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4. Use the variational method with wave functions of the form
φ
α
(
r
)=
r
a
π
2
1
r
2
+
a
2
and
φ
α
(
r
e
−
r/a
√
π
a
3
to estimate the ground state energy of a particle subject to a Coulomb potential
V
(
r
−
e
2
/r.
Note that
the second wave function above reduces, for the correct value of
a,
to the exact ground state wave function
for the Coulomb problem.
5. Evaluate to
f
rst order the energy shifts in the spectrum of the harmonic oscillator Hamiltonian
H
0
=
1
2
~
ω
(ˆ
q
2
+ˆ
p
2
) due to a perturbation
ˆ
V
=
λ
ˆ
q
4
.(H
e
r
eˆ
q
and ˆ
p
are dimensionless position and momentum
variables, such that [ˆ
q,
ˆ
p
]=
i
. See Chapter 3 of the class notes for details, and for useful techniques for
calculating expectation values of powers of ˆ
q
=(
a
+
a
+
)
/
√
2
.
)
6. A system with Hamiltonian
H
(0)
is subject to a perturbation
H
(1)
,
which in a certain ONB can be repre
sented by the following matrices
h
H
(0)
i
=
⎛
⎜
⎜
⎜
⎜
⎝
ε
0
000
0
04
ε
0
00
0
6
ε
0
0
8
ε
0
0
0
1
0
ε
0
⎞
⎟
⎟
⎟
⎟
⎠
h
H
(1)
i
=
⎛
⎜
⎜
⎜
⎜
⎝
γ
∆
0
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 Fall '10
 PaulE.
 mechanics, Energy, Work

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