Perturbation Theory Lecture 5

Interactions of radiations with matter In absence of external perturbation ------(1) In presence of radiations, Hamiltonian will be ---------- -(2)

In absence of electrostatic source Also, using Coulomb Gauge condition Neglecting terms of order of Eq (2) become ------(3) ----(4) To find the effect of external perturbation on atom, we first need to find vector potential . can be calculated classically and quantum mechanically. External perturbation

Classical treatment of radiations We write -----(5) Polarization vector E.F. and M.F. are given by ------(6)

Energy density for single photon is given by --------(7) Averaging over time and writing energy of single photon Per unit volume, Which gives, ----------(8) Thus, from (5), we have ---------(9)

Thus, using (9) in (4), Potential is given by -----(10) which is like harmonic perturbation.

Transition rates for perturbation defined by eq. (10) are given by ---(11) -------(12) Exercise: Derive above equations.

Transition rate within dipole approximation Expanding , we get -------(13) Reason: is small quantity because wavelength of radiations (visible or ultraviolet is large) compared to atomic size (small quantity)

Neglecting higher order terms in (13) i.e. Considering only first term -----------(14) This is called dipole approximation. We know Above eq can be generalized to Which give ------------(15)

Using (14) and (15) in following, we get --------(16) Using (16) in (11), we obtain transition rates within dipole approximation.

Exercise: Discuss selection rules for dipole transitions. e.g. 2p to 1s transition will be allowed but 2s to 1s will be forbidden. http://quantummechanics.ucsd.edu/ph130a/130_notes/node422.html