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Introduction to Quantum and Statistical Mechanics
Homework 1
Prob. 1.1.
To establish the intuitive link between the Hamiltonian mechanics and the wave
equation, we will use the example of the longitudinal vibrations in classical mechanics written as
the wave equation.
Consider
n
equal particles on a line from 0 to
L
, coupled by identical strings
with linear spring constant of
k
.
At equilibrium, all particles are equally spaced at a distance
∆
x
=
L/(n+1)
.
Let
z
j
be the displacement of the
j
th particle from equilibrium along the
x
axis
(measured positive to the right).
(a)
Show that the motions are governed by the differential equations:
(10 pts)
n
j
z
z
z
k
dt
z
d
m
j
j
j
j
,...,
1
),
2
(
1
1
2
2
=
+

=

+
(b)
Let
n
→∞
, m
→
0,
x
0
; obtain the wave equation as the limit.
What is the group velocity
now?
What is the phase velocity?
(10 pts)
.
(c)
What is the role of
L
in the derivation, assuming fixed boundaries at 0 and
L
?
Qualitatively describe the vibration behavior when
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