Physics 463  Final Exam
Work all
f
ve problems on this exam.
You have until 4:30PM of the day after you receive this exam to complete it and turn it in.
1.
Use the variational principle to
f
nd an appropriate bound for the
f
rst excited state energy (
n
=2
)o
fthe
hydrogen atom, using a variational wave function of the form
ψ
(
r
)=
Ae
−
α
r
Y
0
1
(
θ
,
φ
)
,
treating the positive
constant
α
as a variational parameter. Explain why this bound applies most directly to the energy of the
n
=2
level, and is not appropriate for estimating the
n
=1
ground state energy.
2.
A particle moving in one dimension is con
f
ned to the region
x
∈
[0
,L
]
by an in
f
nite onedimensional square
well potential. It is initially in its ground state (at
t
=
−∞
) when a pulsed perturbation of the form
V
=
V
0
cos(
π
x/L
)
e
−
t
2
/
τ
2
is applied to the system. Find, using
f
rst order time dependent perturbation
theory, the probabilty that the pulse causes the system to make a transition to the
f
rst excited state. How
small must
V
0
be in order for the
f
rst order result to be valid?
3.
The total angular momentum of a particle with spin
s
=
1
2
is the sum
~
J
=
~
L
+
~
S
of its orbital and spin
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 Fall '10
 PaulE.
 mechanics, Energy, Work, ground state, total spin, ﬁrst excited state, irreducible invariant subspaces

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