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Physical Review Letters

Variational principle for confined quantum systems

Kenneth R. Brownstein

Published Aug 30, 1993• 38 citations
Publisher

Abstract

A new variational principle for the energy eigenvalues of a confined quantum system is presented. Whereas the exact wave function \ensuremath{\Psi} must vanish on the bounding surface of the region, the trial function \ensuremath{\psi} in this principle need not obey any specific boundary condition. Calculationally, the method is similar to the conventional variational method except that kinetic energy turns out to be a weighted average of -F\ensuremath{\psi}${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\psi} d\ensuremath{\tau} and F(\ensuremath{\nabla}\ensuremath{\psi})\ensuremath{\cdot}(\ensuremath{\nabla}\ensuremath{\psi})d\ensuremath{\tau} in the ratio 2:-1. Although the principle is not a definite (minimum) one, good results are obtained in several examples.

DOI
10.1103/physrevlett.71.1427

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A new variational principle for the energy eigenvalues of a confined quantum system is presented. Whereas the exact wave function \ensuremath{\Psi} must vanish on the bounding surface of the region, the trial function \ensuremath{\psi} in this principle need not obey any specific boundary condition. Calculationally, the method is similar to the conventional variational method except that kinetic energy turns out to be a weighted average of -F\ensuremath{\psi}\ensuremath∇2{\mathrm{\ensuremath{\nabla}}}^{2}\ensuremath∇2\ensuremath{\psi} d\ensuremath{\tau} and F(\ensuremath{\nabla}\ensuremath{\psi})\ensuremath{\cdot}(\ensuremath{\nabla}\ensuremath{\psi})d\ensuremath{\tau} in the ratio 2:-1. Although the principle is not a definite (minimum) one, good results are obtained in several examples.
A new variational principle for the bound state energy eigenvalues of a quantum system is presented. In it, not only ∇ψ(r) but also ψ(r) itself can be discontinuous across a surface. That is, the trial ψ(r) can have both a kink and a jump across a surface. This allows the use of different basis functions in different regions of space without having to impose any matching conditions across the surface. All parameters appearing in the principle can be variational parameters; none are needed to help invoke any continuity conditions on the trial function. Although the principle is not a definite (minimum) one, good results are obtained in several examples.
The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wave functions have been assumed as the product of the ‘‘free’’ solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method. This might constitute a complementary exercise for undergraduate students since usual quantum mechanics textbooks do not explore the use of the variational method to study confined systems.
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.
The extension of the Rayleigh-Ritz variational principle to ensemble states ρw≡∑kwk|Ψk⟩⟨Ψk| with fixed weights wk lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state ρw as well as the individual eigenstates |Ψk⟩ and eigenenergies Ek . To be more specific, we derive linear bounds d−ΔEw≤ΔQ≤d+ΔEw on the errors ΔQ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of d±≡d±(Q)(w,E) provide valuable insights into the optimal choice of the auxiliary weights wk in practical applications.
Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.