The repeated contradiction of classical laws made it necessary to develop

some new concepts and the development of these concepts emerged

“Quantum Theory” that introduced new concepts:

Blackbody radiation

behaviour of radiation Photoelectric effect

Compton effect

behaviour o

f

Particle diffraction

Particle

Wave

Quantization of physical quantities

particle

Electrons at t

he atomic states mwit L nh vr

Quantum physics is a theory describing the properties of matter at the level

of micro phenomena (molecule, atom, nucleus, elementary particle, …).

This theory provides the answers to many questions which remained

unsolved in classical physics.

3

The Quantum Point of View

The concept of a particle doesn’t exist in the quantum world, particles

behave both as a particle and a wave (wave-particle duality).

Unlike the classical physics, nature itself will not allow position and

momentum to be resolved to infinite precision (Heisenberg uncertainty

relation), 0 02

xx t p t where 0x t is the minimum uncertainty in

the measuremen of the position in the x-direction at time 0t and 0xp t is

the minimum uncertainty in the measurement of the momentum in the x-

direction at time 0t . Position and momentum are fundamentally

incompatible observables (the universe is inherently uncertain).

Therefore, if we cannot know the position and momentum of a particle at

0t , we cannot specify the initial conditions of the particle and hence

cannot calculate the trajectory.

Physical quantities (like energy) take some certain values (quantization).

4

Heisenberg uncertainty relations

The standard deviation or uncertainty for two operators A and B are given

by

22 22 22 2ΔA = = 1

ΔA ΔB ,2

A A A AA B

iA A

The quantity 22ΔA= A A measures the spread of values about the

mean for A.

Wave-particle duality

Particles behave both as a particle and a wave in the quantum world

Matter

WavePhoton

Wave

MatterMatter

Physical quantities

for particles

Physical quantities

for waves

E h

p /h

What is the matter wave?

Plane wave in 1D: wt)i(kxe At)ψ(x,

dt

dx

tΔ

xΔlimv

0tΔp

constantwtkx “phase”

v2

1

mv

mv)2/1(

p

h

h

Eνλ

/λ2π

ν 2π

k

w

dt

dxv

2

p

speed particle

speed Broglie de

p v 2

1 v plane wave cannot represent a particle.

5

Wave packet: the superposition of the waves that is approximately

localized in space at any given time).

constantt2

dwx

2

dk v

mdv

mvdv

dp/

dE/

dk

dw

dt

dxvg

speed particle

speed Broglie de

g v v

wave packet represents a particle.

The wave packet is a superposition of waves.

Superposition of waves Fourier integrate

/1( , ) (p)

2

i px Etx t dp e

that is a solution of a partial differential equation called Schrodinger

Equation a free particle of mass m

2

22

x

t)ψ(x,

2m-

t

t)ψ(x, i

Schrodinger Equation in the presence of a potential V(x) will be

t)ψ(x,)x(Vx

t)ψ(x,

2m-

t

t)ψ(x, i

2

22

where (x,t) is the solution of Schrodinger equation, mathematical

description of the wave packet. It is to be physically acceptable solution

(called wave function) if it satisfies square integrable (finite), single-valued

and continuous properties. Any solution of Schrodinger equation that

becomes infinite must be discarded.

5 10 15 20 25 30 35

2

1

0

1

2

6

The Schrödinger equation has two important properties

The equation is linear and homogeneous. An important consequence

of this property is that the superposition principle holds. This means

that if 1ψ (x,t) and

2ψ (x,t) are solutions of the Schrödinger equation,

then the linear combination of these functions is also a solution

( , ) ( , )i ii

x t C x t .

The equation is first order with respect to time derivative (meaning

that the state of a system at some initial time to determines its

behavior for all future times)

2 2

2

( , ) ( , ) -

2first order in time derivative

x t x ti

t m x

It has one initial condition. Thus, if ( , 0)x t is known, ( , )x t can be

found. Firstly, (p) is calculated from - /1( ) ( ,0)

2

ipxp dx x e

. Then,

using (p) , t)ψ(x, is obtained from i(px-Et)/1ψ(x,t) dp (p) e

2 π .

Every solution of Schrodinger equation is not a wave function (physically

acceptable solution). To be an acceptable solution, an eigenfunction

( , )x t and its derivative ( , ) /x t x are required to have the following

properties:

( , )x t and ( , ) /x t x must be continuous.

( , )x t and ( , ) /x t x must be single valued.

( , )x t and ( , ) /x t x must be finite (square integrable).

7

Probability interpretation of the wave function:

Finite property requires the interpretation of probability density. t)ψ(x, is in

general a complex function and is apparently not a measurable quantity.

However, the wave function is a very useful tool for calculating other quite

meaningful, mathematically real quantities. 2|t)ψ(x,| is always real. It is

large where the particle is supposed to be, and small elsewhere. We also

know that 2|t)ψ(x,| is spreading with time (it means that as time passes, it is

less probable to find the particle where it is put at t=0). 2|t)ψ(x,|t)P(x, is

called the probably density and requires:

-

2 1dx|t)ψ(x,| that is called the

normalization condition.

Conservation of Probability

The probability density is defined as , * , ,x t x t x t . We know

that ,x t satisfies the Schrödinger equation. It is also true that * ,x t

satisfies the Schrödinger equation. It satisfies the following equation

called continuity equation

, ( , ) 0x t j x tt x

where *

*2

d dj

mi dx dx

is the flux (probability current density that

is the number of particles per second passing any point x )

In 3D, they will be

, ( , ) 0r t j r tt

and ( , ) * *2

j r tmi

We then find

s as 0t),x(jsdt),x(j xdt) P(x,xdt

S-

3

-

3

constantt)x P(x,d -

3

A change of the density in a region bxa is equal to a net change in the

flux into that region ( , ) ( , ) ( , ) ( , )b b

a a

ddx P x t dx j x t j a t j b t

dt x

8

Expectation value:

How can we calculate the measurable quantities (position, momentum,

energy, etc) from the wave function?

An average value of a measurement (statistical average of a large number

of measurements) is called the expectation value.

The expectation value of the position of a particle:

In general, x-coordinate of a particle cannot have a certain value in a

region if 0t)ψ(x, in that region. It is possible to talk about the average

value of x-coordinate. It is called the expectation value of x-coordinate

and is given by

-

*

-

dx t)ψ(x,x t)(x,ψdx t)x P(x,x

The expectation value of the momentum of a particle:

-

* dx t)ψ(x, xi

t)(x,ψp

where xi

p

is the momentum operator in x-space.

Alternatively, in momentum space

*

-

x dp (p) i (p) p

and *

-

p dp (p) p (p)

where p

ix

is the position operator in momentum space.

For an arbitrary function, the expectation value

-

* dx t)ψ(x, f(x) t)(x,ψx)(f and

-

* dx t)ψ(x, xi

f t)(x,ψp)(f

Example:

-

2*2 dx t)ψ(x, x t)(x,ψx and

-

2

*2 dx t)ψ(x, xi

t)(x,ψp

The uncertainties in x and p

22 xxΔx and 22 ppΔp

The quantity 22 xxΔx measures the spread of values about the

mean for x and 22 ppΔp for p .

9

Hermitian Operator

The operator xi

p

is complex, but its expectation value is real. Such

operators (position, momentum, energy, etc) are called Hermition

operators.

The energy operator : t

iE

Hamiltonian of the system: )x(Vxm2

H2

22

Operators play a central role in Quantum Mechanics. Products of

operators need careful definition, because the order in which they act is

important.

AB-BAB,A is called a commutation relation between operators

Commutation relation between momentum and position: ixp,

This relation is independent of what wave function this acts on. This is a

fundamental commutation relation in quantum physics. Heisenberg

uncertainty relation becomes

2

ΔpΔx x,pi4

1ΔpΔx

222

Important concepts:

The notation of a wave packet as representing a particle

Schrodinger equation as fundamental equation in quantum physics

The wave function t)ψ(x, that has the probability interpretation

Heisenberg uncertainty relation

Statistical average of a large number of measurements as expectation

value

10

Eigenvalues and Eigenfunctions

Time-dependent Schrodinger Equation

t)ψ(x,)x(Vx

t)ψ(x,

2m-

t

t)ψ(x, i

2

22

The method of separation of variables: if the spatial behavior of the wave

function does not change with time, we use some time-varying multiplying

factor in front of the spatial part of the wave function.

Eu(x)V(x)u(x)dx

u(x)d

2m-

CeT(t)ET(t)dt

dT(t)i

)x(u)t(Tt)ψ(x,

2

22

iEt/-

The eigenvalue equation

ˆ O ( )ˆ ˆ O O

ˆ O f(x) λ f(x) eigenvalueoperator

eigenfunction eigenfunctionof wrt f x

of of

The eigenvalue equation states that the operator Q acting on certain

functions f(x) (eigenfunction) will give back these functions multiplied by

a constant λ (eigenvalue).

Energy eigenvalue equation

(x)u E (x)uH EE

Although the wave function -iEt/

E e)x(ut)ψ(x, depends on time, the

probability density 2

E

2 |(x)u||t)ψ(x,| does not depend on time.

The initial spatial wave function as a linear superposition of the energy

eigenstates

ψ(x,0) ( )n n

n

C u x

The evolution of the wave function in time is given by a simple linear

superposition of these eigenstates

n

t/-iE

nnne)x(uCt)ψ(x,

11

Alternatively and equivalently, a time-evolution operator - /iH te is applied

to an initial state0ψ(x,t ) at time

0t to obtain the evaluated state 1ψ(x,t ) at time

1t as

1 0- /

1 0( , ) ( , )iH t t

x t e x t

Example : a particle in an infinite box

0x ,

ax0 ,0

0x ,

V(x)

BcoskxAsinkxu(x)0u(x)mE2

dx

u(x)d22

2

Energy eigenfunction and eigenvalues

1,2,3,...n 2ma

nπE

a

xn πsin

a

2(x)u

2

222

n

n

Orthonormality condition

nm when1

nm when0(x)u(x)udx mn

a

0

m

*

n δ

The expansion postulate: Any function ψ(x) can be expanded in terms of

orthogonal functions.

(x)uAψ(x) n

n

n

where a

0

*

nn ψ(x)(x)udx A is the projection of ψ(x) onto nu (x) .

The expectation value of energy is given by

n

n

2

n E |A|ψ(x)|H|ψ(x)

where 2

n |A| is the probability that a measurement of the energy for the state

ψ(x) yields the eigenvalue nE .

12

Momentum eigenfunction

ipx/

pppope e π 2

1(x)u (x)u p (x)u p

Degeneracy: if more than one eigenfunction corresponds to the same

eigenvalue, this eigenvalue is said to be degenerate.

Parity

1λλψ(x)ψ(-x)ψ(x)P

parity offunction eigen also ision eigenfunctEnergy

motion ofconstant a is P0H,P

Important concepts:

The eigenvalue equation

Particle in a box problem

The expansion postulate

13

One-Dimensional Potentials

The Potential Step

a) When a neutron with an external kinetic energy K enters a nucleus, it

experiences a potential

b) While a charged particle moves along the axis of two cylindrical

electrodes held at different voltages, its potential energy changes very

rapidly when passing from one to the other. Potential energy function

can be approximated by a step potential.

The Potential Well

a) The motion of a neutron in a nucleus can be approximated by assuming

that the particle is in a square well potential with a depth about 50 MeV.

b) A square well potential results from superimposing the potential acting

on a conducting electron in a metal.

14

The Potential Barrier

a) scattering problems

b) Emission of α particles from radioactive nuclei

c) Fusion process

d) Tunnel diode

e) Cold emission electrons

The Harmonic Oscillator

15

The General Structure of Wave Mechanics

Postulates

Postulate 1:

The dynamical state of a particle can be described by a wave function

which contains all the information that can be known about the particle. Postulate 2:

An arbitrary function can be expressed as a linear superposition of a set of

orthonormal functions.

Postulate 3:

The Schrodinger equation describes the behavior of the wave function in

space and time. Postulate 4:

Each observable quantity q can be directly associated with a linear,

Hermitian operator. The value q is an eigenvalue of the operator.

Hermitian property: 2121 ψQψQψψ

Theorem 1: The eigenvalues of a Hermitian operator are real.

Theorem 2: The eigenfunctions of a Hermitian operator are orthogonal

if they correspond to distinct eigenvalues. Postulate 5:

The expectation value of a measurement of a variable q is given

mathematically as

ψψ

ψQ|ψq

Each observable quantity q can be directly associated with a linear,

Hermitian operator. The value q is an eigenvalue of the operator.

Commuting Observables:

The cummutator of two operators A and B is defined by AB-BAB ,A .

If the commutator vanishes when acting on any wave function, the

operators A and B are said to commute, ABBA .

When the operators commute, 0B ,A , their observables A and B are said

to be compatible. Observables are non-compatible if 0B ,A .

If two observables are compatible, their corresponding operators have the

simultaneous eigenfunctions and A and B are said to be simultaneously

measurable. Thus, compatible observables can be measured simultaneously

with arbitrary accuracy, non-compatible observables cannot.

16

nnnn ψBaψABψBA 0B ,A

nψB is an eigenfunction of A belonging to the eigenvalue na . Since na is

non-degenerate, nψB can only differ from nψ by a multiplicative constant

which we can call nb

nnn ψbψB

Thus we see that nψ is simultaneously an eigenfunction of the operators A

and B belonging to the eigenvalues na and nb , respectively.

If . . . ,C ,B ,A are a set of commuting operators, there is a simultaneous

eigenfunction nψ of . . . ,C ,B ,A with the eigenvalues ,...c ,b ,a nnn .

Time Dependence and Classical Limit:

The expectation value of an operator A is

-

*

tdx t)ψ(x, A t)(x,ψA

The expectation value varies with time as

t

tt

A,Hi

t

AA

dt

d

If A has no explicit time dependence, tt

A,Hi

Adt

d

. The observable A is

a constant of motion if the operator A commutes with H.

17

Operator Methods in Quantum Mechanics

One dimensional harmonic oscillator Hamiltonian is

222

xmω2

1

2m

pH where ixp,

The problem is how to find the energy eigenvalues and eigenstates of this

Hamiltonian.

1. Polynomial Method

2 2

1/2

α x /2

n nn

n

αu (x) e H (α x)

π 2 n!

1 E n ω n 0,1,2,....

2

2. Operator Method

Dimensionless position and momentum operators are defined as

q-ip

mω

1p

x mω

q

x

Two non-hermitian operators are introduces in terms of q and p as

p mω

1ix

mω

2

1ip)(q

2

1A

p mω

1ix

mω

2

1ip)(q

2

1A

x

x

Since x and p are Hermition, A is indeed the hermitian conjugate of A .

Commutator A,A gives 1AAAA1A,A .

The Hamiltonian in terms of these operators becomes

2

1AAH ω

where AAN is known as the number operator which is Hermitian.

0NH, They have simultaneous eigenfunctions.

n|nn|N

n|En|H n

18

The other commutation relations become

AωAHHAAωAH,

AωAHHAAωAH,

Consider n|En|H n . Multiply both sides with A as

n|Aω)E(n|AHn|EAn|HA nn

AωAH

n|A is an eigenfunction of H with the eigenvalue ω)E( n such that the

energy nE is lowered by one unit of ω .

Multiply again with A

n|A ω)2E(n|A Hn|Aω)E(An|AHA 2n

2n

AωAH

n|A2 is also an eigenfunction of H with the eigenvalue ω)2E( n such that

the energy nE is lowered by ω2 .

The operator A is called a lowering operator.

Since the harmonic oscillator has only positive energy states including zero,

there must be a lower bound on the energy. There is a state of lowest

energy, the ground state.

00|A

So that energy cannot be lowered any more.

ωωω

ω

2

1E0|

2

10|

2

10|AA

0|2

1AA0|H

0

Now multiply n|En|H n with A

n|Aω)E(n|AHn|EAn|HA nn

AωAH

n|A is an eigenfunction of H with the eigenvalue ω)E( n such that the

energy nE is raised by one unit of ω .

Multiply again with A

n|A ω)2E(n|A Hn|Aω)E(An|AHA2

n

2

n

AωAH

n|A2 is also an eigenfunction of H with the eigenvalue ω)2E( n such that

the energy nE is raised by ω2 .

19

The operator A is called a raising operator.

We obtained the energy spectrum of the harmonic oscillator without solving

any differential equation as

0,1,2,....n ω2

1n En

What are the values of nC and nD ?

1n|Dn|A

1n|Cn|A

n

n

Using AA,N and AA,N , we get 1nCn and nDn .

The eigenstates are

0|A!n

1n|

.

.

.

0|A1.2.3

12|A

3

13| :2n

0|A1.2

11|A

2

12| :1n

0|A1

11| :0n

n|A1n

11n|1n|1nn|A

n

3

2

The explicit for of the eigenstates are

2/qnnn

2/q0

n

n

2/q000

0

0

2

2

2

e)q(HN)q(u

eCq

q2

1

!n

1u|

eCu0qudq

ud

dq

dip ,0u|)ipq(

2

1

00|A

20

The time dependence of operators:

The solution of time dependent Schroidnger equation is ψ(0)|eψ(t)| /iHt

The expectation value of an operator B is

0

/iHt/iHt

t)t(Bψ(0)|)t(B| ψ(0)ψ(0)|eBe| ψ(0)ψ(t)|B| ψ(t)B

Pictures

Schrodinger picture:

Operators are time-independent.

Time evaluation of the system is determined by a time dependent wave

function.

Heisenberg picture:

Operators are time-dependent.

Wave functions are time-independent.

The result is the same whatever picture we use. Time variation of )t(B is

given by

S t ω i

H

S tω i

H

Ae(t)A

A e(t)A(t)BH,

i(t)B

dt

d

As an example: position and momentum operators of a particle

tx(0)sin ω mω- tp(0)cosωp(t)

tsinωmω

p(0) tx(0)cos ωx(t)

21

Levi-Civita Tensor

Coordinate transformation in 3D can be written as

1,2,3i xλx3

1j

jij'i

The magnitude of a vector is invariant under coordinate transformation

(rotation)

'rr

'z'y'x'r

zyxr

222

222

or

22'rr

3 32 2 ' '

i i ij j ik ij ik j

i=1 jk 1

ij ik

' x x x λ x λ x λ λ x x

λ λ "orthogonality condition"

i i k k

i i j k j

jk

j

r r x

ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ij

ijk k

i i j j i j i j i i

i j ij i

i i j j i j i j i k ijk k

i j ij ije

i ijk i j ijk mk i jm im j

jk k

A B A e B e A B e e A B

C A B A e B e A B e e A B e

C A B

1 if i, j, k form an even permutation of 1, 2, 3

1 if i, j, k form an odd permutation of 1, 2, 3

0 when any two indices are the same

ijkε

x

y

'z,z

,z,b

b

'x

θ

θ

z

y

x

1 0 0

0 1 θ-

0 θ 1

z'

y'

x'

z

y

x

1 0 0

0 cosθ sinθ-

0 sinθ cosθ

z'

y'

x'rotation

malinfinitesi

'y

z'

y'

x'

1 0 0

0 cosθ sinθ

0 sinθ- cosθ

z

y

x