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Journal of physics. A, Proceedings of the Physical Society. General

A variational estimate of the lowest eigenvalue of the multiatom, multimode Hamiltonian of quantum optics

B. V. Thompson

Published Oct 1, 1972• 6 citations
Publisher

Abstract

In those cases where a direct diagonalization of the energy matrix is not attempted because of insuperable practical difficulties of computation, it is always possible to estimate the gamut of the eigenvalue spectrum by a variational method. A trial state function similar in some respects to a Glauber state is employed. Certain asymptotically exact results due to Tavis and Cummings (1968) are recovered. Finally, ways in which the accuracy might be improved are outlined.

DOI
10.1088/0305-4470/5/10/010

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In those cases where a direct diagonalization of the energy matrix is not attempted because of insuperable practical difficulties of computation, it is always possible to estimate the gamut of the eigenvalue spectrum by a variational method. A trial state function similar in some respects to a Glauber state is employed. Certain asymptotically exact results due to Tavis and Cummings (1968) are recovered. Finally, ways in which the accuracy might be improved are outlined.
The second order error δF(2), made by employing the Schwartz−Dalgarno−Delves variational principle for the diagonal matrix element φ†Wφ of an arbitrary Hermitian operator W, is examined in the case that φ is the bound ground state eigenfunction of some given Hamiltonian H. This variational principle characteristically involves not only a trial estimate φt of φ, but also a trial estimate Lt(φt) Lt(φt) of a well−defined but generally not exactly known auxiliary function L (φ). It has previously been shown that, for certain special choices of a trial Hamiltonian Ht, the trial Lt(φt) can be found from a minimum principle. The present work finds that for these same special Ht it is possible to express δF(2) in comparatively simple closed form, depending only on known quantities, so that δF(2) should be calculable when the system described by H is not too complicated. However, these results for δF(2) are obtained on the assumption that Lt(φt) is known essentially exactly for any given φt; the practical utility of the formulas derived still must be tested, therefore. If Lt(φt) can be determined to this necessary accuracy, one expects that combining the computed δF(2) with the usual variational estimate of φ†Wφ will be a significant improvement over the usual variational estimate alone. Under the same circumstances, when W is a positive definite operator, the expression for δF(2) can provide nonrigorous but nonetheless potentially useful second order (variational) upper and lower bounds on the exact φ†Wφ.
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.
A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.
In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.
We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.