University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Mechanics C (130C) Winter 2015
Final exam cover sheet
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].
✓ ◆
✓
◆
✓
◆
01
0 i
1 0
x
y
z
=
,
=
,
=
10
i 0
0 1
| "n̂ i = e
i'/2
✓
✓
cos | "ẑ i + e+i'/2 sin | #ẑ i
2
2
~ · n̂| "n̂ i = | "n̂ i
i↵n̂·~
= 1 cos ↵ in̂ · ~ sin ↵.
◆ 1 ✓
◆
cos ✓ sin ✓
cos ✓ sin ✓
=
sin ✓ cos ✓
sin ✓ cos ✓
e
✓
satisfies
✓
◆
1
p2
1
HSHO = ~! a a +
=
+ m! 2 q2
2
2m 2
r
r
~
1 ~m!
q=
a + a† , p =
a a† ; [q, p] = i~ =) [a, a† ] = 1.
2m!
i
2
†
1