Physics 3316, Spring 2010
Basics of Quantum Mechanics
Problem Set 4
(Due in lecture, Feb 22, 2010)
Required Readings :
•
French and Taylor Ch. 3, Griffiths Ch. 1
1
Propagation of wavepackets in an external potential
In lecture, we considered the correspondence between particles moving in free space according to an energy
function
E
(
p
) =
p
2
2
m
and wavepackets propagating under a dispersion relation given by
ω
(
k
) =
E/
planckover2pi1
=
planckover2pi1
k
2
2
m
.
The link between the particle’s momentum
p
and the wavelength
λ
of the wavepacket is provided by de
Broglie’s relation
p
=
h/λ
=
planckover2pi1
k
. In this problem you will consider the same correspondence but for a particle
in a system with potential energy.
Consider a particle whose energy is given by
H
=
p
2
2
m
+
U
(
x
)
.
Written this way, as a function of the momentum
p
, the energy function of the particle has a special name. It
is known as the “Hamiltonian,” and so we call the energy here “H” instead of the more familiar “E”.
Imagine that an external force
F
ext
(
t
) is applied to the particle in this system in such a way that the
momentum of the particle as a function of time is given by
p
(
t
).
a)
Determine
dp
dt
(
t
) in terms of the force acting on the particle
F
ext
(
t
), the position of the particle
x
(
t
) and
a derivative of the potential function
V
(
x
). (Note that this amounts to applying the concept of conservation
of momentum.)
b)
Explain
(briefly) why
d
dt
H
=
F
ext
˙
x
(
t
). Thus, show
that
˙
x
=
∂H
∂p
.
(1)
Is this relation true even when the external force is zero?
c)
Show that when the external force is zero we also have
˙
p
=
−
∂H
∂x
.
(2)
Eq. (1) and Eq. (2) taken together are known as the canonical equations of Hamilton.
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 Spring '08
 FLANAGAN
 mechanics, Momentum space, PAB, momentum space representation, de Broglie Hypothesis

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