TIME DEPENDENCE in QUANTUM MECHANICS

Time-dependent perturbation theory is formulated in a systematic way. The focus is shifted from the wavefunction to the unitary transformation that evolves the wavefunction from one slice of time to another. Perturbation theory is formulated in terms of a systematic iterative expansion of the unitary transformation in terms of the perturbing hamiltonian, expressed in the interaction representation. Two standard results are obtained in first order time-dependent perturbation theory. These are the Fermi Golden Rule for transition rates and the Lorentz line shape for radiative transitions, as formulated by Wigner and Weisskopf. Timedependent perturbation theory is approached systematically in higher orders for a very specific perturbation of a very specific physical system, the simple harmonic oscillator subjected to a decaying exponential dipole driving term. Expansions are carried out to third order. These calculations suggest that a useful bookkeeping system be introduced for keeping track of the terms appearing in the perturbation series. This “diagrammatic technique” evolves, in a direct line, to the Feynman diagrams which keep the books for the perturbation theory called Quantum Electrodynamics. Some Lie Group theory is introduced and used to find an analytic expression for all transition amplitudes for any dipole-like perturbing potential. The analytic expression is compared with the systematic approximations at low orders for the exponentially decaying dipole forcing term where perturbation theory can be carried out explicitly to any order.