Physics 451

Quantum mechanics I

Fall 2012

Sep 10, 2012

Karine Chesnel

Announcements

Quantum mechanics

• Homework 4: T Sep 11 by 7pm

Pb 1.9, 1.14, 2.1, 2.2

• Homework 5: Th Sep 13 by 7pm

Pb 2.4, 2.5, 2.7, 2.8

Homework

No student assigned to the following transmitters:

Quantum mechanics

Please register your i-clicker at the class website!

2214B6817A790201E5C6E2C1E71A9C6

Uncertainty principle

Quiz 4a

Which statement is accuratefor these electronic wave functions?

A. Both the position x and the momentum are fairly well defined

B. The position of the particle is fairly well defined but the momentum is poorly defined

C. The momentum of the particle is fairly well defined but the position is poorly defined

D. Both the position and the momentum are poorly defined.

p

Quantum mechanics Ch 1.6

Uncertainty principle

Position

x

2x p

Heisenberg’s uncertaintyPrinciple 1927

2p m E V

particle

De Broglie formula

1924

2 hp

wave

Momentum

p ix

Quantum mechanics Ch 1.6

Uncertainty principle

Quantum mechanics Ch 1.6

Pb 1.9

How to check the uncertainty principle?

• Calculate and x 2xx

• Calculate and p 2pp

• Estimate the product x p

• Compare to 2

Quantum mechanics Ch 1

Probability current

Pb 1.14

Density of probability2

( , ) ( , ) *x t x t

Probability between two points ( , )b

ab

a

P x t dx

, ( , )abdPJ a t J b t

dt

where *, *

2

iJ a t

m x x

Quantum mechanics Ch 2.1

Time-independentSchrödinger equation

2 2

22i V

t m x

In general ( , )V x t Here function of x only

The potential is independent of time

( )V x

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

Quantum mechanics Ch 2.1

Time-independentSchrödinger equation

2 2

2

1 1

2

d di V

dt m dx

Plugging the general solution: ( , ) ( ) ( )x t x t in the Schrödinger

equation

Function of time only

Function of space only

E

Quantum mechanics Ch 2.1

Time-independentSchrödinger equation

1 di E

dt

d iE

dt

• Time dependent part:

General solution: ( )iEt

t e

Quantum mechanics Ch 2.1

Time-independentSchrödinger equation

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

• Space dependent part:

2 2

2

1

2

dV E

m dx

2 2

2( )

2

dV x E

m dx

Global solution:/( , ) ( ) iEtx t x e Stationary state

Quiz 3b

“If the particle is in one stationary state, its expectation value for position is not changing in time.”

A. True

B. False

Quantum mechanics

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

Quantum mechanics Ch 2.1

Stationary states

Pb 2.1

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

a) En must be realb) n(x) can always be real

c) n(x) is either real or odd, when V(x) is even

Pb 2.2 minnE VClassical analogy: The kinetic energy is always positive!

However, in QM, it is possible that at some locations x( )nE V x