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Chapter 8 TIME DEPENDENT PERTURBATIONS: TRANSITION THEORY 8.1 General Considerations The methods of the last chapter have as their goal expressions for the exact energy eigen- states of a system in terms of those of a closely related system to which a constant pertur- bation has been applied. In the present chapter we consider a related problem, namely, that of determining the rate at which transitions occur between energy eigenstates of a quantum system of interest as a result of a time-dependent, usually externally applied, perturbation. Indeed, it is often the case that the only way of experimentally determining the structure of the energy eigenstates of a quantum mechanical system is by perturbing it in some way. We know, e.g., that if a system is in an eigenstate of the Hamiltonian, then it will remain in that state for all time. By applying perturbations, however, we can induce transitions between di¤erent eigenstates of the unperturbed Hamiltonian. By probing the rate at which such transitions occur, and the energies absorbed or emitted by the system in the process, we can infer information about the states involved. The calculation of transition rates for such situations, and a number of others of practical interest are addressed in this chapter. To begin, we consider a system described by time-independent Hamiltonian H 0 to which a time-dependent perturbation ^ V ( t ) is applied. Thus, while the perturbation is acting, the total system Hamiltonian can be written H ( t )= H 0 + ^ V ( t ) : (8.1) It will be implicitly assumed unless otherwise stated in what follows that the perturbation ^ V ( t ) is small compared to the unperturbed Hamiltonian H 0 ; if we want to study the eigenstates of H 0 we do not want to change those eigenstates drastically by applying a strong perturbation. In fact, we will often write the perturbation of interest in the form ^ V ( t ¸V ( t ) (8.2) where ¸ is a smallness parameter that we can use to tune the strength of ^ V . We will denote by fj n ig a complete ONB of eigenstates of H 0 with unperturbed energies " n ; so that ,byassumpt ion H 0 j n i = " n j n i X n j n ih n j =1 h n j n 0 i = ± n;n 0 : (8.3) Our general goal is to calculate the amplitude (or probability) to …nd the system in a given …nal state j à f i at time t if it was known to be in some other particular state j à i i at time t = t 0 : Implicit in this statement is the idea that we are going to let the system evolve from j à i i until time t and then make a measurement of an observable A of which j à f i is an eigenstate (e.g., we might be measuring the operator P f = j à f ih à f j ). A little less generally, if the system was initially in the unperturbed eigenstate j n i i of H 0 at t 0 ;
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240 Time Dependent Perturbations: Transition Theory we wish to …nd the amplitude that it will be left in (or will be found to have made a transition to) the eigenstate j n f i at time t>t 0 ; where now the measurement will be that of the unperturbed Hamiltonian itself. We note in passing that if we could solve the full Schrödinger equation i ~ d dt j à i ( t ) i = H ( t ) j à i ( t ) i (8.4)
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