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arXiv:1605.01974

Convergence of transition amplitudes obtained with the Schwinger variational principle

V D Rodríguez

Published May 6, 2016• 0 citations
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Abstract

An exactly solvable time-dependent quantum mechanical problem is employed to study the convergence properties of transition amplitudes calculated by using the Schwinger variational principle. A detailed comparison between the amplitudes approximated by the perturbative series and by their associated Schwinger variational principles is performed. The much better performance obtained by the variational principle is documented through different case studies. For a given order of the Schwinger principle, it is observed that the transition amplitudes do not converge to the exact one for large perturbations. The latter is true even though large combinations of unperturbed states with constant coefficients are taken as trial wave functions. As a matter of fact, it is shown that the improvement of the method comes from using better trial wave functions and increasing the order of the Schwinger principle employed.

DOI
10.48550/arxiv.1605.01974

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An exactly solvable time-dependent quantum mechanical problem is employed to study the convergence properties of transition amplitudes calculated by using the Schwinger variational principle. A detailed comparison between the amplitudes approximated by the perturbative series and by their associated Schwinger variational principles is performed. The much better performance obtained by the variational principle is documented through different case studies. For a given order of the Schwinger principle, it is observed that the transition amplitudes do not converge to the exact one for large perturbations. The latter is true even though large combinations of unperturbed states with constant coefficients are taken as trial wave functions. As a matter of fact, it is shown that the improvement of the method comes from using better trial wave functions and increasing the order of the Schwinger principle employed.
The probability amplitude for the transition of a many particle system between given initial and final states in some finite time interval is considered where these states are products of single particle functions. It is pointed out that, if this amplitude is calculated by approximating the intermediate states by means of the usual time dependent Hartree theory, a spurious time dependence of the amplitude is obtained. An alternative time dependent extension of Hartree theory, more suited to this problem, is obtained by employing suitable trial functions in a variational principle for the transition amplitude. Some simple properties of the approximation are discussed.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.
The paper discusses how the variationally inspired perturbation theory (VIPT) scheme of approximation in quantum mechanics can be improved convergencewise if one uses for variational trial functions the perturbative series with variational parameter, so that one is effectively doing variational calculations directly on the high-order perturbative series for energy. The result optimizes the high-order energy directly and thus represents a significant improvement over the VIPT procedure. When applied to a double potential in which even the VIPT is badly divergent, we saw that the result is still very much convergent.
In chapters 13 and 14 we focused on perturbation theories for time-independent and time-dependent perturbations. For time-independent perturbations we obtained first-order and second-order corrections to energy eigenvalues and eigenfunctions. For time-dependent perturbations we obtained the probability for a system to make a transition from an ith initial state to fth final state. We have also discussed how to study the effect of adiabatic and sudden perturbations. In the present chapter we consider the variational method. The variational method can be used to construct an approximate ground state eigenfunction. In some cases the method can be used for finding approximately the energy of first few discrete states of a bound system. The perturbation theory uses a set of unperturbed states and the perturbation Hamiltonian H(1) to construct the eigenfunctions and eigenvalues through highly formalized approximations. In contrast, the variational method is based on a guess for the state-a trial state which is used to determine the corresponding energy. Any well behaved trial state introduced will yield an appropriate energy, although some trial states will be better than other, but whatever the guess, we can always make systematic improvements. We can construct a reasonably accurate eigenfunction often by a physical insight and ingenuity.