16. In the spin space of a particle of spin 1
/
2
,
suppose the particle is in a spin state which is spinup along the
z
direction:

ψ
s
i
=

1
2
,
1
2
i
z
,
so that
S
z

ψ
s
i
=
1
2

ψ
s
i
.
Consider the component
S
u
=
~
S
·
ˆ
u,
of the spin operator
~
S
along a direction ˆ
u
=s
in
θ
ˆ
x
+cos
θ
ˆ
z
in the
xz
plane, making an angle
θ
with respect to the
z
axis.
(a) In the standard representation

1
2
,
±
1
2
i
z
of eigenstates of
S
2
and
S
z
,
construct the 2
×
2matr
ix[
S
u
]
representing the operator
S
u
.
We have
[
S
u
]=
u
x
[
S
x
]+
u
y
[
S
y
u
z
[
S
z
]
i
n
θ
[
S
x
]+cos
θ
[
S
z
]
[
S
u
μ
1
2
cos
θ
1
2
sin
θ
1
2
sin
θ
−
1
2
cos
θ
¶
.
(b) If
S
u
is measured on the state

ψ
s
i
,
what values can be obtained, and with what probability will those
values be obtained.
The only values that can be obtained are the eigenvalues
m
u
of
S
u
.
These are found by solving the
characteristic equation det (
S
u
−
m
u
)=0
,
which gives
m
u
=
±
1
2
,
the same as it is for any of the spin
components. To
f
nd the probability we must solve for the eigenvectors. Putting
m
u
=
±
1
/
2intothe
eigenvalue equation gives, after normalization, the eigenvectors

1
2
,
1
2
i
u
=
r
1+cos
θ
2
∙

1
2
,
1
2
i
z
−
sin
θ
θ

1
2
,
−
1
2
i
z
¸

1
2
,
−
1
2
i
u
=
r
1
−
cos
θ
2
∙

1
2
,
1
2
i
z
+
sin
θ
1
−
cos
θ

1
2
,
−
1
2
i
z
¸
The probability that a measurement of
S
u
on

ψ
i
=

1
2
,
1
2
i
z
will yield
m
u
=
1
2
is
P
μ
m
u
=
1
2
¶
=
¯
¯
¯
¯
u
h
1
2
,
1
2

1
2
,
1
2
i
z
¯
¯
¯
¯
2
=
θ
2
and
P
μ
m
u
=
−
1
2
¶
=
¯
¯
¯
¯
u
h
1
2
,
−
1
2

1
2
,
1
2
i
z
¯
¯
¯
¯
2
=
1
−
cos
θ
2
.
(c) What is the mean value
h
S
u
i
for this state?
We evaluate
h
ψ

S
u

ψ
i
=
¡
10
¢
μ
1
2
cos
θ
1
2
sin
θ
1
2
sin
θ
−
1
2
cos
θ
¶μ
1
0
¶
=
1
2
cos
θ
.
17. A quantum system is in an eigenstate

j,m
i
of
J
2
and
J
z
.
(a) Showthatisitalsoinaneigenstateof
J
2
z
and of
J
2
x
+
J
2
y
,
(but not, generally of
J
x
or
J
y
) and determine
the associated eigenvalues.