1

Quantum MechanicsPostulate 4

Describes expansionIn the eigenvalue equation the eigenfunctions {ui} constitute a complete, orthonormal set

The set of eigenfunctions {ui} form a basis: any wave function Ψ can be expanded as a coherent superposition of the complete set

iii uouO =ˆ

)()(as written becan it ,)(any For

xucxFx

ii

i∑=Ψ

∈Ψ

( ) ( ) relationlity orthonorma thecalled is *ijji xuxudx δ=∫

∞

∞−

2

ExpansionThus you can write any wave function in terms of wave functions you already know. The coefficients can be evaluated as

j

jii

i

iji

i

iiijj

c

c

uudxc

ucudxudx

=

=

=

=Ψ

∑

∫∑

∫ ∑∫∞

∞−

∞

∞−

∞

∞−

*

**

δ

3

ExpansionThe expectation value can be written using the {ui} basis

Thus the probability of observing oj when O is measured is |cj|2

jj

j

jjij

jii

jjj

iii

oc

uoudxcc

ucOucdx

OdxO

2

**

**

*

ˆ

ˆˆ

∑

∫∑∑

∑∫ ∑

∫

=

=

=

ΨΨ=

∞

∞−

∞

∞−

∞

∞−

4

Quantum Mechanics

Postulate 5Describes reductionAfter a given measurement of an observable O that yields the value oi, the system is left in the eigenstate uicorresponding to that eigenvalueThis is called the collapse of the wavefunction

5

Reduction

Before measurement of a particular observable of a quantum state represented by the wave function Ψ, many possibilities exist for the outcome of the measurementThe wave function before measurement represents a coherent superposition of the eigenfunctions of the observable Ψ = (ucat dead + ucat alive)

6

Reduction

The wave function immediately after the measurement of O is ui if oi is observed

|cj|2 = δij

Another measurement of O will yield oisince the wave function is now oi, not Ψ

If you look at the cat again it’s still dead (or alive)

7

Quantum MechanicsPostulate 6

Describes time evolutionBetween measurements, the time evolution of the wave function is governed by the Schrodinger equation

VmpVTHH

Ht

i

Vxmt

i

ˆ2ˆˆˆˆoperator n Hamiltonia theis ˆ

ˆ

2

2

2

22

+=+=

Ψ=∂Ψ∂

Ψ+∂Ψ∂

−=∂Ψ∂

h

hh

8

Time Independent Schrodinger Equation

How do we solve Schrodinger’s equation?In many cases, V(x,t) = V(x) onlyApply the method of separation of variables

Vdxd

mdtdf

fi

fVfdxd

mdtdfi

fdxd

xdtdf

t

tfxtx

+−=

+−=

=∂Ψ∂

=∂Ψ∂

=Ψ

2

22

2

22

2

2

2

2

12

12

readsnow equation sr'Schrodinge and

and then

)()(),(

ψψ

ψψψ

ψψ

ψ

hh

hh

9

Time Independent Schrodinger Equation

The last equation shows that both sides must equal a constant (since one is a function of t only and the other a function of x only)

Thus we’ve turned one PDE into two ODE’susing separation of variables

ψψψ EVdxd

fiEdtdf

Edtdf

fi

=+−

−=

=

2

22

2m (right)

or

1 (left)

h

h

h

10

Time Independent Schrodinger Equation

Time dependent piece

Note the e-iωt has the same time dependence as the free particle solution

Thus we see

h

h

h

iEt

tiiEt

etf

CCeCef

fiEdtdf

−

−−

=

==

−=

)(

so theinto absorb can We solution has

ψ

ω

ωh=E

11

Time Independent Schrodinger Equation

Time independent piece

Comment 1Recall our definition Hamiltonian operator H

ψψψ EVdxd

m=+− 2

22

2h

ψψ EH

Vxm

VmpVTH

=

+∂∂

−=+=+=

ˆhave weThen

ˆ2

ˆ2ˆˆˆˆ

2

22 h

12

Time Independent Schrodinger Equation

Comment 2

The allowed (quantized) energies are the eigenvalues of the Hamiltonian operator HIn quantum mechanics, there exist well-determined energy states

rseigenvecto theare seigenvalue theare

equation eigenvaluean is ˆ

ψ

ψψE

EH =

13

Time Independent Schrodinger Equation

Comment 3Wave functions that are products of the separable solutions form stationary states

The probability density is time independentAll the expectation values are time independentNothing ever changes in a stationary state

( ) ( )

( ) ( )2**2,

densityy probabilit theNote,

isfunctionwavefullheremember tAlways

xψψeeψtxΨ

exψtxΨ

iEtiEt

iEt

==ΨΨ=

=

−+

−

hh

h

14

Time Independent Schrodinger Equation

Comment 4

In stationary states, every measurement of the total energy is a well-defined energy E

( )

0 So

ˆ Then

ˆˆˆˆˆ Also

ˆ

222H

2*22*2

22

**

=−=

===

====

===

∫∫

∫∫

∞

∞−

∞

∞−

∞

∞−

∞

∞−

HH

EdxEHdxH

EHEEHHHH

EdxEHdxH

σ

ψψψψ

ψψψψ

ψψψψ

15

Time Independent Schrodinger Equation

Comment 5A general solution to the time dependent Schrodinger equation is

( ) h/

1),( tiE

nnn

nexctx −∞

=∑=Ψ ψ

16

Free ParticleFree particle means V=0 everywhereYou’d think that this would be the easiest problem to solve

lenormalizabnot isit solutiona is function wave thisWhile

as writtenbe can solution General

2 where

2

22

2

2

22

ikxikx BeAe

mEkkdxd

Edxd

m

−+=

=−=

=−

ψ

ψψ

ψψ

h

h

17

Free ParticleLet’s look at the momentum eigenvalueequation

( ) ( ) ( )

( )

( ) ( ) ( )

( )

( ) ( ) ( )ppxuxudx

C

ppC

edxCxuxudx

CeCexu

xpuxxu

ixup

pp

xppipp

ikxipxp

pp

p

′−=

=

′−=

=

==

=∂

∂−=

∫

∫∫

∞

∞−′

′−∞

∞−

∞

∞−′

δ

π

δπ

*

2

/2*

/

21 letting

2

conditionlity orthonorma esatisfy th toionseigenfunct seexpect the We is solution The

continuous are seigenvalue theso p on constraint no is There

ˆ

h

h

h

h

h

18

Properties of the δ Function

( ) { }

( )

( ) ( ) ( )

( ) ( )∫

∫

∫

∞

∞−

−

∞

∞−

∞

∞−

=−

=−

=

=∞≠=

00

1

0xfor ,0for 0

xxikdkexx

afaxxdxf

xdx

xx

δ

δ

δ

δ

19

Free ParticleThis is the orthonormality condition for continuous eigenfunctions

( ) ( )

( ) ( ) ( ) s)(continuou

(discrete)

*

*

ppxuxudx

xuxudx

pp

ijji

′−=

=

∫

∫∞

∞−′

∞

∞−

δ

δ

20

Free ParticleBack to our eikx normalization dilemma

Solution 1 - Confine the particle in a very large box

We’ll work this problem next time

21

Free ParticleBack to our eikx normalization dilemma

Solution 2 – Use wave packets

Look familiar? Remember

Thus by using a sufficiently peaked momentum distribution, we can make space distribution so broad so as to be essentially constant

( )

( ) ( ) h

h

/

21

set completea form theSince

ipx

p

epadpx

xu

πψ ∫

∞

∞−

=

( )

( )( ) ikx

ikx

exdxkA

xkA

ekdkAx

−

∞

∞−

∫

∫

Ψ=

Ψ

=Ψ

0,)(

0, of ansformFourier tr theis )(then

)(0,