cphys351 c5:1

Chapter 5: Quantum MechanicsLimitations of the Bohr atom necessitate a more general approach

de Broglie waves –> a “new” wave equation“probability” waves classical mechanics as an approximation

Wave Function probability amplitude

P(x) 2 is generally complex

A iB Re i A iB Re i

A2 i2B2 A2 B2 R 2e i i R2

P(x),2 real, non- negative quantities

cphys351 c5:2

Mathematical properties of the wave function

P(x) 2

2dV

all space finite, non- zero

convention: choose P(x) 2

PdVall space

2dV

all space 1

"normalize" '

N 2dV

all space ,' 1

N

cphys351 c5:3

More mathematical properties of the wave function

must be continuous and single valuedx

,y

,z

must be continuous and single valued

must be normalizable 0 as x , y , z

cphys351 c5:4

The classical wave equation as an example of a wave equation:

2 yx 2

1v 2

2 yt 2

solutions of the form: y F t xv

* verify

y Ae i (tx v) (plane wave solution)Re[y]physically relevant for classical wavesRe[y]Re[A]cos (t x v) Im[A]sin( t x v)

A'cos{ (t x v) }

cphys351 c5:5

22

222

2

22

2

)(

),(2

),(2

particle particle) icrelativist(non for

:esmatter wavfor

2,2||

wavesplanefor form ealternativ

vk

CWEvstxUmk

EtxUm

p

hEkhp

vk

iffCWEtosolutionverifyyityyik

xy

kAey tkxi

cphys351 c5:6

plane wave type solutions x

ik t

i

with 2k 2

2mU (x, t)

2

2m2x 2

U i t

1 d

2

2m2x 2

2y 2

2z 2

U i

t

2

2m2U i

t

3 d

Time dependent Schrödinger Equationlinear (in ) partial differential equation

cphys351 c5:7

Expectation values (average values)

P(x) | (x) |2

| (x) | 2 dx

1 normalized

x xP(x)dx

x | (x) |2 dx

G(x) G(x)P(x)dx

G(x) | (x) | 2 dx

but, statistics and averages for momentum? will look at

G(x) * (x)G(x)(x)dx

cphys351 c5:8

If the potential energy U is time independent, Schrödinger equation can be simplified by “factoring”

separation of variablesTotal energy can have a constant (and well defined) valueConsider plane wave:

Aei(kx t) Aei ( px Et) Aei (px)e i (Et)

(x)e i (Et) so i t

(x)e i (Et) E (x)e i (Et) in the S.E.

it

2

2m2

x 2 U

Ee i (Et) e i (Et) 2

2m2

x 2 U

or

2

2m2

x 2U

E

An eigenvector, eigenvalue problem!

cphys351 c5:9

The time independent Schrödinger equation

2

2m2

x 2

2

y 2

2

z 2

U

E

2

2m2 U

E

Allowed values for (some) physical quantities such as energy are related to the eigenvalues/eigenvectors of differential operators

eigenvalues will depend on the details of the wave equation (especially in U) and on the boundary conditions

cphys351 c5:10

V0

L

U

x

Particle in a box: (infinite) potential well

LxnAx

mLhn

mLnE

LnknnkLkLAL

BBABC

EmkkxBkxA

Edxd

m

UU

EUdxd

m

nn

sin)(;82

),2,1(0sin)(

000cos0sin)0(:2

,cossin

2

0 outside, ,0 inside,2

22222

22

2

22

2

22

cphys351 c5:11

Wavefunction normalization

,2,1sin2

2

221

2cos121sin

1)(

,0

0,sin

22

0

2

0

22

2

nL

xnL

LAchoose

LALA

dxL

xnAdxL

xnA

dxdxxP

otherwise

LxL

xnA

n

LL

n

n

cphys351 c5:12

Example 5.3 Find the probability that a particle trapped in a box L wide can be found between .45L and .55L for the ground state and for the first excited state.

Example 5.4 Find <x> for a particle trapped in a box of length L

cphys351 c5:13

V0

L

U

x

Particle in a box: finite potential well

axIII

axIIIIII

axI

axII

II

eDeC

eDeC

EVmawithEVdxd

m

EVU

EmkkxBkxA

Edxd

m

U

EUdxd

m

)(2)(2

outside,2

,cossin

2

,0 inside,2

022

02

22

0

22

2

22

2

22

E

I II III

cphys351 c5:14

V0

L

U

x

Boundary Conditions

I C I e ax DI e ax

II Asinkx Bcoskx

III C III e ax DIII eax

0 as x C I 0 0 as x D III 0 continuous I (0) II (0) DI B II (L) III (L) AsinkL BcoskLC III e aL

' continuous I '(0) II '(0) aDI kA

II '(L) III '(L) kAcoskL kBsinkL aC III e aL

E

I II III

cphys351 c5:15

DI ka

A B

AsinkL ka

AcoskL C III e aL

AkcoskL k ka

AsinkL aC III e aL '

sinkL ka

coskLka

coskL ka

2

sinkL

ka

2

12ka

cotkL

kLcotkLaL2

kLaL

2

1

cphys351 c5:16

kLcotkLaL2

kLaL

2

1

a2 2mV0

2 k 2 aL (kL)2 ,

V0

2 (2mL2 )

kLcotkL2(kL)2

2 (kL)2 ucotu

2u2

2 u2

= 4

2 4 6 8 10

-6

-4

-2

2

4

6

= 100

10 20 30 40

-20-15-10-5

5101520

= 1600

cphys351 c5:17

TunnelingL

U

x

E

I II III

V0

2

2md 2

dx 2U

E

outside, U 0,

2

2md 2dx 2

E

I Aeikx Be ikx

III Feikx,

2k 2

2mE

inside, U V0 E

2

2md 2dx 2

(V0 E) with a2 2m 2

(V0 E)

II Ceax De ax

cphys351 c5:18

L

U

x

E

I II III

V0

Boundary Conditions

I Aeikx Be ikx I I

II Ceax De ax

III Feikx

continuous I (0) II (0) A B C D

II (L) III (L) CeaL De aL FeikL

' continuous I '(0) II '(0) ik(A B) a(C D)

II '(L) III '(L) a(CeaL De aL ) ikFeikL

transmission/reflection related to group velocities

T | III |2 vIII

| I |2 v I

R | I |2 v I

| I |2 v I

cphys351 c5:19

LnVE

eeak

kaTaL

aLaik

kaaLAFT

DeFeaik

CeFeaik

DeCeaFike

DeCeFe

DikaC

ikaA

DCaBAikDCBA

II

aLaL

aLikL

aLikL

aLaLikL

aLaLikL

nransmissioresonant tsincos,sinhcosh,:

411:large

sinh41cosh

2)1(

2)1(

)(

)1()1(2)()(

0

22

1222

1

2222

22

cphys351 c5:20

Example 5.5: Electrons with 1.0 eV and 2.0 eV are incident on a barrier 10.0 eV high and 0.50 nm wide.

(a) Find their respective transmission probabilities.(b) How are these affected if the barrier is doubled in width?

cphys351 c5:21

Harmonic Oscillator: classical treatment

F kx U 12 kx 2

F ma md 2xd 2 t

kx x Acos( 0 t )

0 k m

as an approximation for any system with equilibriumequilibrium positionx0

U (x)has a minimum at x0

dU (x)

dx xx0

U '(x 0 ) 0

Taylor Series expansion/approximationU (x) U (x 0 )U '(x0 )(x x 0 ) 1

2U "(x 0 )(x x0 )2

U 0 12kxr

2

cphys351 c5:22

222222

2

2

222

2

221

22

22

2

22

22

22

22

22

22

22

) (large for behavior cAssymptoti

221

2

mkaexaexaaedxed

eψtryψxmkdx

ψd

xkxE

ψmExmkdx

ψd

Eψψkxdx

ψdm

EPEKE

xxxx

x

aaaa

a

Quantum Oscillator

cphys351 c5:23

)energypoint zero(0

)()(2

)12(

2,1,0,12:sEigenvalue

0)(

22zationparameteri essdimensionl convenient

021

0

0

21

0210

22

2

0

02

EhnE

hnnnE

nn

ydyd

EExmxmky

n

cphys351 c5:24

Quantum Harmonic Oscillator Solutions

2432210

sPolynomial Hermite

)()!2(

22

2

22141

0

0

02

2

yy

HnH

yHenm

EExmxmky

n

n

nyn

n

cphys351 c5:25

Operators

Momentum in a plane wave ( = e i(kx t) ) is "well defined"x

ik ipx

ˆ p x

ix

For general wave functions, general operators

px * ˆ p x dx i

*ix

dx

G * ˆ G dx

Energy Hamiltonian operator

E KE PE ˆ H ˆ p 2

2mU (x)

Time independent Schrödinger equation as ev problemˆ H E

cphys351 c5:26

Example 5.6: An eigenfunction of the operator d 2 /dx 2 is y = e2x. Find the corresponding eigenvalue.

cphys351 c5:27

Chapter 5 exercises: 4, 5, 6, 11, 23