Question

1) A hydrogen atom, initially in the ground state, is placed in a time-dependent electric field that turns on suddenly at t 0 E(t) Eo exp(-yt) e t>0 Use first order time-dependent perturbation theory to find the probability that the hydrogen atom will be found in the n 2 level for t (You will need to consider transitions to each of the (1, m) substates separately; use the Wigner-Eckart theorem to help you decide which matrix elements you need to worry about.)

Answer 1

Given that A hydrogen atom is placed in a spatially homogenous time dependent electric field is given by E (t) = {0 t < 0 epsilon_0 exp (- gamma t) t > 0 at t = 0 it is in the ground state The potential due to the interaction of hydrogen electrons with the external field E (t) The transition probability is given by the evaluting P_1 - 2 = 1/h^2 integral^infinity_- infinity < psi_2 Vertical-bar v (t) Vertical-bar psi_1 > e^i w_2 t dt Vertical-bar ^2 rightarrow (1) where psi_1 = 1s state means Vertical-bar 00 > psi_2 = 2p state means Vertical-bar 210 > therefore < 210 Vertical-bar v (t) Vertical-bar 100 > = < 210 Vertical-bar (- er^vector middot E^vector) Vertical-bar 100 > = -e E_0 e^- E_0 e^- rt (210 Vertical-bar 3 Vertical-bar 100 > rightarrow (2) since 3 = cos theta psi_1s = R_10 (ohm) = 1/squareroot pi a^3 e^- r/a_0 \r\n psi_2p = R_21 1/10 (ohm) = 1/squareroot 8 pi a_0^3 r/2 a_0 e^- r/2 a_0 cos theta < 210 Vertical-bar 3 Vertical-bar 100 > = integral^infinity_0 r^3 R_21