Midterm 2
PHY 465 - Spring 2014
Thursday, April 9, in class exam
Please sign a pledge on your exam stating that you have abided by the Duke honor code.
Problem 1: [20 pts.]
For the 1-D simple harmonic oscillator,
H=
1
p2
+ mω 2 x2 ,
2m 2
choose as a variational trial wave function
(
ψ(x) =
a2 − x 2
0
|x| ≤ a
.
|x| > a
Calculate E(a), minimize with respect to a and compare to h̄ω/2.
Problem 2: [20 pts.]
Use the WKB approximation to compute the energy levels for a particle moving in the
potential
V (x) = k|x|.
Repeat for the potential
(
V (x) =
kx
∞
x≥0
.
x<0
Problem 3: [20 pts.]
An electron is in the ground state of an isotropic 3-D harmonic oscillator potential with
2
2
~
angular frequency ω0 at t = −∞. A time-dependent electric field, E(t)
= ẑE0 e−t /τ is applied until t = ∞. Using first order time-dependent perturbation theory, find the probability
that the atom ends up in each of the first excited states.
Problem 4: [20 pts.]
Recall the expression that you derived in homework for the propagator for a particle
moving in a constant force field.
i m(x − x0 )2 1
f 2 t3
U (x, t; x , 0) = A(t) exp
+ f t(x + x0 ) −
h̄
2t
2
24m
"
0
#!
,
Find A(t) by demanding that U (x, t; x0 , 0) obeys the convolution property
U (x, t; x0 , 0) =
Z ∞
dyU (x, t; y, t0 ) U (y, t0 , x0 , 0)
−∞
Choose x = x0 = 0 to simplify the calculation.
Problem 5: [20 pts.]
For the simple harmonic oscillator, find the equation obeyed by the annihilation and
creation operators in the Heisenberg picture, âH and â†H :
d
âH = ???
dt
d
ih̄ â†H = ??? .
dt
ih̄
Here the subscript H indicates that these are Heisenberg picture operators. Use your result
to calculate the expectation value hz|x(t)|zi, where |zi is a coherent state. This means
that |zi satisfies
â|zi = z|zi
hz|↠= hz|z ∗ .
Here, z is a complex number, and â is the Schrödinger picture operator: â = âH (t = 0).
Gaussian Integral
r
π b2 /4a
−ay 2 −by
e
dye
=
a
−∞
Z ∞
Variational Principle
hψλ |H|ψλ i
E(λ) =
≥ E0
hψλ |ψλ i
WKB Approximation
I
q
dx 2m(E − V (x)) = 2πh̄ n +
1
2
Simple Harmonic Oscillator
1
p
+ mω 2 x2
H|ni = h̄ω(n + 1/2)|ni
H =
2m 2
s
s
1
h̄
h̄mω
x =
(a + a† )
p=
(a − a† )
2mω
i
2
√
√
a|ni = n|n − 1i
a† |ni = n + 1|n + 1i
2
Heisenberg Picture
ih̄
d
OH = [OH , H]
dt
OH (t) = eiHt/h̄ OS e−iHt/h̄
Time-Dependent Perturbation Theory (to first order)
iZt 0 0 1
0
dt hf |H (t)|i0 ieiωf i t
df i = δf i −
h̄ 0
Gauge Transformations
~ = ∇
~ ×A
~
B
~ → A
~ − ∇Λ
~
A
~
~ = −∇φ
~ − 1 ∂A
E
c ∂t
1 ∂Λ
φ→φ+
c ∂t
Propagator, Path Integral X
0
U (x, t; x , t0 ) = hx|e−iH(t−t )/h̄ |x0 i =
eiS/h̄
0
paths
ψ(x, t) =
Z
dx0 U (x, t; x0 , t0 )ψ(x0 , t0 )