Question

[4.129]

Apply the operator as defined in equation 4.129, and the operator   as defined in equation 4.132 to the hydrogen state   to show that they have the eigenvalues given in equation 4.133.

Answer 1

For hydrogen atom Psi_2, 1 (r, theta, d) = 1/8 squareroot pi a_0^3/2 (r/a_0) e^-r/2a_0 sin theta e^+ i Phi H Psi = E Psi E_2 = -13.6/(212) eV E = -3.4 eV L_z = h/i partial differential/partial differential Phi [1/8 squareroot pi a_0^3/2 (r/a_0) e^-r/2a_0 sin theta e^+ i Phi] = h/i [1/8 squareroot pi a_0^3/2 (r/a_0) e^-r/2a_0 sin theta] partial differential/partial differential Phi (e^+ i Phi) = h/i (1) Vertical-bar Psi > = 1 h Vertical-bar Psi > \r\n So eigen value = 1 h L^2 Vertical-bar Psi > = -h^2 [1/sin theta (partial differential/partial differential theta) (sin theta partial differential/partial differential theta) + 1/sin^2 theta partial differential^2/partial differential Phi^2] Vertical-bar Psi > = -h^2 [-1 + 2 cos^2 theta/sin^2 theta - 1/sin^2 theta] Vertical-bar Psi > = + h^2 [2 (1 - cos^2 theta)/sin^2 theta] Vertical-bar Psi > = 2h^2 [sin^2 theta/sin^2 theta] Vertical-bar Psi > = 2h^2 Vertical-bar Psi > So eigen values = 2h^2