Question

An electron in the Coulomb field of a proton is in a state described by the wave function 1/6[4 psi_100(r) + 3psi_211(r) - psi_210(r) + Squareroot 10 psi_21 - 1(r)] (a) What is the expectation value of the energy? (b) What is the expectation value of L^2? (c) What is the expectation value of L_z?

Answer 1

psi = 1/6 [4 psi_100 (r) + 3 psi_2m (r) - psi_210 (r) + squareroot 10 psi_21-1 (r)] P_1 (100) = (4/6) ^2 p_2 (211) = (3/6) ^2 p_3 (210) = (1/6) ^2 p_4 (21, -1) = (squareroot 10/6) ^2 E_1 = -13.6/12 = 13.6 E_2 = -3.4 E_3 = -3.4 E_4 = -3.4 < E > = P_1E_1 + P_2 E_2 + P_3 E_3 + P_4 E_4 = - {(4/6) ^2 times 13.6 + [(3/6) ^2 + (1/6) ^2 + (squareroot 10/6) ^2]_x 3.4} = - (16/36 times 13.6 + 20/36 times 3.4) = -21 times 13.6/36 = -7.9 eV (b) L^2 psi = l (l + 1) h^2 psi L_1 = 0 l_2 = 1, l_3 = 1 l_4 = 1 < L^2 > = sigma p_1 l^2_i = ((4/6) ^2 times 0 + (3/6) ^2 + (1/6) ^2 + (squareroot 10/6) ^2) 2h^2 = 40/36 h^2 = 10/9 h^2 \r\n (c) L_2 psi mh psi < L_2 > = [Pi (mi h) = (p_1 times 0