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
y
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
.
(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.
(c) What is the mean value
h
S
u
i
for this state?
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
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 Fall '10
 PaulE.
 mechanics, Angular Momentum, Work, Hilbert space, Jx, rotation operator Uu

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