Question

Please: Show work, Carry all parts, Use clear writing. Thanks for help. Quantum Statistical Mechanics

2) The quantum rotor in two dimensions has the Hamiltonian h2 d2 21 de2' with 0 θ < 2π . a. Find the eigenstates and energy levels of the system. b. Write an expression for the density matrix (0'Ipl0) in a canonical ensemble of temperature T.

Answer 1

Given that Quantum rotor in two dimensions has the Hamiltonian H = hstrok^2/2I d^2/d theta^2 with 0 lessthanorequalto theta lessthanorequalto 2 pi Eigen states and energy values of the system - hstrok^2/2I d^2/d theta^2 (f(theta) = lambda f(theta) - hstrok^2/2I d/ d theta (f prime (theta))^2 = lambda d/d theta (f(theta))^2 f prime(theta) = squareroot - lambda 2I/ hstrok^2 f(theta) f(theta) = A e^ik theta where K = squareroot lambda 2 i/ hstrok^2 Energy is lambda = k^2 hstrok^2/2I As e^ik theta = 1 = e^ik(2 pi) KE integer \r\n So eigen states are psi(theta) = A e^-k theta with energy levels lambda = hstrok^2 k^2/2I where K = 0, plusminus 2 + . . . . . Expression for the density matrix < theta prime Vertical-bar P Vertical-bar theta > in a canonical ensemble of temperature T Density matrix is given by P = sigma/K e^F - Ex/K T Vertical-bar K > < K Vertical-bar