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PHYS664_HW1

# PHYS664_HW1 - Robert Szoszkiewicz PHYS 664 HW 1 DUE on(or...

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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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PHYS664_HW1 - Robert Szoszkiewicz PHYS 664 HW 1 DUE on(or...

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