Department of Mathematics

IDEA: Internet Differential Equations Activities

Solving Differential Equations of Quantum Mechanics

Differential Equations of Quantum Mechanics

In general, the differential equations (DE) of quantum mechanics are special cases of eigenvalue problems. These pages offer an introduction to the mathematics of such problems for students of quantum chemistry or quantum physics. Several illustrative examples are given to show how the problems are solved using various methods. Exercises, some with solutions and some without, are presented to give students practice with solving typical quantum mechanics problems.

Parallel to the DE eigenvalue problems is an equivalent class of eigenvalue problems expressed in linear algebra (LA). Often the connection between a given DE eigenvalue problem and its LA equivalent is given.

Map of the site

References   Introduction and Background
...Quantum Mechanics Links
Eigenvalue Problems from Classical Physics
...Mass on a Spring and Electrical Analog
...Simple Pendulum
...Sound Waves
...Vibrating String   		Orthonormal Basis Functions
...Primary example: Fourier Series
...Example: Legendre Series
...Example: Hermite Polynomials
Sturm-Liouville Theory
...General S-L eigenvalue problem
...Example: Legendre's equation   		Numerical Integration Method for Eigenvalues
...Two Point Boundary Value Problems
Quantum Mechanical Systems. I. Analytical Solutions
...Particle in a Box
...Particle in a Finite Well
...Simple Harmonic Oscillator
...Rigid Rotator in 3 dimensions
...Hydrogen Atom		 Quantum Mechanical Systems. II. Numerical Solutions
...Particle in a Box, PIB
...PIB with central well
...PIB with spring
Quantum Mechanical Systems. III. LA solutions
...Variation Principle
.. Particle in a Box (PIB) via polnomial Basis
.. PIB with central well via Fourier Basis		 Time dependent Schroedinger Equation
(under construction) see a preview

Created or up-dated 08/03/99   by R.D. Poshusta
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The IDEA site is undergoing renovations, in conjunction with the renewed activities of CODEE (as described below). Thank you for your patience.
CODEE, the Consortium for Differential Equations Experiments, has been revitalized. CODEE was quite active in the 1990s in spreading differential equations activities, information, and software tools. In particular, CODEE formed the organization for the ODE Architect software. Recently, an NSF project headed by Darryl Young of Harvey Mudd College has reinvigorated CODEE.

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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.
Copyright © 1996-2008 Thomas LoFaro and Kevin Cooper