Physics 463  Final Exam
Your grade will be based upon your answers to 4 of the following 6 problems. If you turn in more than 4
solutions, your 4 highest scores will be used to compute your
f
nal exam grade.
1.
Let
{
n
x
,n
y
,n
z
i}
denote the eigenstates of an isotropic 3D oscillator of frequency
ω
centered at the origin,
where
n
x
denotes the number of vibrational quanta associated with Cartesian coordinate
x
.
(i) Determine, at least to
f
rst order, the splitting of the threefold degenerate
n
=1
states

1
,
0
,
0
i
0
,
1
,
0
i
0
,
0
,
1
i
due to a weak perturbation
V
=
α
m
ω
2
xz.
(Recall, in
1
D,
q
=
x
p
m
ω
/
¯
h
.)
2.
A particle in one dimension is in the single bound state
φ
0
(
x
)=
√
qe
−
q

x

of an attractive delta function
potential
V
=
−
αδ
(
x
)
,
in which
q
=
m
α
/
¯
h
2
, and the bound state energy is
ε
B
=
−
¯
h
2
q
2
/
2
m.
Aharmon
ic
perturbation of the form
V
=
V
0
cos(
k
0
x
)cos(
ω
t
)
is applied to the system. Find the transition rate for
ionizing transitions to free particle states
φ
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 Fall '10
 PaulE.
 Physics, mechanics, Angular Momentum, 2m, εijk Wk, irreducible invariant subspaces

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