Question

a transition in a hydrogen atom is forbidden if

$\int \varphi _{f}(-er)\varphi _{i} d\tau$

Where r is a vector in cartesian coordinates. Do the radial integral in spherical coordinates for transitions l=0 to l=0 and (l=1,$m_{l}$=0) to l=0. show which are forbidden or permitted.

Answer 1

Psi_100 (r, theta, d) = R_10 Y_00 (theta, d) R_10 = 2/ squareroot a^3 e^-r/a, Y_00 = 1/ squareroot 4 pi Psi_210 (r, theta.) = R_21 (r) Y_10 (theta, d) R_21 (r) = 2 Y_10 (theta, d) = Squareroot3/4 pi cos theta = 1 squareroot 4 a^3 (r/a) e^- r/2a Note Transition langle i rangle doubleheadarrow langle f rangle possible if only if Delta l = plusminus 1 [So (i) rightarrow forbidden (ii) rightarrow allowed I_fi_0 rightarrow 0 = integral Psi_f (- e r) Psi_i d f = integral^infinity_0 R_f (r) R_i (r) r^3 d r integral^pi_0 integral^2 pi_0 1/f (theta, phi) Y_f (theta, phi) sin theta d theta d phi = I (r) integral^pi_0 integral^2 pi_0 [Y_00 (theta, phi)]^* , [Y_00 (theta phi)] sin theta d theta d phi