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Physical Review A

Gaussian effective potential: Computation of eigenvalues

Arjendu K. Pattanayak, William C. Schieve

Published Jul 1, 1996• 2 citations
Publisher

Abstract

The Gaussian effective potential (GEP) is an approximation to quantum mechanics that semiquantitatively estimates quantum effects such as zero-point fluctuations and tunneling. We show how to use the GEP to compute semiclassical eigenvalues. It is well known that the GEP provides very accurate variational approximations to the ground-state energy of simple quantum systems; our results show that it also provides accurate excited-state energies. We also show how this method is related to the first-order Bargmann-Voros coherent-state representation of quantum mechanics, and that our results improve upon those derived from the latter method. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.947

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The Gaussian effective potential (GEP) is an approximation to quantum mechanics that semiquantitatively estimates quantum effects such as zero-point fluctuations and tunneling. We show how to use the GEP to compute semiclassical eigenvalues. It is well known that the GEP provides very accurate variational approximations to the ground-state energy of simple quantum systems; our results show that it also provides accurate excited-state energies. We also show how this method is related to the first-order Bargmann-Voros coherent-state representation of quantum mechanics, and that our results improve upon those derived from the latter method. \textcopyright{} 1996 The American Physical Society.
We apply the variational method to the ground-state and first-excited-state energies of a Gaussian well and show that the expansion of the approximate energy when the potential strength approaches zero does not agree with the exact result. We argue that a recent semiclassical approach to tunneling through a Gaussian barrier does not agree with an accurate calculation of the transmission probability. We point out the advantages of converting the Schrödinger equation into a dimensionless eigenvalue equation.
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.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
Abstract: In this work, we present a detailed study of a one-dimensional Schrödinger equation in the presence of quantum Gaussian well interaction. Further, we investigate the approximate solutions by using the harmonic oscillator approximation, variational principle, four-parameter potential fitting and numerical solution using the finite-difference method. The parabolic approximation yields an excellent energy value compared with the numerical solution of the Gaussian system only for the ground state, while for the excited states, it provides a higher approximation. Also, the analytical bound-state energies of the four-parameter potential under the framework of the Nikiforov-Uvarov (NU) method have been used after getting the suitable values of the potential parameters using numerical fitting. The present results of the system states are found to be in high agreement with the well-known numerical results of the Gaussian potential. Keywords: Gaussian potential, One-dimensional Schrödinger equation, Nikiforov- Uvarov (NU) method, Four-parameter potential. PACS: 03.65.−w; 02.90.+p; 12.39.Pn.
Generalized eigenvalue problems (GEPs) play an important role in the variety of fields including engineering, machine learning, and quantum chemistry. Especially, many problems in these fields can be reduced to finding the minimum or maximum eigenvalue of GEPs. One of the key problems to handle GEPs is that the memory usage and computational complexity explode as the size of the system of interest grows. This paper aims at extending sequential quantum optimizers for GEPs. Sequential quantum optimizers are a family of algorithms that iteratively solve the analytical optimization of single-qubit gates in a coordinate descent manner. The contribution of this paper is as follows. First, we formulate the GEP as the minimization or maximization problem of the fractional form of the expectations of two Hermitians. We then show that the fractional objective function can be analytically minimized or maximized with respect to a single-qubit gate by solving a GEP of a $4\ifmmode\times\else\texttimes\fi{}4$ matrix. Second, we show that a system of linear equations characterized by a positive-definite Hermitian can be formulated as a GEP and thus be attacked using the proposed method. Finally, we demonstrate two applications to important engineering problems formulated with the finite-element method. Through the demonstration, we have the following bonus finding; a problem having a real-valued solution can be solved more effectively using quantum gates generating a complex-valued state vector, which demonstrates the effectiveness of the proposed method.
A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or upper bounds. A semiclassical interpretation of the generic formula obtained for the eigenvalues supports a new definition of the effective particle mass used in solid state physics. An analytical toy model with a Gaussian dependence in the momentum is studied in order to check the validity of the method.