Physics 401---Due November 28
1. Suppose we have two physical observables S and T which in quantum mechanics are
associated with Hermitian operators
S
ˆ
and
T
ˆ
. These operators do not necessarily
commute. Further, suppose that the system is in an eignenstate of
S
ˆ
with eigenvalue
n
s
:
ψψ n
sS =
ˆ
. Suppose one were to make two consecutive measurements of
the system, say first measure T and then S. There is some probability that the
measurement would yield
m
t
when measuring T and then
n
s
when measuring S.
We can denote this probability
nmst
P
. Alternatively one could first measure S and
obtian
n
s
and then measure T and obtain
m
t
. We can denote this probalitiy
mn ts
P
.
Show that
2
mnnm tsst PP =
. Recall that after a measurement the system is an
eigenstate of the measured operator; the probability that one is in this state is the
absolute value squared of the inner product state of the system and the eigenstate
in question.
2. Consider the infinites square-well from 0 to a that we studied earlier in the semester.
The state
n
represents the nth energy eigenstate of the system.
a. Show that the operator
x
ˆ
is given by
( ) ( )
a
xk
a
xj
a
a
kj
xdxkjx ππ sinsin
ˆ
0
2
,∫
∑
=
b. Show that operator
2
ˆ
x
is given by
( ) ( )
a
xk
a
xj
a
a
kj
xdxkjx ππ sinsin
ˆ2
0
2
,
2∫
∑
=
c. Show that the matrix elements for
x
ˆ
are given by
( )
≠
−
−+−
=
=+
jia
kj
kj
jka
kxj
kj
for
)(
)1(14
for
ˆ
2222
2
1
π
.
d. Show that the diagonal matrix elements for
2
ˆ
x
are given by
2
22
2
2
1
3
1
ˆa
j
jxj
−= π
.
3. This problem explores the relationship between the diagonal matrix elements of the
square of an operator to the off-diagonal matrix elements
