Time Dependent Perturbation Theory

1. Time Dependent Perturbation Theory • Many possible potentials. Consider one where • V’(x,t)=V(x)+v(x,t) • V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions • let perturbation v(x,t) be small compared to V(x) • examples:finite square well plus a pulse that starts at t=0 or atoms in an oscillating electric field • write wavefunction in terms of known eigenfunctions but allow the coefficients (cn) vary with time  fraction with any eigenvalue (say energy) changes with time P460 - perturbation 2

2. Time Dependent • start from unperturbed with |n> time independent • add on time-dependent perturbation and try to find new, time-dependent wavefunctions. l keeps track of the order. Gives Schrodinger Eq.: • write out y in terms of eigenfunctions |n> • substitute into (2) and eliminate part that is (1) P460 - perturbation 2

3. Time Dependent • take dot product with <m| • this is exact. How you solve depends on the potential and initial conditions • an example. start out at t<0 in state k. “turn on” perturbation at t=0. See what the values of cm are at a later time • reminder. cm gives probability to be in state m • for first order, just do transitions from km. ignore knm and higher (as the assumption that all are in state k won’t hold for t>0) P460 - perturbation 2

4. Time Dependent – example • try to solve. assume in particular state k at t=0 • and so get diff. eq. for each state m and solution • if there isn’t any time dependence to V (or if it changes slowly compared to DE/hbar) then can pull the matrix element out of the integral • notice DE and matrix element dependence P460 - perturbation 2

5. sidenote • if the perturbing potential is sinusoidal (e.g. photons of a given energy) then have • then the results are similar (see book for doing the integral) • which has a large value when the energy difference is equal to the frequency of the perturbation P460 - perturbation 2

6. Time Dependent Probability • The probability for state k to make a transition into any other state is: • evaluate P by assuming closely spaced states and replacing sum with integral • define density of states (in this case final; after the perturbation) in the same way as before P460 - perturbation 2

7. Fermi Golden Rule • Assume first 2 terms vary slowly.Pull out of integral and evaluate the integral at the pole • this doesn’t always hold--ionization has large dE offset by larger density • n are states near k. Conserve energy. Rate depends on both the matrix element (which includes “physics”) and the density of states. Examples later on in 461 P460 - perturbation 2

8. 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 P460 - perturbation 2

9. 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. • b includes EM terms. 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: P460 - perturbation 2

10. 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 P460 - perturbation 2

11. Spherical Harmonics-reminder • The product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M. • They hold whenever V is function of only r. Seen related to angular momentum P460 - perturbation 2

12. 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 P460 - perturbation 2

13. 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 (sin and cos terms). • 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 P460 - perturbation 2

14. 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 P460 - perturbation 2