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HW #3 - Prof Jennifer Curtis School of Physics Georgia Tech...

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Prof. Jennifer Curtis School of Physics, Georgia Tech Spring 2012 Statistical Mechanics 4142: Problem Set 3 Due date: Friday, Feb. 3, 2:05pm . Comments: You may use the following definite integral of a Gaussian function : [1] Quantum Harmonic Oscillator, Multiplicity and Total Energy. (K&K #2.3) (a) (2 pts.) Find the entropy of a set of N oscillators of frequency ω as a function of the total quantum number n. Use the multiplicity function for a quantum harmonic oscillator (given in class or in K.K. 1.55, where n=q). You will need to use typical approximations, assuming N is large. (b) (2 pts.) Let U denote the total energy q ω of the oscillators. Express the entropy as σ U , N ( ) . Show that the total energy at temperature t is: U = N ω exp ω τ ( ) 1 . This is the Planck result; it is derived again in Ch. 4 (K&K) by a powerful method that does not require us to find the multiplicity function. [2] Multiplicity of a Large Two-state Paramagnet. (Schroeder 2.24) For a single, large two-state paramagnet, the multiplicity function is very sharply peaked about N = N 2 . (a) (2 pts.) Use Stirling’s approximation to estimate the height of the peak in the multiplicity function. (b) (2 pts.) Use the methods introduced in class to derive a formula for the multiplicity function in the vicinity of the peak, in terms of x N N 2 . Check that your formula agrees with your answer to part (a) when x=0. (c) (2 pts.) How wide is the peak in the multiplicity function?
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