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Physics 137B, Fall 2007
Problem set 3
: more perturbation theory; review time dependence and identical particles
Assigned Friday, 14 September. Due Friday, 21 September.
1. Consider a spin
s
= 1
/
2 that starts off at
t
= 0 in the eigenstate of
S
z
with eigenvalue ¯
h/
2. In our usual
basis, this eigenstate can represented by the spinor (1
,
0).
(a) Suppose that the Hamiltonian is
H
=
BS
z
(1)
where
B
is some constant with units of (energy/angular momentum). How does this state evolve in time?
(b) Now suppose that the Hamiltonian is
H
=
BS
x
(2)
where again
B
is some constant. Is the initial state (1
,
0) an eigenstate of this Hamiltonian? How does this
initial state evolve in time?
Give the value
S
z
as a function of time.
You may wish to use the matrix
representation of
S
from the last problem set.
2. Consider the nonrelativistic infinite square-well potential, with
V
= 0 between
x
= 0 and
x
=
L
, and
V
=
∞
elsewhere. Calculate the first- and second-order energy shifts for all eigenstates from the potential
shift
H
=
Aδ
(
x
-
L/
2)
.
(3)
Some energy shifts may be zero.
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- Fall '07
- MOORE
- Physics, mechanics, Angular Momentum, Fundamental physics concepts, one-dimensional Hamiltonian H=, second-order energy shift, second-order energy shifts
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