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The Journal of Chemical Physics

Quantum dynamics via mobile basis sets: The Dirac variational principle

David Hsu, D. F. Coker

Published Mar 15, 1992• 20 citations
Publisher

Abstract

We propose a variation on the wave-packet interpretation of quantum mechanics introduced by Heller, here using the Dirac variational principle to optimize the time evolution of all the parameters that enter our basis set. We solve the general one-dimensional problem, and test our method on the Morse oscillator and the symmetric quartic double well.

DOI
10.1063/1.462820

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We propose a variation on the wave-packet interpretation of quantum mechanics introduced by Heller, here using the Dirac variational principle to optimize the time evolution of all the parameters that enter our basis set. We solve the general one-dimensional problem, and test our method on the Morse oscillator and the symmetric quartic double well.
The use of a time-dependent basis set for solving the time-dependent Schrödinger equation is proposed. The basis set is a generalization of the harmonic oscillator functions. The proposed method is similar to the approaches formulated by Lee and Heller and by Coalson and Karplus. However, in the present method the time dependence of the basis functions is determined differently. Numerical studies on the quartic oscillator and on the Morse potential are carried out to investigate the performance of the various methods.
In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on (i) the definition of a linearly independent working space and (ii) a variational construction of the propagator over finite time steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are carried out on a quartic double-well potential with Gaussian basis functions whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.
The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.
We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.
Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.