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4. Consider a classical particle at temperature T. Suppose the Hamilton (i.e. the total energy) function H for the particle can be written as a sum of independent quadratic terms in the variables on which H depends. That is, if H -H(31,£2... ), then Here,5 could be a position or a momentum coordinate, and the a's are constants. As an example, 2 2 Px for a ID classical harmonic oscillator with independent variables χ and px, aIŠ!--and 2m a2ξ22-kx2, where m is the mass and k is the force constant. Show that for a system with such a 2 n that depends on any number of variables, each independent quadratic term (whether potential or kinetic) in H of the form αξ2 contributes a factor of kBT/2 to the average classical energy. In so doing, you will prove the celebrated equipartition theorem. This explains why the average translational kinetic energy for a classical structureless particle free to move in 3D is 3kgT/2

Answer 1

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