1
Ph 12b
Homework Assignment No. 2
Due: 5pm, Thursday, 21 January 2010
1. Quantized rotor
.
A wheel spinning in a plane can be described as
a Hamiltonian dynamical system with one degree of freedom:
the
coordinate is the angular orientation
θ
taking values in the interval
[0
,
2
π
), and the conjugate momentum is the angular momentum
L
.
The Hamiltonian
H
is
H
=
L
2
/
2
I,
where
I
is the moment of inertia.
a
)
What are the Hamilton equations of motion for this system?
Is
there a conserved constant of the motion? What is the associated
symmetry?
In quantum mechanics, the Hilbert space for this system is the space
of squareintegrable periodic functions of
θ
, i.e.
functions with the
properties
ψ
(
θ
+ 2
π
) =
ψ
(
θ
)
,
integraldisplay
2
π
0
dθ

ψ
(
θ
)

2
<
∞
.
The angular momentum operator becomes
ˆ
L
=

i
¯
h
d
dθ
.
b
) Find the eigenvalues and normalized eigenfunctions of the operator
ˆ
L
. That is, find all values of
λ
and functions
ψ
λ
(
θ
) such that
ˆ
Lψ
λ
(
θ
) =
λψ
(
θ
)
, ψ
λ
(
θ
+ 2
π
) =
ψ
λ
(
θ
)
,
integraldisplay
2
π
0
dθ

ψ
λ
(
θ
)

2
= 1
.
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 Spring '09
 DUDKO
 mechanics, Angular Momentum, Work, Hilbert space, Hermitian, dθ ψλ

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