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Unformatted text preview: 1 Physics 219  Problem Set 2 Due Date: February 5, 2008 1. Equipartition Theorem. An ideal gas has a Hamiltonian that depends only on particle momenta, H = N ∑ i = 1 ~ p 2 i 2 m where the momentum ~ p i is a ddimensional vector. Consequently, each particle has average energy d 2 k B T . Consider a system with N particles with ddimensional coordinates ~ x i and momenta ~ p i , i = 1 ,..., N . Show that if the Hamiltonian is: H = N ∑ i = 1 a  ~ p i  n p + b  ~ x i  n x Then the energy per particle is (convince yourself that it doesn’t matter whether or not they are distinguishable) : E N = d 1 n p + 1 n x k B T This is the generalized equipartition theorem. 2. Consider a classical diatomic gas in three dimensions. We will ignore the vibrational degrees of freedom of the molecule and concentrate on its translational and rotational degrees of freedom. We will assume that it has total mass M and moment of inertia I about either of the two axes perpendicular to the axis joining the two atoms (i.e. the symmetry axis of thethe two axes perpendicular to the axis joining the two atoms (i....
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 Spring '08
 NAYAK
 mechanics, Momentum, Statistical Mechanics, Fundamental physics concepts, partition function, Einstein Phonons

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