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Proceedings of the Physical Society

Angular change of state and bounds on transition probabilities for two-state quantal systems

L Aspinall, Ian Colin Percival

Published Feb 1, 1967• 5 citations
Publisher

Abstract

Simple rigorous upper bounds are obtained for the transition probability from the initial state of a two-state quantal system with a time-dependent Hamiltonian. The bounds are compared numerically with the exact transition probability for some special cases. The total integrated angular change of state of the system is defined, and is used to obtain the inequalities.

DOI
10.1088/0370-1328/90/2/302

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Simple rigorous upper bounds are obtained for the transition probability from the initial state of a two-state quantal system with a time-dependent Hamiltonian. The bounds are compared numerically with the exact transition probability for some special cases. The total integrated angular change of state of the system is defined, and is used to obtain the inequalities.
Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.
A formalism of time-smoothed total transition probabilities and their rates is developed which employs Laplace averages of these quantities. Under conditions pertinent to scattering processes, the Laplace-average formalism is shown to yield results equivalent to those obtained from a stationary-state formalism. Rigorous lower and upper bounds are obtained for the Laplace-averaged quantities which reduce to equalities for two-level systems. The lower bounds appear to be potentially useful estimating lower bounds for total cross sections of various processes.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.
Work on the reduction of independent variables in the Schr\"odinger equation for two-particle systems from six to three is extended to give a closed expression of the variational equation for states of arbitrary total angular momentum. The method consists of writing the Hamiltonian in six coordinates, by a suitable choice of transformation equations, and evaluating integrals involving real angular-momentum functions. An existing variational equation for P states follows as a special case of the present formalism.