8. A system with Hamiltonian
H
(0)
is subject to a perturbation
H
(1)
,
which in a certain ONB can be repre
sented by the following matrices
h
H
(0)
i
=
⎛
⎜
⎜
⎜
⎜
⎝
ε
0
00 0 0
0
ε
0
00
0
ε
0
0002
ε
0
0
000 02
ε
0
⎞
⎟
⎟
⎟
⎟
⎠
h
H
(1)
i
=
⎛
⎜
⎜
⎜
⎜
⎝
0
∆
0
∆
0
∆
0
∆
0
000 0
i
∆
000
−
i
∆
0
⎞
⎟
⎟
⎟
⎟
⎠
Find the new energies of this system, correct to
f
rst order in the perturbation.
9. An otherwise free particle of charge
e
is constrained to move on the circumference of a circle of radius
R
.I
f
θ
denotes the angular position of the particle, its (rotational) kinetic energy is associated with the operator
H
0
=
L
2
z
2
mR
2
where
L
z
is the component of angular momentum perpendicular to the plane in which the particle moves.
In the (angular) position representation, in which the particle’s state is represented by the wave function
ψ
(
θ
)=
h
θ

ψ
i
the operator
L
z
has the action
h
θ

L
z

ψ
i
=
−
i
~
∂ψ
(
θ
)
∂θ
.
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 Fall '10
 PaulE.
 mechanics, Angular Momentum, Work, Fundamental physics concepts, wave function, angular momentum operator

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