Question

PLEASE ANSWER :

Find the eigenvalues given the Eigen functions and operators Operator: the hydrogen atom Hamiltonian -h^2/2m {1/r^2 partial differential/ partial differential r (r^2) partial differential Psi(r, theta, phi )/ partial differential r + 1/r^2 sin(theta) partial differential / partial differential theta sin(theta) partial differential Psi (r, theta phi)/ partial differential theta + 1/r^2 sin^2 (theta) partial differential^2 Psi(r, theta, phi)/ partial differential phi ^2 } Ze^2/4 pi epsilon_0 r Psi(r, theta, phi) = E Psi(r, theta, phi) Eigen function: Psi(r, , phi)_100 = 1/ Squareroot pi (Z/a_0)^3/2 e^-Zr/a_0

Answer 1

{ 1/r^2 partialdifferential/ partialdifferential x (r^2) partialdifferential/ partialdifferential r {1/squareroot pi 82/ e^] + 1/r^2 sin theta partialdifferential/ partialdifferential theta sin theta partialdifferential/ partialdifferential theta [1/squareroot pi z/(a_0) 3/2 (e^- zr/a_0)] + 1/r^2 sin^2 theta partialdifferential^2/ partialdifferential phi^2 (1/ squareroot pi (z/a_0) 3h e^- zr) } - zr^2/4 pi epsilon_0 r (1/ squareroot pi (1/squareroot pi (z/a_0 3h e^2r/a_0) = -h^2 2m1/squareroot n z/(a_0) .3/2 { 1/r^2 partialdifferential/partialdifferential r (r^2) (- z/a_0) e^- zr/a_0 + 0 + 0} - z e^2/4 pi epsilon_0 r (1/ squareroot pi z/(a_0) 3h e^- zr/a_0) = h^2/2m z(a_0) 3h { 1/r^2 (2r e^- zr /a_0 = z/(a_0) r62 e^- zr/a_0 } - z e^2/4 pi epsilon_0 r (1/ squareroot pi z/(a_0) 3h e^- zr/a_0) = + h^2/2m { 2z/a_0 r - z^2/a_0^2 - z e^2/4 pi epsilon_0 r} 1/ squareroot pi z/(a_0) 3h e^- zr /a_0 EIGEN VALUE = { h^2/2m (2z/a_0 r - z^2/a_0 ^2 - z r^2/4 pi epsilon_0 r)}