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ps8rev - Statistical Physics I(8.044 Spring 2009 Assignment...

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Statistical Physics I (8.044) Spring 2009 Assignment 8 April 8, 2009 Due April 15, 2009 Please remember to put your name and section number at the top of what you turn in. Your SECOND TEST is Wednesday April 22 at the usual lecture time, 1-2 pm , in an unusual location: 3RD FLOOR OF WALKER MEMORIAL, AKA 50-340 . Readings The reading for Chapter VI of 8.044 is the Notes on the Canonical Ensemble, the Notes on Rotational Raman Spectrum of a Diatomic Molecule, which are both available on the 8.044 web page, and Baierlein Chapters 5, 13 and 14. Problem Set 8 1. Interpolating Between a Two-State System and a Harmonic Oscillator (10 points) (a) As a prelude to this problem, show that the heat capacity of a quantum harmonic oscillator is unchanged if one ignores the zero point energy and writes the energy eigenvalues as ε n = n ¯ instead of ε n = ( n + 1 2 . (b) Consider N localized noninteracting particles, each of which has a finite number, n , of energy levels. The n levels are evenly spaced in energy, each separated from the next by an energy ε . Define θ ε/k B . Show that the partition function is Z N 1 where Z 1 = Z h . o . ( θ/T ) Z h . o . ( nθ/T ) with Z h . o . ( x/T ) being the partition function for a harmonic oscillator with ¯ hω/k B = x and with the zero point energy ignored. (c) Show that the heat capacity of the system in (b) is the difference between the heat capacities of two harmonic oscillators. (d) Sketch a plot of the heat capacity of the system in (b) as a function of temper- ature for n = 2, n = 5 and n 1. 1
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2. A Three-State System (10 points) A solid contains N ions with magnetic moments due to an angular momentum J = 1. That means that each ion can exist in one of three possible states labelled by a quantum number m J which can take on the values -1, 0 or 1. The magnetization of the solid is M = ( m J ) . In the absence of any applied magnetic field, electric effects within the solid 1 cause the states with m J = ± 1 to have an energy Δ lower than the state with m J = 0. Turning on a magnetic field H induces a further shift
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