Transcript Mod6QM2

Modern Physics 6b
Physical 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

p̂  i
x

E i
t
Apply to the Schrödinger eqn:
E(x,t) = T (x,t) + V (x,t)


i

 V
2
t
2m x
2
Find the wavefunction
for a given potential V(x)
2

 ( x, t )   cn n e
n 1
 i En t
Expectation values

f    * f  dx

Most likely outcome of a measurement of position, for a system
(or particle) in state (x,t):

x 

x  ( x, t ) dx
2
where
   *
2

Order matters for operators like momentum – differentiate (x,t):
d x
p m
 i
dt

 * 
  x

 dx

Expectation values
Exercise: Consider the infinite square well of width L.
 n ( x) 
2
 n
sin 
L
 L
(a) What is <x>?
(b) What is <x2>?
(c) What is <p>? (Guess first)
(d) What is <p2>? (Guess first)

x

Expectation values
Exercise: Consider the infinite square well of width L.
 n ( x) 
2
 n
sin 
L
 L

x

(a) What is <x>?
(b) What is <x2>?
A: L/2
B: x
2
L2
L2
  2 2
3 2n 
(c) What is <p>? (Guess first)
C: <p>=0
(d) What is <p2>? (Guess first)
D: <p2>=2mE
Harmonic oscillator
This is one of the classic potentials for which we can analytically
solve Sch.Eqn., and it approximates many physical situations.
Simple Harmonic oscillator (SHO)
W   F dx    Kx dx  _______   V
2
p
dx
E  Tmax 
and p  mv, v  , x  A cos(t )
2m
dt
Solve for Etot (m,  , A)  ________
Then, E  mv  V  mv  ______
1
2
2
1
2
2
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
d 2 1
2 2
E  
 m x 
2
2m dx
2
2
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:
 n  Cn e
 m x 2
2
H n ( x)
En  ( n  12 )  , n  0,1, 2,...
•
•
•
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
QHO: 1. Substitution: Verify solution to SE:
d 2 1
2 2
E  

m

x
2
2m dx
2
2
2. QHO analytically: solve the diffeq directly:
Rewrite SE using x  x
m
,
d 2
2

x
 K ) ,
(
2
dx
K
d 2
2

x

2
dx
* At large x~x,
has solutions
-x 2 / 2
 (x )=h(x )e
* Guess series solution h(x)
* Consider normalization and BC to find that hn=an Hn(x)
where Hn(x) are Hermite polynomials
* The ground state solution 0 is the same as before:
 0 (x )=A0e
-ax 2 / 2
* Higher states can be constructed with ladder operators
2E

3. QHO algebraically: use a± to get n
Ladder operators a± generate higher-energy wavefunctions from the ground state 0.
Griffiths Quantum Section 2.3.1
Result:
1  d

a 

im

x


2m  i dx

 n  An (a ) e
n
 m 2
x
2
, with En  ( n  12 ) 
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:
k2

2m
Applications of Quantum mechanics
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), a-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...
Choose your Minilectures for Ch.7
Scanning Tunneling Microscope
Alpha Decay
Ammonia Atomic Clock