Physics 137B, Fall 2007
Quantum Mechanics II
Midterm I
10/9/2007, 9:4011:00 a.m.
No books, notes, or calculators are allowed. Please start each of the 4 problems on a fresh side.
You should make an eﬀort to answer every problem.
I. Consider a onedimensional harmonic oscillator:
H
=
p
x
2
2
m
+
kx
2
2
.
(1)
Some useful facts: the energy levels of the
onedimensional
harmonic oscillator are
E
= (
n
+1
/
2)¯
hω
,
n
= 0
,
1
, . . .
, and the lowest two wavefunctions are of the form
ψ
0
=
α
1
exp(

βx
2
)
, ψ
1
=
α
2
x
exp(

βx
2
)
(2)
where
α
1
, α
2
, β
are positive constants. All the following questions are for a single spinless particle.
(a) Using the exact values for the energy levels and
ω
=
p
k/m
, ﬁnd the change in the ground state
energy induced by a small increase in the mass
m
→
m
+
dm
, to ﬁrst order in
dm
.
The ground state energy is ¯
hω/
2 = ¯
h
p
k/
(
m
+
dm
)
/
2
≈
¯
h
p
(
k/m
)(1

dm/m
)
/
2
≈
(¯
h/
2)
p
k/m
(1

dm/
2
m
), where
≈
means that we have kept only linear order in
dm
. So the change is ¯
h
p
k/m
(

dm/
4
m
).
(b) Explain how you would use ﬁrstorder perturbation theory for the change in (a) by writing the
change induced by
dm
as a perturbation. Write, but do not calculate, the integral that will give
the ﬁrstorder perturbation theory result for the energy change.
The perturbation Hamiltonian is

(
p
2
/
2
m
)(
dm/m
). The ﬁrstorder energy correction is then
h
ψ
0
 
(
p
2
/
2
m
)(
dm/m
)

ψ
0
i
(3)
. You can check that this is indeed equal to the above result, since the expected kinetic energy in
the harmonic oscillator is half the total energy.
(c) Is the second order correction to the ground state energy positive, negative, or zero?
The secondorder correction from the ground state is always either negative or zero. Since all
even states are connected by the perturbation Hamiltonian in this case, the secondorder shift is
negative.
II. (a) Consider the 2p levels of the hydrogen atom and ignore ﬁne structure corrections. Including
spin, how many levels are there (for a single electron)?
There are six levels with
`
= 1 and
s
= 1
/
2, which we can label by two quantum numbers
m
`
=

1
,
0
,
1 and
m
s
=

1
/
2
,
1
/
2. Without ﬁne structure, these are all degenerate.
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 Fall '07
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 mechanics, ground state

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