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Continuity equation and wavefunction properties

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

Abstract This chapter discusses the continuity equation and continuity properties of the wavefunction and its spatial derivatives. In addition, it discusses other theorems which are useful for calculations, such as the variational theorem.
DOI
10.1093/acprof:oso/9780198749349.003.0007

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This chapter examines the interpretation of the wavefunction, and specifically what it reveals about the location of a particle. In quantum mechanics, all the properties of a system are expressed in terms of a wavefunction, which is obtained by solving the equation proposed by Erwin Schrödinger. Indeed, wavefunctions provide the essential foundation for understanding the properties of electrons in atoms and molecules, and are central to explanations in chemistry. The chapter then considers how, according to the Born interpretation, the probability density at a point is proportional to the square of the wavefunction at that point. A wavefunction is normalized if the integral over all space of its square modulus is equal to 1. Ultimately, a wavefunction must be single-valued, continuous, not infinite over a finite region of space, and have a continuous slope. The quantization of energy stems from the constraints that an acceptable wavefunction must satisfy.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
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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.