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
Choose for next quarter: EM, QM, Gravity? 2/3. Vote on Tuesday.
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
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)
A: L/2
2 22
2 2:
3 2
L LB x
n
2( ) sinn
nx x
L L
C: <p>=0
D: <p2>=2mE
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
