University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Mechanics C (130C) Winter 2014
Final exam — frontmatter
Please remember to put your name on your exam booklet. This is a closed-book exam.
6
There are
problems, each with several parts, of varying levels of difficulty; make sure you
try all of the parts. None of the problems require very extensive calculation; if you find
yourself involved in a morass of calculation, step back and think. Good luck.
Possibly useful information:
U(t) = e−iHt/~ satisfies i~∂t U = [H, U].
0 1
0 −i
1 0
x
y
z
σ =
, σ =
, σ =
0 −1
1 0
i 0
θ
θ
~ · n̂| ↑n̂ i = | ↑n̂ i
σ
| ↑n̂ i = e−iϕ/2 cos | ↑ẑ i + e+iϕ/2 sin | ↓ẑ i satisfies
2
2
~ sin α.
e−iαn̂·~σ = 1 cos α − in̂ · σ
−1 cos θ sin θ
cos θ − sin θ
=
− sin θ cos θ
sin θ cos θ
1
1
p2
†
HSHO = ~ω a a +
+ mω 2 q2
=
2
2m 2
r
r
~
1 ~mω
q=
a + a† , p =
a − a† ; [q, p] = i~ =⇒ [a, a† ] = 1.
2mω
i
2
~ ·A
~ = 0, Φ = 0):
~ = −∂t A,
~ B
~ =∇
~ × A.
~
In Coulomb gauge, in vacuum (∇
E
r
X
X
~
†
−i~k·~
r
i~k·~
r
?
~ r) =
A(~
a
~
e
(
k̂)e
+
a
~
e
(
k̂)e
,
~
s
~k,s s
20 ωk L3 s=1,2 k,s
~k
r
X
~ωk X †
i~k·~
r
?
−i~k·~
r
~ r) = i
E(~
a
~
e
(
k̂)e
−
a
~
e
(
k̂)e
~
s
~k,s s
20 L3 s=1,2 k,s
~k
Chain of coupled springs:
r
~ X 1 ikx
−ikx †
q(x) =
e ak + e
ak ,
√
2m k
ωk
1
p(x) =
i
1
r
~m X √ ikx
−ikx †
ωk e ak − e
ak .
2 k
1. Short answers and conceptual questions [4 points each, except as noted]
For true or false questions: if the statement is false, you must explain what is wrong or
correct it or give a counterexample; if the statement is true, you can simply say ‘true’.
2