Physics 451

Quantum mechanics I

Fall 2012

Review 2

Karine Chesnel

EXAM II

Quantum mechanics

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

When: Tu Oct 23 – Fri Oct 26Where: testing center

EXAM II

Quantum mechanics

1. The delta function potential

2. The finite square potential

(Transmission, Reflection)

3. Hermitian operator, bras and kets

4. Eigenvalues and eigenvectors

5. Uncertainty principle

Quantum mechanics

Square wells and delta potentials

V(x)

x

Bound statesE < 0

ScatteringStates E > 0

Symmetry considerations

even evenx x

odd oddx x

Physical considerations

ikxreflected x Be

ikxincident x Ae

ikxtransmitted x Fe

Quantum mechanics

Square wells and delta potentials

Continuity at boundaries

Delta functions

Square well, steps, cliffs…

dx

d

is continuous

is continuous except where V is infinite

022

m

dx

d

dx

d

is continuous

is continuous

Quantum mechanics

The delta function potential

V x x

2 2

2( )

2

dx E

m dx

2

2 2

2d mE

dx

For 0x

Quantum mechanics

The delta function well

Bound state

0E

2/

xme

mx

2

2

2m

E

Quantum mechanics

Scattering state 0E

Ch 2.5

0

ikx ikxleft x Ae Be ikx

right x Fe

A F

Bx

Reflection coefficient Transmission coefficient

2 2

1

1 2 /R

E m

2 2

1

1 / 2T

m E

The delta function well/ barrier

V x x

“Tunneling”

Quantum mechanics

The finite square well

V(x)

x

-V0

Symmetry considerations

The potential is even function about x=0

cos

kx

kx

Ae

D lx

Ae

even

The solutions are either even or odd!

Bound state

0E

Quantum mechanics

The finite square well

Bound states

2

0tan 1z

zz

z la 0 02a

z mV

where

Quantum mechanics

The finite square well

Scattering state 0E

V(x)

x

-V0

A

B

FC,D

-a +a

(1)

(2)

ikx ikx

ilx ilx

ikx

Ae Be

Ce De

Fe

(1)

(3)

(2)

(3)

Quantum mechanics

The finite square well

1

220

00

21 sin 24

V aT m E V

E E V

V(x)

x

-V0

A

B

F

Coefficient of transmission

2 22

0 22 (2 )nE V nm a

The well becomes transparent (T=1)

when

Formalism

Quantum mechanics

ˆijH H Linear transformation

(matrix)Operators

Wave function Vector

Observables are Hermitian operators †Q Q

Quantum mechanics

Eigenvectors & eigenvalues

For a given transformation T, there are “special” vectors for which:

T a a

a is an eigenvector of T

is an eigenvalue of T

Quantum mechanics

Eigenvectors & eigenvalues

0T I a

det 0T I

To find the eigenvalues:

We get a Nth polynomial in : characteristic equation

Find the N roots 1 2, ,... N Spectrum

Find the eigenvectors 1 2, ,... Ne e e

Quantum mechanics

Hilbert space

N-dimensional space

1 2 3, , ,... Ne e e e

Infinite- dimensional space

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

Hilbert space: functions f(x) such as 2

( )b

a

f x dx

*( ) ( )f g f x g x dx

Inner product

Schwarz inequality f g f g

2 2*( ) ( ) ( ) ( )f x g x dx f x dx g x dx

Orthonormalitym n nmf f

Quantum mechanics

The uncertainty principle

2

2 2,

2A B

A B

i

Finding a relationship between standard deviations for a pair of observables

Uncertainty applies only for incompatible observables

Position - momentum 2x p

Quantum mechanics

The uncertainty principle

Energy - time

2E t

Special meaning of t

Qtd Q

dt

,d Q i Q

H Qdt t

Derived from the Heisenberg’s equation

of motion

Quantum mechanics

The Dirac notation

Different notations to express the wave function:

• Projection on the energy eigenstates

• Projection on the position eigenstates

• Projection on the momentum eigenstates

/niE tn n

n

c e

( , ) ( )y t x y dy

/1( , )

2ipxp t e dp

Quantum mechanics

The Dirac notation

Bras, kets

= inner product

= matrix (operator)

Operators, projectors

n ne e projector on vector en