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Acta Physica Sinica

THE STEEPEST DESCENT PERTURBATION THEORY FOR THE EXCITED STATE OF A QUANTUM SYSTEM

WEN GEN-WANG, 湖南师范大学物理系,长沙,410081

Published Jan 1, 1991• 1 citation
Publisher

Abstract

The steepest descent perturbation theory is extended to the calculation of the energy and eigenfunction of the excited state of a quantum system. In case of the orthogonality of the trial function for the excited state to those for lower-energy state or ground state in the same symmetry class is preserved, the variational collapse to lower energy state can be avoided in this proposal. An iterative procedure is given for generating better eigenvalue and eigenfunction of the excited state without requiring an infinite summation over reference states as in conventional perturbation theory. This new perturbation method can be applied to calculate the excitation energy and wave function of excited states for any many-body quantum system to a high degree of accuracy without so much computational effort as in conventional method.

DOI
10.7498/aps.40.1388

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The steepest descent perturbation theory is extended to the calculation of the energy and eigenfunction of the excited state of a quantum system. In case of the orthogonality of the trial function for the excited state to those for lower-energy state or ground state in the same symmetry class is preserved, the variational collapse to lower energy state can be avoided in this proposal. An iterative procedure is given for generating better eigenvalue and eigenfunction of the excited state without requiring an infinite summation over reference states as in conventional perturbation theory. This new perturbation method can be applied to calculate the excitation energy and wave function of excited states for any many-body quantum system to a high degree of accuracy without so much computational effort as in conventional method.
A new Monte Carlo method for computing excited state properties of quantum systems is introduced. It is a generalization of the transient estimate method used for fermion Green’s function Monte Carlo and of subspace projection methods used for computing eigenstates of matrices. The time dependent autocorrelation function of a vector of trial functions is calculated for a random walk generated by the imaginary-time Schrödinger equation and estimates of energy levels are determined by the eigenvalues of the matrix of correlation functions. This method is especially useful for treating states with the same symmetry as it automatically keeps higher states orthogonal to lower states. The estimated energy converges to the exact eigenvalue with a rate which decreases with increasing excitation energy, thus limiting the method to relatively low-lying states. The method is zero variance in the sense that as better trial functions are introduced, the statistical error decreases to zero. The method has a nontrivial bias which is analyzed. As an illustration, the eigenvalues of a double well are computed.
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.
We provide a very efficient procedure for obtaining the excited states of a quantum operator H, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator A=(H-Et{\mathit{E}}_{\mathit{t}}Et​)2{)}^{2})2, and hence the eigenvalue of H nearest in energy to Et{\mathit{E}}_{\mathit{t}}Et​, where Et{\mathit{E}}_{\mathit{t}}Et​ is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2\ifmmode\times\else\texttimes\fi{}2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T\ensuremath{\bigotimes}\ensuremath{\tau} Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.
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 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 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.
A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.
Recently developed neural network-based wave function methods are capable of achieving state-of-the-art results for finding the ground state in real space. In this work, a neural network-based method is used to compute excited states. We train our network via variational principle, along a further penalty term that imposes the orthogonality with lower-energy eigenfunctions. As a demonstration of the effectiveness of this approach, results from numerical calculations for one-dimensional and two-dimensional harmonic oscillators are presented.