Question

2. Problem 21.2 The vibrational potential energy of an atom in the Einstein model is ik(a2+2+2), where q (y) is displacement. Find the classical probability density pyx) for displace- ment.

Answer 1

Potential energy of atom is given by P.E = 1/2 k (q^2_1 + q^2_2 + q^2_3) & K.E of atom is given by K.E = 1/2 m (v^2_x + v^2_y + v^2_z) so total energy is given by E = K.E + P.E = 1/2 m (v^2_x + v^2_y + v^2_z) + 1/2 k (q^2_x + q^2_y + q^3_z) writing E in canonical variables (p, q) E (p^bar, q^bar) = p^2_x/2 m + p^2_y/2m + p^2_z/2m + 1/2 k(q^2_x + q^2_y + q^2_z) 1 so we know that probability is given by p = e^-beta E (p, q) where E(p, q) is given by 1 1 = integral e^-beta E (p, q) d^3 p d^3 q