Exercise 6.4 Consider an anisotropic three-dimensional harmonic oscillator potential V (x, y, z) = = m(w² x² + w} y² +w? z²). (a) Evaluate the energy levels in terms of @x, @y, and wz.

Exercise 1: Consider N free particles in a box without interaction, the classical Hamiltonian is given by: H = E; (+ V(ri)), where V(r¡) = 0 in the box of volume N 2m We assume that this system belongs to canonical set. 1- Give the expression of partition function (Z). 2- Give the volume element (dtn) in the space. 1 3- a- Prove that: Z. - (Z,)N where Zz is the partition function of one particle. N! b- Give the expression of Z1. 4- Calculate the partition function Z,. 5- Deduce the partition function of the perfect gas. azc 6- Calculate aß 7- Calculate the mean value of energy of the system.

Problem 2.13 A particle in the harmonic oscillator potential starts out in the state ¥ (x. 0) = A[3¥o(x)+ 4¼1(x)]. (a) Find A. (b) Construct ¥ (x, t) and |¥(x. t)P. (c) Find (x) and (p). Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ¥2(x), instead of Vi(x)? Check that Ehrenfest's theorem (Equation 1.38) holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?

Problem #2 Calculate the Legendre transform (F1) of y = x². For your answer, give the new function F1 and its derivative dF1. (a(f(x)) Note that dy = C dx, C, = f(x), and dC, = (0) dx. dx