Problem Set ---Due February 17
1) Consider a system whose angular state in the standard
lm
basis is given by
( )
2122
2
1
γ
ψ
i
e
+=
where γ is a phase. Show that
( )
γψψ
cos
ˆ
=
x
L
. The fact
that the expectation value depends on the phase γ is a demonstration of the general
principle that while the overall phase of a states is unphysical, the relative phase
between different components has physical significance.
2) Consider a system with a central potential. As noted in class the energy eigenstates
are of the form
mln
where n is the radial quantum number and l and m describe the
angular degrees of freedom. Suppose the system is in the state
( )
122122112012002001
10
3
10
1
2
1
2
1
2
1
−−++−−=
i
ψ
.Verify that the
state is normalized and calculate the following expectation values:
ψψ
z
L
ˆ
ψψ
2
ˆ
L
ψψ
2
ˆ
z
L
ψψ
y
L
ˆ
ψψ
x
L
ˆ
3) In class we showed how to construct explicit matrices for the operators
x
L
ˆ
,
y
L
ˆ
,
z
L
ˆ
and
2
ˆ
L
and that the matrices were block diagonal.
a) Working in the block with l=1 explicitly construct the 3 by 3 matrices associated
with these four operators.
b) Denote the matrices associated with
x
L
ˆ
,
y
L
ˆ
,
z
L
ˆ
as
x
L
,
y
L
,
z
L
show that they
satisfy the same commutation relations as the operators:
[ ]
zyx
LiLL
=
,
[ ]
xzy
LiLL
=
,
[ ]
yxz
LiLL
=
,
c) Show that as matrices
2
222
LLLL
xxx
=++
where
2
L
is the matrix associated with
2
ˆ
L
.
4) We arbitrarily choose to work with a basis of states which were eigenstates of
z
L
ˆ
and
2
ˆ
L
we
equally well could have picked
x
L
ˆ
and
2
ˆ
L
. Let us denote the new basis as the primed
basis. The primed basis states are normalized and have the following properties
''
ˆ
mlmmlL
x
=
(i)
