University of Central Florida
Department of Physics
Fall 2009
PHY 4604
Problem Set # 3
Due: October 2, 2009
There are still some problems from Chapter 3.
The book gives the answer to the most
interesting problems from Chap.
4.
Study them carefully.
The list below contains other
problems which are to be handed in for grading.
1.
Liboff
, 3.15
2.
Liboff
, 3.23
3.
Liboff
, 3.18 in a slightly modified (more logical) form:
Assuming
that
ˆ
H
(
t
)
ˆ
H
(
t
′
) =
ˆ
H
(
t
′
)
ˆ
H
(
t
) for any two times
t
and
t
′
, show that
ψ
(
r
,t
) = exp
bracketleftbigg
−
i
¯
h
integraldisplay
t
0
dt
′
ˆ
H
(
t
′
)
bracketrightbigg
ψ
(
r
,
0)
.
(1)
4. When the Hamiltonian is time dependent but
ˆ
H
(
t
)
ˆ
H
(
t
′
)
negationslash
=
ˆ
H
(
t
′
)
ˆ
H
(
t
), the dynamics
cannot be solved as in Eq. (1). Another strategy is the following. Assume that
ψ
(
r
,t
) =
summationdisplay
n
a
n
(
t
)
e
−
iE
n
t/
¯
h
φ
n
(
r
)
,
where
{
E
n
}
and
{
φ
n
(
r
)
}
are the eigenvalues and eigenfunctions of the Hamiltonian at
t
= 0, respectively.
(a) Find a set of differential equations which the timedependent coefficients
{
a
n
(
t
)
}
need to satisfy.
Hint
: Recall that if
ˆ
H
(0) is Hermitian, than its set of eigenfunctions
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 Spring '09
 Mucciolo
 Physics, mechanics, Energy, Schrodinger Equation, Eigenvalue, eigenvector and eigenspace, wave function, University of Central Florida Department of Physics

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