Physics 137B, Fall 2007
Problem set 2
: perturbation theory and some identical particles review
Assigned Friday, 7 September. Due (in 251 LeConte box) 5 pm Friday, 10 September.
1.
(from Ohanian) Suppose that the electron in a hydrogen atom is perturbed by a repulsive
potential concentrated at the origin. Assume that the potential has the form of a delta function,
so that the perturbed Hamiltonian is
H
=
p
2
2
m

1
4
π
0
e
2
r
+
Aδ
(
r
)
.
(1)
(a) To first order in the constant
A
, find the change in the energy of the state with quantum
numbers
n
≥
1,
l
= 0. Hint:
ψ
n
00
(0) =
2
√
4
π
(
na
0
)

3
/
2
.
(b) Find the change in the wavefunction to first order in
A
, using the nondegenerate perturbation
theory formula. You may leave the answer in the form of an infinite series, but make sure that all
the terms in your series are necessary.
2. (from Ohanian) A particle of mass
m
is confined to a onedimensional infinite square potential
well that extends from
x
= 0 to
x
=
L
(i.e.,
V
= 0 in the well,
V
=
∞
outside). Impose appropriate
boundary conditions to show that the energy eigenstates for the (nonrelativistic) Hamiltonian are
E
n
=
n
2
π
2
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 Fall '07
 MOORE
 Physics, mechanics, inner product, ground state, Perturbation theory

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