Abstract There is a vast class of interesting physical problems, however, which involve time-dependent Hamiltonians and which are not accessible to perturbative treatment. In this chapter, we discuss some of these systems.
This chapter presents a modernized version of a method originally due to Dirac. This approach depends on a straightforward expansion in powers of the perturbing interaction. The chapter examines a problem where the unperturbed Hamiltonian had a discrete spectrum, and the perturbation was explicitly time dependent. The situation where the spectrum is continuous and the perturbation is independent of time is encountered much more frequently in physics. The chapter explores the derivations of some rather general formulas for collision cross sections. The photo-effect in hydrogen constitutes a very interesting example of a perturbation problem. The dramatic behavior of the matrix element demonstrates that at low energy the Coulomb attraction is very effective in keeping the slow photo-electron near the origin. The chapter examines the result of a collision experiment. It aims to illustrate the power and limitations of the technique, and to establish its relationship to the treatment of elastic scattering.
Abstract This chapter begins the discussion of the time evolution of an active quantum system. From the earlier chapters, time dependent physics has been observed through the presence of the time evolution of the phase of each eigenstate. The Hamiltonian itself is time independent. This represents the same kind of evolution of a classical system like the vibration of a tuning fork when it has been struck or the oscillation of an LC circuit if the capacitor is charged to some voltage and then the switch is closed. In the quantum case, the Hamiltonian has also been time independent. The time evolution evolves according the full-time dependent Schrödinger equation, depending only on a single initial condition of the state vector or wave function and the corresponding time evolution of the phase factor for each eigenstate. However in this chapter, we consider the case of when there is a time dependent Hamiltonian such as a sinewave generator or laser. As in the case of resonant tunneling, we see the importance in dynamics of resonant coupling. With an oscillating potential energy term, we see the presence of Rabi oscillations in the probability amplitude of a two-state system on resonance, which can be viewed as a quantum flip-flop between two states of a quantum bit (qubit).
How to deal with time-dependent potentials, what kinds of physical problems having time-dependent potentials can be solved exactly and which are to be solved with the use of perturbation theory, what are different possible forms of time-dependent perturbating fields, and semi-classical treatment of the interaction of fields with matter are the subject matters of this chapter. First order perturbation theory, computation of transition probability, transition rates and Fermi's Golden Rule for harmonic and constant perturbations, absorption, stimulated emission and dipole approximation when electromagnetic fields interact with matter, spontaneous emission and Einstein coefficients, and dipole selection rules are other topics which are dealt with in detail in the chapter. Discussions with properly justified mathematical treatment on time-dependent potentials and on the exactly solvable time-dependent two-state systems are the special foci of this chapter.