Physics 451

Quantum mechanics I

Fall 2012

Sep 12, 2012

Karine Chesnel

Homework remaining this week:

• Extended Friday Sep 14 by 7pm: HW # 5 Pb 2.4, 2.5, 2.7, 2.8

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Quantum mechanics

Note:Penalty on late homework: - 2pts per day

Credit for group presentations:Homework 2: 20 points

Quiz 5: 5 points

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Quantum mechanics

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2214B6817A790201E5C6E2C1E71A9C6

Quantum mechanics Ch 2.1

Time-independentSchrödinger equation

Solution (x) depends on the potential function V(x).

• Space dependent part:

/( , ) ( ) iEtx t x e Stationary stateAssociated to energy E

2 2

2( )

2

dV x E

m dx

Quantum mechanics Ch 2.1

Stationary states

Properties:

• Expectation values are not changing in time (“stationary”):

/( , ) ( ) iEtx t x e * ( , )Q Q x dxi x

with

* ( , )Q Q x dxi x

Q is independent of time

0d x

p m v mdt

The expectation value for the momentum is always zero

In a stationary state!

p(Side note: does not mean that and x are zero!)

Quantum mechanics Ch 2.1

Stationary states

Properties:

2 2

2( )

2

dV x E

m dx

• Hamiltonian operator - energy

^ ^* *H H dx E dx E

^ ^2 * 2 2 * 2H H dx E dx E

^

H

0H

Quantum mechanics Ch 2.1

Stationary states

• General solution

1

( , ) ( , )n nn

x t c x t

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

where

• Associated expectation value for energy2

1n n

n

H c E

Quiz 6a

n nn

c EA.

B.

C. one of the values

D.

Quantum mechanics

A particle, is in a combination of stationary states:

What will we get if we measure its energy?1

( , ) ( , )n nn

x t c x t

H

nE

nn

E

Quiz 6b

2

nc

2

1

nc

A. 0

B.

C.

D.

E.

Quantum mechanics

A particle, is in a combination of stationary states:

What is the probability of measuring the energy En?1

( , ) ( , )n nn

x t c x t

nc

n

n

c

c

Quantum mechanics Ch 2.2

Time-independent potential

Expectation value for the energy:

^

n n nH E

*^

1 1m m n n

m n

H c H c

^ ^* *

1 1m n m n

m n

H c c H dx

^ ( )*

1 1

n mt

i E E

m n n nmm n

H c c E e

^ 2

1n n

n

H c E

Quantum mechanics Ch 2.2

Infinite square well

x0 a

The particle can only exist in this region

V(x)=0 for 0<x<a

V=∞ else

Shape of the wave function?

Quantum mechanics Ch 2.2

Infinite square well

Solutions to Schrödinger equation:

Simple harmonic oscillator differential equation

2 2

22

dE

m dx

22

2

dk

dx

2mEk

with

Quantum mechanics Ch 2.2

Infinite square well

Solutions to Schrödinger equation:

Boundary conditions:

( ) sin cosx A kx B kx

At x=0: (0) 0

At x=a: ( ) 0a

( ) sinx A kx with n

nk

a

Quantum mechanics Ch 2.2

Infinite square well

Possible states and energy values:

2 2 2

22n

nE

ma

Quantization of the energy

Each state n is associated to an energy En

2sinn

nx

a a

^

n n nH E

Quantum mechanics Ch 2.2

Infinite square well

Properties of the wave functions n:

x0 a

1.They are alternativelyeven and odd

around the center

2. Each successive statehas one more node

Ground state 1 1,E

Excited states

2 2,E

3 3,E

3. They are orthonormal

*m n nm

4. Each state evolves in time with the factor /niE te

Quantum mechanics Ch 2.2

Infinite square well

Pb 2.4 Particle in one stationary state

Pb 2.5 Particle in a combination of two stationary states

1 2,0 ( )x A

x p evolution in time? 2, ( , )x t x t

oscillates in time H expressed in terms of E1 and E2

x 2x p 2p2x p

Quantum mechanics Ch 2.2

Infinite square well

Expectation value for the energy: ^ 2

1n n

n

H c E

The probability that a measurement yields to the value En is 2

nc

Normalization 2

1

1nn

c