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Pure and Applied Mathematics Journal

Derivation of Schrõdinger Equation from a Variational Principle

Sami H. Altoum

Published Jan 1, 2013• 0 citations
Publisher

Abstract

The aim of this research is to derive Schrödinger equation from calculus of variations (variational principle), so we use the methodology of calculus of variations. The variational principle one of great scientific significance as they provide a unified approach to various mathematical and physical problems and yield fundamental exploratory ideas.

DOI
10.11648/j.pamj.20130204.12

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The aim of this research is to derive Schrödinger equation from calculus of variations (variational principle), so we use the methodology of calculus of variations. The variational principle one of great scientific significance as they provide a unified approach to various mathematical and physical problems and yield fundamental exploratory ideas.
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Research has been carried out to determine the physical meaning and derivation of the Schrodinger equation with the literature study research method. Based on the research results obtained through data processing in the form of information obtained from various literature sources, it can be concluded that the general solution of the Schrodinger equation is a wave function. In general, based on mathematical calculations by applying the technique of solving differential equations to the Schrodinger equation, it can be seen that this equation has many solutions, but not all of these solutions can be said to be functions that meet the wave function criteria. This wave function must be able to interpret the real physical criteria of a particle.
Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
S u m m a r y . A rigorous theory of variational principles in quantum mechanics is formulated. This result is obtained by generalizing the standard theory of the inverse problem of the variational cahmlus, according to new concepts of differentiability on complex Banach spaces. Our theory provides sound mathematical foundations to all known variational principles of quantum mechanics and a powerful tool to find new ones. As a simple application, we derive rigorously two variational principles for the time-dependent SchrSdinger equation: the conventional one, requiring both an initial and a final condition, and a new one, requiring an initial condition only.
Abstract It has been demonstrated in Chapter 5 that Schrodinger’s original derivation of the wave equation can be generalized to provide a quantum description of an atom in a molecule. His derivation uses the calculus of variations as applied to an energy functional of Ψ and its extension to an atom in a molecule entails two essential steps: the introduction of a particular type of operator through its identification with the variation of the state function and a variation of the surface bounding the atomic subsystem.