Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 XM with an ener...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 ..xM with an energy given by: the Ci's and xs are constants. The Ci's are all positive MkBT A) Prove that (E) M regardless of the value of the constants. This result is known as the equipartition theorem each classical quadratic degree of freedom contributes to the average thermal energy of an equilibrium system. kBT B) Use the equipartition theorem to derive the familiar...
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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...
4. Suppose you have a system of N-10 quantum harmonic oscillators described by the Boltzmann distribution. The total energy in the system is M-40ho a) What is the average oscillator energy? b) What is the probability that an oscillator has twice the average energy?