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Birkhäuser Boston eBooks

Dispersive Waves and the Schrödinger Equation

Jeffery Cooper

Published Jan 1, 1998• 0 citations
Publisher

Abstract

After a short section in which we discuss the method of stationary phase, we treat dispersive waves. Then we turn to quantum mechanics, using the Fourier transform to establish the Heisenberg uncertainty principle. Finally we derive the Schrödinger equation. We solve the free Schrödinger equation and the Schrödinger equation with a square well potential. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-1754-1_7

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After a short section in which we discuss the method of stationary phase, we treat dispersive waves. Then we turn to quantum mechanics, using the Fourier transform to establish the Heisenberg uncertainty principle. Finally we derive the Schrödinger equation. We solve the free Schrödinger equation and the Schrödinger equation with a square well potential. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
To construct the Schrödinger processes we must prescribe a state preparation or a pair of the initial and terminal distributions. In the first case we solve the equation of motion (or the Schrödinger equation) and apply the transformation of Markov processes by a multiplicative functional. In the second case we employ the stochastic variational principle. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract A discussion of the physical meaning of the Schrödinger wave equation can not only constitute an exciting introduction to some of the more abstract ideas of quantum mechanics, but serves more generally as a useful demonstration of the application of mathematics to modern physics. This frontline uses physical concepts and mathematical techniques that would be accessible to a sufficiently interested secondary school student, in order to derive the simplest, one-dimensional case of the time-independent Schrödinger equation: the derivation relies only upon aspects of Newtonian/wave mechanics, quantum theory, complex numbers and calculus that would be covered in an advanced secondary school syllabus.