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Unformatted text preview: PHYSICS 420 THERMAL PHYSICS J. D. Maynard READING ASSIGNMENT: Reif, Chapter 6 Homework #7 Due Lecture 37 You are on your honor to work on this on your own, or get hints from Professor Maynard or the Teaching Assistant only. 1. A simple harmonic onedimensional oscillator has energy levels given by k = ( k + 1 / 2) ¯ hω, where ω is the characteristic (angular) frequency of the oscillator and where the quantum number k can assume the possible integral values k = 0, 1, 2, ··· . Suppose that such an oscillator is in thermal contact with a heat reservoir at temperature T low enough so that kT/ (¯ hω ) << 1 . (a) Find the ratio of the frequency of occurence of the oscillator being in the first excited state to that of its being in the ground state. (b) Assuming that only the ground state and first excited state are appreciably occupied, find the mean energy of the oscillator as a funcion of the temperature T. 2. Consider a system of N weakly interacting particles, each of spin 1/2 and magnetic moment μ , located in an external field H, so that the possible energy states of one particle are = μH and 1 = + μH. Suppose that this system is in thermal contact with a heat reservoir at the absolute temperature T. Calculate its mean total energy E as a function of T and H. 3. A solid at absolute temperature T is placed in an external magnetic field H = 30,000 gauss. The solid contains weakly interacting paramagnetic atoms of spin 1/2 so that the energy of each atom is = μH or 1 = + μH. (a) If the magnetic moment μ is equal to one Bohr magneton, i.e., μ = 0.927 × 10 20 ergs/gauss, below what temperature must one cool the solid so that more than 75 percent of the atoms are polarized with their spins parallel to the external magnetic field?...
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