1. P460 - transitions1 Transition Rates and Selection Rules Electrons in a atom can make transitions from one energy level to another not all transitions have the same probability (and some are “forbidden”). Can use time dependent perturbation theory to estimate rates. The atom interacts with the electric field of the emitted photon. Use time reversal (same matrix element but different phase space): have “incoming” radiation field. It “perturbs” the electron in the higher energy state causing a transition to the lower energy state

2. P460 - transitions2 EM Dipole Transitions Simplest is electric dipole moment. It dominates and works well if photon wavelength is much larger than atomic size. The field is then essentially constant across the atom. Can be in any direction. Other electric and magnetic moments can enter in, usually at smaller rates, but with different selection rules Use Fermi Golden Rule from Time Dep. Pert. Theory to get transition rate phase space for the photon gives factor: and need to look at matrix elements between initial and final states:

3. P460 - transitions3 Parity Parity is the operator which gives a mirror image Parity has eigenfunctions and is conserved in EM interactions The eigenfunctions for Hydrogen are also eigenfunctions of parity A photon has intrinsic parity with P=-1. See by looking at the EDM term So must have a parity change in EM transitions and the final-initial wave functions must be even-odd combinations (one even and one odd). Must have different parities for the matrix element to be non- zero

4. P460 - transitions4 Matrix Elements: Phi terms Calculate matrix elements. Have 3 terms (x,y,z) look at phi integral (will be same for y component) no phi dependence in the z component of the matrix element and so m=m’ for non-zero terms gives selection rules

5. P460 - transitions5 Matrix Elements: Theta terms The (x,y,z) terms of the matix elements give integrals proportional to: these integrals are = 0 unless l-l’=+-1. The EM term of the photon is (essentially) a l=1 term. Legendre polynomials are power series in cos(theta). The extra sin/cos “adds” one term to the power series. Orthogonality gives selection rules. Relative rates: need to calculate theta integrals. And calculate r integral. They will depend upon how much overlap there is between the different states’ wave functions

6. P460 - transitions6 First Order Transitions Photon has intrinsic odd parity (1 unit of angular momentum, S=1). For: n \ l 0 1 2 3 4 4S 4P 4D 4F 3 3S 3P 3D 2 2S 2P 1 1S Selection Rues