Physics 451

Quantum mechanics

Fall 2012

Karine Chesnel

Announcements

• Today: Review - Monday: Practice testBe prepared to present the solution of your chosen problem during class (~ 5 to 10 min)

Test 1 next weekMo Sep 24 – Th Sep 27

Phys 451

EXAM I

• Time limited: 3 hours• Closed book• Closed notes• Useful formulae provided

Review lectures,Homework

and sample test

Phys 451

EXAM I

1. Wave function, probabilities and expectation values

2. Time-independent Schrödinger equation

3. Infinite square well

4. Harmonic oscillator

5. Free particle

Phys 451

Review I

What to remember?

Phys 451

Review IQuantum mechanics

1. Wave function and expectation values

2( , ) ( , )x t x t Density of probability

Normalization: 2

( , ) 1x t

Review IQuantum mechanics

1. Wave function and expectation values

*x x dx

“Operator” x

*p i dxx

“Operator” p

* ,Q Q x i dxx

Quiz 9a

22

2m x

22 21

2m x

m x

2 2

22m x

What is the correct expression for the operator T= Kinetic energy?

2 / 2x m2

1

2m x

A.

B.

C.

D.

E.

Review IQuantum mechanics

1. Wave function and expectation values

2x p

Heisenberg’s Uncertainty principle

Variance: 22 2

22 2

x

p

x x

p p

Review IQuantum mechanics

2. Time-independent Schrödinger equation

2 2

22i V

t m x

Here ( )V xThe potential is independent of time

General solution: ( , ) ( ) ( )x t x t “Stationary state”

Review IQuantum mechanics

2. Time-independent Schrödinger equation

2 2

2

1 1

2

d di V

dt m dx

Function of time only

Function of space only

E

( )iEt

t e

Solution:/( , ) ( ) iEtx t x e Stationary state

Review I

^

H E

Quantum mechanics

2. Time-independent Schrödinger equation

Q is independent of time

0d x

p m v mdt

for eachStationary state

^ 2

n nn

H c E

1

( , ) ( , )n nn

x t c x t

/( , ) ( ) niE tn nx t x e

where

A general solution is

Review IQuantum mechanics

3. Infinite square well

x0 a

The particle can only exist in this region

22

2

dk

dx

2mEk

with

Review IQuantum mechanics

3. Infinite square well

2 2 2

22n

nE

ma

Quantization of the energy

2sinn

nx

a a

x0 a

Ground state 1 1,E

Excited states

2 2,E

3 3,E

Review IQuantum mechanics

3. Infinite square well

^

n n nH E

*m n nm

* 2( ) ( ,0) sin( ) ( ,0)n n

n xc x x dx x dx

a a

/ /

1 1

( , ) ( ) sin( )n niE t iE tn n n

n n

xx t c x e c n e

a

^ 2

1n n

n

H c E

x0 a

Quiz 9b

The particle is in this sinusoidal state.What is the probability of measuring the energy E0 in this state?

A. 0

B. 1

C. 0.5

D. 0.3

E. 1

9

Review IQuantum mechanics

4. Harmonic oscillator

x

V(x)

1

2a ip m x

m

1

2H a a

or1

2H a a

2 2 21 1( )

2 2V x kx m x

• Operator position 2

x a am

• Operator momentum 2

mp i a a

Review IQuantum mechanics

4. Harmonic oscillator

Ladder operators:

0

1

!

n

n an

nnE

Raising operator: 11n na n nE

a1n

Lowering operator: 1n na n nE

a

1n

1

2nE n

Quantum mechanics

5. Free particle

22

2

dk

dx

with2mE

k

2

2

pE

m

Review I

/( , ) ( ) i kx t i kx tiEtx t x e Ae Be

( )1( , ) ( )

2i kx tx t k e dk

Wavepacket

Quantum mechanics

Free particle

Method:

( ,0)x1. Identify the initial wave function

1( ) ( ,0)

2ikxk x e dx

2. Calculate the Fourier transform

( )1( , ) ( )

2i kx tx t k e dk

3. Estimate the wave function at later times

Review I

Quantum mechanics

5. Free particle

( )1( , ) ( )

2i kx tx t k e dk

Dispersion relation ( )k2

2

k

m

here

Physical interpretation:

• velocity of the each wave at given k: phasevk

• velocity of the wave packet: group

dv

dk

here

2phase

kv

m

group

kv

m

Review I