Modern Physics 6bPhysical Systems, week 7, Thursday 22 Feb. 2007, EJZ

Ch.6.4-5:

• Expectation values and operators

• Quantum harmonic oscillator → blackbody

• applications

week 8, Ch.7.1-3: Schrödinger Eqn in 3D, Hydrogen atom

week 9, Ch.7.4-8: Spin and angular momentum, applications

Review energy and momentum operators

Apply to the Schrödinger eqn:

E(x,t) = T (x,t) + V (x,t)

p̂ ix

E i

t

2 2

22i V

t m x

1

( , )ni E t

n nn

x t c e

Find the wavefunction

for a given potential V(x)

Expectation values

2 2 *( , )x x x t dx where

Most likely outcome of a measurement of position, for a system (or particle) in state x,t:

Order matters for operators like momentum – differentiate (x,t):

*d xp m i dx

dt x

*f f dx

Expectation values

Exercise: Consider the infinite square well of width L.

(a) What is <x>?

(b) What is <x2>?

(c) What is <p>? (Guess first)

(d) What is <p2>? (Guess first)

2( ) sinn

nx x

L L

This is one of the classic potentials for which we can analytically solve Sch.Eqn., and it approximates many physical situations.

Harmonic oscillator

Simple Harmonic oscillator (SHO)

2

max

2 21 12 2

_______

, , cos( )2

( , , ) ________

, ______

tot

W F dx Kx dx V

p dxE T and p mv v x A t

m dtSolve for E m A

Then E mv V mv

What values of total Energy are possible?

What is the zero-point energy for the simple harmonic oscillator?

Compare this to the finite square well.

Solving the Quantum Harmonic oscillator

2 22 2

2

1

2 2

dE m x

m dx

0. QHO Preview

• Substitution approach: Verify that y0=Ae-ax^2 is a solution

2. Analytic approach: rewrite SE diffeq and solve

3. Algebraic method: ladder operators a±

QHO preview:

• What values of total energy are possible?• What is the zero-point energy for the Quantum Harmonic

Oscillator?• Compare this to the finite square well and SHO

2

2

12

( )

, 0,1,2,...

m x

n n n

n

C e H x

E n n

2 22 2

2

1

2 2

dE m x

m dx

QHO: 1. Substitution: Verify solution to SE:

2. QHO analytically: solve the diffeq directly:

Rewrite SE using

* At large ~x, has solutions

* Guess series solution h()

* Consider normalization and BC to find that hn=an Hn() where Hn() are Hermite polynomials

* The ground state solution 0 is the same as before:

* Higher states can be constructed with ladder operators

2

22

2, ,

m d Ex K K

d

2-a / 20 0( )=A e

22

2

d

d

2- / 2( )=h( )e

3. QHO algebraically: use a± to get n

Ladder operators a± generate higher-energy wave-functions from the ground state 0.

Griffiths Quantum Section 2.3.1

Result:

2

122

1

2

( ) ,m

xnn n n

da im x

i dxm

A a e with E n

Griffiths Prob.2.13 QHO Worksheet

Free particle: V=0

• Looks easy, but we need Fourier series

• If it has a definite energy, it isn’t normalizable!

• No stationary states for free particles

• Wave function’s vg = 2 vp, consistent with classical particle:

2

2

k

m

Applications of Quantum mechanics

Choose your Minilectures for Ch.7

Blackbody radiation: resolve ultraviolet catastrophe, measure star temperatures http://192.211.16.13/curricular/physys/0607/lectures/BB/BBKK.pdf

Photoelectric effect: particle detectors and signal amplifiers

Bohr atom: predict and understand H-like spectra and energies

Structure and behavior of solids, including semiconductors

STM (p.279), -decay (280), NH3 atomic clock (p.282)

Zeeman effect: measure magnetic fields of stars from light

Electron spin: Pauli exclusion principle

Lasers, NMR, nuclear and particle physics, and much more...

Scanning Tunneling Microscope

Alpha Decay

Ammonia Atomic Clock