Problem Set 3
Physics 341
Due October 29
Some abbreviations: S  Shankar.
1
. Let’s start with a straightforward warmup exercise. Consider four operators
A, B, C, D
.
Show that
[
AB, CD
] =
A
[
B, C
]
D
+
AC
[
B, D
] + [
A, C
]
DB
+
C
[
A, D
]
B.
Now consider
e
λA
e
λB
=
e
C
where
A, B, C
are operators and
λ
is a small parameter. Find
an expression for
C
in terms of
A
and
B
including terms of
O
(
λ
4
). Evaluate this expression
for
A
=
x
and
B
=
p
.
2
. For a density matrix
ρ
, please show that
ρ
†
=
ρ,
Tr
ρ
= 1
,
Tr
ρ
2
≤
1
.
For a pure ensemble, show that
ρ
2
=
ρ,
Tr
ρ
2
= 1
.
3
. A particle of mass
m
moves in one dimension in an infinite square well potential extending
from

a
to +
a
. At time
t
= 0, the system is described by the wavefunction
Ψ(
x,
0) =
c
(Ψ
0
(
x
) + 2Ψ
1
(
x
))
where Ψ
0
and Ψ
1
are the normalized eigenfunctions for the ground state and first excited
state, respectively.
(i) Find the value of
c
for which the wavefunction is normalized.
(ii) Compute the probability that the particle is found in the interval

a < x <
0 at time
t
.
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 Spring '10
 Sghy
 Physics, mechanics, ﬁrst excited state, normalized energy eigenfunctions, straightforward warmup exercise, quantum expectation values

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