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PHY4221
Quantum Mechanics I
Fall 2004
Examples in One Dimension:
1. Free Particle
The Hamiltonian operator
ˆ
H
of a free particle consists of the kinetic energy term
only:
ˆ
H
=
ˆ
p
2
2
m
.
(1)
Then what are the eigenvalues and eigenvectors of
ˆ
H
? Since any eigenvector of ˆ
p
is also
an eigenvector of
ˆ
H
, we can easily see that the eigenvalues of
ˆ
H
are simply given by
E
p
=
p
2
2
m
(2)
where
p
∈
(
∞
,
∞
) and the corresponding eigenvectors are
{
p
i}
. It should be noted
that there exists a twofold degeneracy for each eigenvalue
E
p
; in other words, there are
two diﬀerent eigenvectors, namely
{ ±
p
i}
, associated with the same eigenvalue
E
p
.
Next, we consider the question: “How does the state vector of the free particle evolve
in time?” To answer this question, we need to solve the timedependent Schr¨
odinger
equation:
ˆ
H

ψ
(
t
)
i
=
i
¯
h
∂
∂t

ψ
(
t
)
i
(3)
whose solution is formally given by

ψ
(
t
)
i
= exp
‰

i
t
¯
h
ˆ
H
±

ψ
(0)
i
=
Z
∞
∞
dp

p
ih
p

exp
‰

i
t
¯
h
ˆ
H
±

ψ
(0)
i
=
Z
∞
∞
dp

p
i
exp
(

i
p
2
t
2
m
¯
h
)
h
p

ψ
(0)
i
=
⇒ h
p

ψ
(
t
)
i
= exp
(

i
p
2
t
2
m
¯
h
)
h
p

ψ
(0)
i
.
(4)
1
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View Full DocumentIf the initial state vector

ψ
(0)
i
, or equivalently the initial wave function
h
p

ψ
(0)
i
in the
momentum space, of the free particle is given, then we can determine the subsequent time
evolution of the system, namely determining the state vector

ψ
(
t
)
i
. It is interesting to
note that the wave function
h
p

ψ
(
t
)
i
at any later time
t >
0 diﬀers from the initial wave
function simply by a phase factor. Furthermore, the evolution operator exp
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 Spring '11
 CFLO
 Energy, Kinetic Energy

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