Chap 3. Formalism

1. Hilbert Space

2. Observables

3. Eigenfunctions of a Hermitian Operator

4. Generalized Statistical Interpretation

5. The Uncertainty Principle

6. Dirac Notation

Vector Space & Inner Product

Vector space : Linear space closed under vector addition & scalar multiplication.

Inner product :

: V V

satisfying*

0 0

Conjugate symmetric.

Positive.with 0

b c b c Linear.

which implies * *b c b c

b c

, V a b V ,a b F

orF=R C

Dual Space

Dual space V* of a vector space V Set of all linear

maps

: V F

V* can be associated with an inner product on V by setting

V* is itself a vector space and isomorphic to V.

Thus, the dual | V* of a vector | V is defined as the linear mapping such that

3.1. Hilbert Space

State of system : Wave functionObservables : Operators

( -D ) VectorsLinear transformations

Linearity

Hilbert space Complete inner product space.( Cauchy sequence always converges )

E.g., Set of all square-integrable functions over a domain :

2L

*d f f r r exists 2f L

with inner product :

*f g d f g

r r

Quantum state space is a Hilbert space.

L2 L2 :

Schwarz inequality :2

f g f f g g ( cos 1 )

[ Guarantees inner product is finite in Hilbert space. ]

*f g g f

m n m mnf f A orthogonal

1

n nn

f c f

r r

*1n n

n

c d f fA

r r

f fr r

1n n

n

f c f

r r

1n n

n

c f fA

1

1n n

n n

f f IA

completeness

r component of vector f

1n n

n

f c f

1

1n n

n n

f f f fA

conjugate symmetric

Read Prob 3.1, Do Prob 3.2

3.2. Observables

1. Hermitian Operators

2. Determinate States

3.2.1. Hermitian Operators

* ˆ ˆQ d Q Q

r rExpectation value of Q :

Outcomes of experiments are real :*

Q Q

*ˆ ˆQ Q Q̂ Q̂

ˆ ˆQ Q Q is hermitian (self-adjoint)

Observable are represented by hermitian operators.

E.g. : *ˆ

dp d x

i d x

*d

d xi d x

*

dd x

i d x

p̂

Read Prob 3.3, 3.5 Do Prob 3.4

3.2.2. Determinate States

Determinate state : A state on which every measurement of Q gives the same value q.

220 Q Q

i.e., ˆQ Q q

2Q q

Q̂ q

2Q̂ q ˆ ˆQ q Q q

Determinate states are eigenstates.

Spectrum of an operator Set of all of its eigenvalues

If two or more independent eigenfunctions share the same eigenvalue, the spectrum is degenerate.

E.g., solutions to the Schrodinger eq. H E

are determinate states of the total energy,

as well as eigenfunctions of the hamiltonian.

Example 3.1.

Let ˆ dQ i

d , where is the polar coordinate in 2-

D.

Is Q hermitian ?

Find its eigenfunctions and eigenvalues.

Ans. Consider the Hilbert space of all functions 2f f 0, 2

2*

0

ˆ d gf Q g d f i

d

2 *

2

00

d fi f g d i g

d

Q̂ f g

Q is hermitian.

*2

0

d fd i g

d

d fi q f

d has eigenfunctions i qf A e Eigenequation

2f f 0, 1, 2, ...q 2 1i qe

Spectrum of Q is the set of integers, & it’s non-degenerate.

3.3. Eigenfunctions of a Hermitian Operator

1. Discrete Spectra

2. Continuous Spectra

Phys : Determinate states of observables.Math : Eigenfunctions of hermitian operators.

Discrete spectrum : n L2 normalizable & physically realizable.

Continuous spectrum : k not normalizable & not physically realizable.Can be used to form wave packets.

Examples:

Purely discrete spectrum : Harmonic oscillator.

Purely continuous spectrum : Free particle.

Mixed spectrum : Finite square well.

3.3.1. Discrete Spectra

Theorem 1 : Eigenvalues of hermitian operators are real.

Proof :

Q̂ f q fLet

ˆ ˆf Q f Q f f

( f is eigenfunction of Q with eigenvalue q )

( Q is hermitian )

*q f f q f f

*q q QED

Theorem 2 : Eigenfunctions belonging to distinct eigenvalues are orthogonal.

Proof :

Q̂ f q fLet

ˆ ˆf Q g Q f g

( f , g are eigenfunctions of Q with eigenvalue q and r )

( Q is hermitian )

*r f g q f g

0f g r q QED

Q̂ g r g

q f g ( Theorem 1 )

Using the Gram-Schmidt orthogonalization scheme on the degenerate subspaces, all eigenfuctions of a hermitian operator can be made orthonormal.

Axiom (Dirac) : Eigenfunctions of an observable operator are complete.

Note:

Eigenfunctions of a hermitian operator on a finite dimensional space are complete.

Not necessarily so if the space is infinite dimensional.

( Required to guarantee every measurement has a result. )

3.3.2. Continuous Spectra

Eigenfunctions not normalizable.

Example 3.2. Momentum Operator

Example 3.3. Position Operator

Example 3.2. Momentum Operator

Find the eigenfunctions & eigenvalues of the momentum operator.

p p

df x p f x

i d x

expp

i p xf x A

2* expp p

i p p xd x f x f x A d x

2

2A p p

1

2i k xd x e k

expp

i p xf x

* 2p pd x f x f x p p

Set

2p p p p

Ans.

( Dirac orthogonality )Placement of the 2 is a matter of taste.

;i k xe k R is complete ( Fourier transform )

2 p

d pf x c p f x

For any real function f :

/

2i p xd p

c p e

//

2i p p xi p x d p

d x f x e c p d x e

d p c p p p

c p

i.e. /i p xc p d x f x e

Example 3.3. Position Operator

Find the eigenfunctions & eigenvalues of the position operator.

y yx g x y g x

Ans. Let gy be the eigenfunction with eigenvalue y.

yg x A x y

Dirac orthonormality : y yg g y y

2*y yd x g x g x A d x x y x y

2A y y

2

1A

yg x x y

f x d y f y y x

Completeness:For any real function f

yd y c y g x

c y f ywith

Preferred Derivation

2p p p p 12

d pp p

2

d pp p

2

d px x p p

2

ip xd p

x e p

p d x p x x

d x x x

i

p xp d x e x

x x x x 1d x x x

3.4. Generalized Statistical Interpretation

Generalized statistical interpretation :1.Measurement of an observable Q(x, p) on a state (x, t) always gets one of the eigenvalues of the hermitian operator Q(x, i d / dx).

22

n nc f

2.a) Discrete eigenvalues, orthonormalized eigenfunctions :

2 2

zc z dz f dz

2.b) Continuous eigenvalues, Dirac orthonormalized eigenfunctions :Probability of getting an eigenvalue q(z) with z between z and z + dz is

Probability of getting the eigenvalue qn is

3. Upon measurement, collapses to fn or fz .

Proof for Discrete Eigenvalues

, n nn

x t c f x n nc f * ,nd x f x x t

21n

n

c If is normalized.

m m n nm n

c f c f *m n m n

m n

c c f f *m n mn

m n

c c

ˆQ Q m m n n nm n

c f c q f *m n n m n

m n

c c q f f *m n n mn

m n

c q c 2

n nm

q c

| cn |2 = Probability of getting the eigenvalue qn .

| cn |2 could be a probability.

Position Eigenfunctions

yg x x y Eigenfunction of position operator :

yc y g * ,yd x g x x t

,d x x y x t

,y t

2 2,c y dy y t dy = probability of finding particle within ( y, y+dy ).

Momentum Eigenfunctions

/i p xpf x e

Eigenfunction of momentum operator :

pc p f * ,pd x f x x t

/ ,i p xd x e x t

2 2,

2 2

dp dpc p p t

= probability of finding particle with momentum within ( p, p+dp ).

Momentum space wave function. /, ,i p xp t d x e x t

/, ,2

i p xd px t e p t

Position space wave function.

2 2, ,

2

d pd x x t p t

Note :

Example 3.4.

A particle of mass m is bound in the delta function well V(x) = ( x).

What is the probability that a measurement of its momentum would yield a value greater than p0 = m / .

2 2, exp exp

m m ix t x E t

Ans. 2

22

mE

2 2, exp exp

m i i mp t E t d x px x

00

1exp exp

p iE t d x i px p x

20

2

p

m

0

0 0 0

0

1 1 1exp exp expd x i px p x d x i p p x d x i p p x

0 0i p p i p p

0

2 20

2 p

p p

00

1, exp exp

p ip t E t d x i px p x

3/20

2 20

, 2 expp i

p t E tp p

0

2

0 ,2p

d pP p p p t

0

30 22 2

0

2 1

p

p d pp p

0

102 2

0 0

1tan

p

p p p

p p p

12 3 2 22 2

1 1tan

2

ax xd x

a x a ax a

1 1

2 2 4

1 1

2 4 0.0908

Read Prob 3.12

Do Prob 3.11

3.5. The Uncertainty Principle

1. Proof of the Generalized Uncertainty Principle

2. The Minimum Uncertainty Wave Packet

3. The Energy-Time Uncertainty Principle

2x p

3.5.1. Proof of the Generalized Uncertainty Principle

22A A A

2A f f

ˆ ˆA A A A

A hermitian, A real.

2

* ˆd A A r r System in state .

*

ˆ ˆd A A A A r r

ˆf A A where

2B g g ˆg B B where

2 2A B f f g g 2

f g Schwarz inequality

22 2A B f g

2Im f g

2*1

2f g f g

i

2

1

2f g g f

i

ˆf A A ˆg B B

ˆ ˆf g A A B B

ˆ ˆA A B B A, B hermitian, A , B real.

ˆ ˆˆ ˆA B A B B A A B

ˆ ˆA B A B

ˆ ˆf g AB A B f g , A B ˆˆg f B A A B

ˆ ˆˆ ˆf g g f AB B A ˆ ˆ,A B ˆ ˆ ˆˆ ˆ ˆ,A B AB B A where

22 2 1 ˆ ˆ,

2A B A Bi

Generalized Uncertainty Principle

1 ˆ ˆ,2A B A Bi

or

1 ˆ ˆ,2A B A Bi

1ˆ ˆ,

2x p x pi

2

ˆ ˆ,x p i

Observables A and B are incompatible if ˆ ˆ, 0A B

Stationary states of a system can be specified by the eigenvalues of a maximal (complete) set of observables compatible with H.

Measuring A collapses state to an eigenstate of A, and similarly for B.

If A and B are incompatible, repeated measurements of A, B, A, B, ..., will never, except by accident, get the same values.

If A and B are compatible, repeated measurements of A, B, A, B, ..., will get the same values provided A and B are also compatible with H.

Read Prob 3.13Do Prob 3.15

3.5.2. The Minimum Uncertainty Wave Packet

E.g., ground state of a harmonic oscillator ,Gaussian wave packets of a free particle.2x p

Minimum Uncertainty

g c f

Setting the Schwarz inequality to equality

c i a

2f f g g f g c = constant

22 2 ImA B f g

Minimal uncertainty thus implies

Re 0f g a = real

g i a f

ˆB̂ B i a A A

i.e.

minimal uncertainty state

Uncertainty principle keeps only Im < f | g > to give

Re 0c f f

2 22 2 ImA B f g f g

or

ˆB̂ B i a A A

For position-momentum uncertainty : dp i a x x

i d x

2exp exp

2

a ix A x x p x

Prob 3.16

3.5.3. The Energy-Time Uncertainty Principle

Position-momentum uncertainty :2x p x p

Special relativity suggests energy-time uncertainty :2

t E 4-vectors:

( c t, x ), (E / c, p )

Non-relativistic theory :

1. t is a parameter, not a dynamic variable.

2.t t .

3. t = time for system to change appreciably.

Energy-time uncertainty is NOT like the other uncertainty pairs.

ˆd Q dQ

dt d t

Let Q( x, p, t ) be some characteristic observable of the system.

ˆˆ ˆQ

Q Qt t t

ˆ1 1ˆ ˆˆ ˆQH Q Q H

i t i

ˆi H

t

ˆ1 ˆ ˆ,Q

Q Hi t

1 ˆ ˆ,d Q Q

Q Hdt i t

H hermitian

1 ˆ ˆ,2A B A Bi

1 ˆˆ ,2H Q H Qi

2

d Q

dt

Q = Q( x, p )

2E t

DefineQt

d Q

d t

HE

Example 3.5.

For a stationary state, 0d Q

dt for all observable Q.

E 0, t

Time dependence occurs only for linear combinations of stationary states.

E.g., 1 2/ /1 2, i E t i E tx t a x e b x e

a, b, 1 , 2 real.

2 2 22 2 2 11 2 1 2, 2 cos

E Ex t a x b x ab x x t

E E2 E12 1

2 2t

E E

2E t 2

Example 3.6.

How long does it take for a free-particle wave packet to pass by a particular point ?

Roughly, x m x

tv p

2

2

pE

mNote : p p

Em

E t p x 2

c.f. Prob 3.19

x width of wave packet.

Example 3.7.

The particle lasts about 1023 s before spontaneously disintegrate.

A histogram of all measurements of its mass gives a bell-shaped curve centered at 1232 MeV/c2, with a width about 120 MeV/c2.

Why does the rest energy ( mc2 ) sometimes come out larger than 1232, and sometimes lower? Is this experimental error?

Ans.

231120 10

2E t MeV s

226 10 MeV s

while 166.58 10 eV s 223.29 102

MeV s

Spread in measured mass is close to minimum uncertainty allowed. It’s not experimental error.

Measured data :

2t E

Caution :

does NOT mean you can violate energy conservation by

borrowing energy E and paying it back within t / (2E).

3.6. Dirac Notation

The set of all eigenfunctions of any observable is complete.

It can be used as a basis for the system’s Hilbert space.

State of system ( vector in Hilbert space ) : t t is a parameter

r-representation : Basis = Eigenstates of the position operator.

, t t r r

3 ˆd r I r r completeness

3t d r t r r 3 ,d r t r r

r r r r orthogonality, Dirac normalization

p - Representationp-representation : Basis = Eigenstates of the momentum operator.

, t t p p

3

3ˆ

2

d pI

p p

completeness

3

32

d pt t

p p

, t t r r

3

3 ,2

d pt

r p p

32 p p p p orthogonality, Dirac

normalization

3

32

d p

r p p r r r

r r

3

3 exp2

d p i

p r r

expi

r p p r

exp

i

p r p r

3 ˆd r I r r 3d r p r r p p p 3 expi

d r p p r

3

3 exp ,2

d p it

p r p

Let ˆnH n E n | n is an energy eigenstate.

ˆn

n n I completeness

n

t n n t

n

t n n t r r

mnm n orthonormality

, n nn

t c t r r nc t n t

ˆexp 0i

t H t ˆi t H t

t

ˆexp 0n

ic t n H t

ˆexp 0n

iH t

exp 0n

iE t n

expn n n

ic t E t c

0nc n , expn n n

n

it c E t

r r

where

Operators: Discrete Basis

Operators are linear transformations :

Q̂

Orthonormal basis ne n nn

a e n na e

n nn

b e n nb e

ˆn n n n

n n

a Q e b e ˆn m n n m n

n n

a e Q e b e e mb

Or mn n mn

Q a b ˆmn m nQ e Q ewhere matrix elements

ˆ m ne Q e

Operators: Continuous Basis

Operators are linear transformations :

Q̂

Dirac-orthonormal basis z d z a z z a z d z z

d z b z z b z d z z

ˆd z a z Q z d z b z z ˆd z a z z Q z d z b z z z b z

Or ,d z Q z z a z b z ˆ,Q z z z Q z where matrix elements

,Q z z Q z z z Q z a z b zIf Q is diagonal, i.e., then

Example: x-representation

x x x x x̂ x x x

ˆx x x x x x Matrix elements : x x x x x x

ˆ ˆ ˆ ˆ ˆ ˆ,x x p x x x p px x ˆ ˆx x p x x x p x ˆx x x p x

i x x i x x

Since ( prove it ! )

ˆd

x p x i x xd x

di x x

d x

d f xdf x x a x a

d x d x

we have dx x x x x x

d x

ˆd

x p x i x xd x

ˆ ˆx p d x x p x x d x xd x i x

d x

d xd x i x x

dx

ˆ

d xx p

i dx

ˆd

pi d x

ˆ ˆp d x x x p d x

d x xi dx

ˆx x x x x x

2ˆˆ ˆ

2

pH d x x x V x

m

2ˆˆ ˆ

2

px H d x x x x V x

m

22

22

d xd x x x V x x

m d x

22

22

d xV x x

m d x

di t

d t

di x t

d t

,i x tt

Example 3.8

Consider a system with only 2 independent states1

10

02

1

General normalized state : 1 2S a ba

b

with

2 21a b

Most general Hamiltonian :1*

2

h g

g h

H with h1, h2 real.

Assumeh g

g h

H with h, g real.

If the system starts out ( at t = 0 ) in state | 1 , what is its state at time t ?

Ans :

Time-dependent Schrodinger eq.

Time-independent Schrodinger eq.

S Sd

i Hd t

s sH E

s sH Eh g

g h

H

det 0

h E gE

g h EH I Characteristic eq.

2 2 0 h E g h E g

E h g Eigenenergies are :

Corresponding eigenvectors are given by: 0

h E g

g h E

0

0

h E g

g h E

Eigenenergies are :

E h

g

1

1s

Normalized :11

12s

11

12s

System starts out ( at t = 0 ) in state | 1 : 10

0S

1

2s s

1

2S s s

i i

E t E tt e e

1 11

1 12

i i

h g t h g te e

1

2

i ih g t h g t

i ih g t h g t

e e

e e

E h g

cos

sin

ih t

g t

eg t

i

neutrino oscillation: e .

ket : | f is a vector in L2 .bra : f | is in its dual L2

*.

Dirac notation :

* f d f

For L2 infinite dimensional,| f is a function, f | a linear functional:

so that*f g d f g

For L2 finite dimensional,| is a column vector, | is a row vector :

* * *1 2 na a a

1

2

n

a

a

a

so that*

1

n

j jj

a b

Let | be normalized, then

ˆ P is the projection operator onto the 1-D space spanned by | .

ˆ PE.g.

is a vector in the direction | with magnitude | .

Let { | en ; n = 1, ..., N } be an orthonormal basis, i.e., m n mne e

then1

ˆ

N

n nn

e e I = unit operator in N-dimensional space spanned by { | en } .

1 1

N N

n n n nn n

e e e eE.g.

If { | ez } is a Dirac orthonormalized continous basis, i.e., z ze e z z

then ˆ z zd z e e I = unit operator in function space spanned by { | ez }.

ne magnitude of | along | en

Read Prob 3.21, 3.24 Do Prob 3.22, 3.23