Physics 828: Problem Set I
Due Wednesday, January 16 at 11:59 PM
Note: all problems are worth 10 points unless otherwise stated
1. (20 pts.) In class, I discussed the translation operator, and explained
how it can be expressed in terms of the momentum operator
p
. In
this problem, you will consider rotation operators, and show that they
can be expressed in terms of components of the
angular
momentum
operator
L
.
Start with a wave function
ψ
expressed in spherical coordinates as
ψ
(
r,θ,φ
). We now consider a rotation about the z axis by an angle
φ
0
.
Let
R
z
(
φ
0
) be the operator which has the following effect on the wave
function
ψ
:
R
z
(
φ
0
)
ψ
(
r,θ,φ
) =
ψ
(
r,θ,φ
−
φ
0
)
.
(1)
(a). By expressing
ψ
(
r,θ,φ
−
φ
0
) in a Taylor series about
ψ
(
r,θ,φ
),
show that
ψ
(
r,θ,φ
−
φ
0
) = exp
parenleftBigg
−
φ
0
∂
∂φ
parenrightBigg
ψ
(
r,θ,φ
)
,
(2)
and hence that
R
z
(
φ
0
) = exp
parenleftBigg
−
φ
0
∂
∂φ
parenrightBigg
.
(3)
Hence, show that
R
z
(
φ
0
) = exp(
−
iφ
0
L
z
/
¯
h
)
(4)
where
L
z
=
−
i
¯
h
(
∂/∂φ
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 Winter '08
 STROUD
 Physics, mechanics, Angular Momentum, Fundamental physics concepts, Bloch, parity operator

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