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Journal of Chemical Education

Exploring the Rayleigh-Ritz Variational Principle

Emili Besalú, Josep Martí

Published Jan 1, 1998• 9 citations
Publisher

Abstract

The particle-in-a-box (PIB) system is used as a simple example in order to illustrate the basis of the quantum variational principle. After the general exposition of the variational principle demonstration, a quadratic polynomial trial function is applied to the PIB hamiltonian and the different steps of the demonstration are followed and reproduced for this particular case.

DOI
10.1021/ed075p105

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arXiv:1605.09374•May 29, 2016•0 citations
The particle-in-a-box (PIB) system is used as a simple example in order to illustrate the basis of the quantum variational principle. After the general exposition of the variational principle demonstration, a quadratic polynomial trial function is applied to the PIB hamiltonian and the different steps of the demonstration are followed and reproduced for this particular case.
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.
The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The x n ( a − x ) n and x n ( a /2 − x )( a − x ) n polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a ] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.
The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues E1{E}_{1}E1​\ensuremath{\le}E2{E}_{2}E2​\ensuremath{\le}...\ensuremath{\le}EM{E}_{M}EM​ of a Hamiltonian H, and given M real numbers w1{w}_{1}w1​\ensuremath{\ge}w2{w}_{2}w2​\ensuremath{\ge}...\ensuremath{\ge}wM{w}_{M}wM​>0, an upper bound for the weighted sum w1{w}_{1}w1​E1{E}_{1}E1​ +w2{w}_{2}w2​E2{E}_{2}E2​+...+wM{w}_{M}wM​EM{E}_{M}EM​ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (w1{w}_{1}w1​=w2{w}_{2}w2​=...=wM{w}_{M}wM​). Applications of the generalized principle are discussed.