PHY4221: Homework 5
Due Oct. 9, 2008
1. In the 1D infinite square well problem defined in the region

x
 ≤
L
, an extra potential
V
=
kx
is added. Express the Hamiltonian of the system as a form of matrix in the basis
defined by the eigenvectors of the original square well Hamiltonian.
2. The Hamiltonian of a system is given by
H
=
αp
2
+
βx,
where
α
and
β
are constants.
Find and solve the Heisenberg equation of motion of the
position and momentum operators. The initial state is a Gaussian wave packet, centered
at zero speed and
x
= 0, with a position width given by
σ
. Find expectation value of the
position and its variance as a function of time.
3. (a) Show that
1
A

B
=
1
A
+
1
A
B
1
A
+
1
A
B
1
A
B
1
A
+
1
A
B
1
A
B
1
A
B
1
A
+
.....
where
A
and
B
are two operators whose inverse exist. Then calculate the expectation value
of 1
/
(
A

B
) with respect to the state

0
i
, given that
B

0
i
=

1
i
and
B

1
i
=

0
i
, and
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 Spring '11
 CFLO
 Work, Hilbert space, expectation value, Heisenberg equation, Gaussian wave packet

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