Robert Szoszkiewicz, PHYS 664, HW # 1, Aug 30, 2009
DUE: on (or before) 10:30 am, Sept 7, 2009.
1.
Multiplicity function for harmonic oscillators.
(a) Follow the notes of the Lecture 3 and prove (via a method described in the notes) that
g
(
N,n
= 0) =
g
(
N,
0) = 1.
(b) Then, prove that
g
(
N,n
= 1) =
g
(
N,
1) =
N
.
(c) Next, prove that
g
(
N,
2) =
N
(
N
+1)
2!
.
(d) Next, prove that
g
(
N,
3) =
N
(
N
+1)(
N
+2)
3!
.
(e) Finally, prove that
g
(
N,
4) =
N
(
N
+1)(
N
+2)(
N
+3)
4!
.
(f) Out of a) to e) we generalize:
g
(
N,n
) =
N
(
N
+1)(
N
+2)(
N
+3)
...
(
N
+
n

1)
n
!
=
(
N
+
n

1)!
n
!(
N

1)!
.
But, such a prove is not complete. Why? Explain when we would prove it completely.
(Hint: mathematical induction theorem).
(g) Our proof contains also another ”tiny” mistake. The way we sum the energies for each
g
(
N,n
) is not correct. Why? (Hint: do a few exact calculations using a complete form
of energies for each harmonic oscillator.)
2.
Multiplicity function for a model of a polymer.
Consider a freely jointed chain (FJC) model for a polymer, as presented in the Example 5,
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 Spring '10
 RobertSzoszkiewicz
 Physics, 1D, multiplicity function, 2D square lattice, Robert Szoszkiewicz

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