JCE Software, 4D1: Schroedinger.m

Schroedinger.m: A Mathematic Package for Solving the Time-Independent Schrödingier Equation

John C. Hanse
Southwest State University, Marshall, MN 56258

Note:
This program is included in the Advanced Chemistry Collection (SP-28).

To Order Advanced Chemistry Collection

Schroedinger.m enables the user of Mathematica, a software system available from Wolfram Research, Inc. (1), to define a potential and determine bound-state energy eigenvalues for a one-dimensional Hamiltonian. Students new to both quantum mechanics and Mathematica find Schroedinger.m easy to use. It can be used in a junior-level physical chemistry course to provide examples or to provide the means for students in such a course to carry out numerical experiments. The commands defined in the package require only elementary knowledge of Mathematica. However, the more Mathematica one knows, the further one can go with the applications. Knowing even a very few basic Mathematica functions enables a student to use solutions from this package to do some fairly sophisticated computations with one-dimensional wave functions.
Schroedinger.m defines commands in Mathematica to set the potential, solve the Schrödinger equation to obtain both energies and wave functions, display the solutions graphically, and calculate certain integrals involving the wave functions. It is limited to one-dimensional bound-state problems with the boundary conditions (a) = (b) = 0; it cannot be applied to other one-dimensional problems with periodic boundary conditions. It is well suited to solving the one-dimensional R-dependent Schrödinger equation that describes most diatomic molecular potentials.
A student does not need to know the details of Schroedinger.m in order to use it. However, it helps to be able to understand how the computer represents the solutions, in order to move beyond the simplest applications or to understand the limitations of the package. The numerical method used is described in detail in the program documentation.
The user has direct control over parameters that define the extent of the grid, its spacing, and the range of solutions. In many cases, however, it is not necessary to worry about these numerical parameters. Simply using built-in defaults gives an adequate answer. However, to apply the method to an unusual or complex problem, it may be necessary to carefully consider how to set these parameters. Fortunately, the methods for doing so are fairly straightforward.

This screen from the first example notebook shows the n=4 and n=10 wave functions calculated by Schrödinger.m.
Hardware and Software Requirements
Programs in Series D of JCE: Software require Microsoft Windows version 3.1 or later and associated hardware as defined by Microsoft. We recommend an IBM-PC or compatible computer with a 80386 or higher processor, a minimum of 4 MB of memory, a hard disk, one floppy disk drive to install the software, a mouse, and a Windows-compatible graphics card, such as an IBM VGA or Super VGA adapter with a compatible color monitor. DOS 5.0 or later is highly recommended. The issue is supplied on 3.5-in. disks. Additional requirements are noted below:
Schroedinger.m requires a version of the Mathematica program with notebook-type interface.

Literature Cited

Mathematica, Wolfram Research, Inc., P.O. Box 6059, Champaign, IL 61820-6059.

First Published: July 1996
Citation: Hanse, J. C. . Schroedinger.m: A Mathematic Package for Solving the Time-Independent Schrödingier Equation J. Chem. Educ. Software 4D1
Keywords: Lecture Aid; Computer Room; Physical; Quantum chemistry; Mathematic

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Last Updated: July 19, 2001
Created: December 3, 1996 Created by: J. L. Holmes
Comments to: jceonline@chem.wisc.edu

© 1997 Division of Chemical Education, Inc., American Chemical Society. All rights reserved.



Schroedinger.m: A Mathematic Package for Solving the Time-Independent Schrödingier Equation
John C. Hanse Southwest State University, Marshall, MN 56258

Note:	This program is included in the Advanced Chemistry Collection (SP-28). To Order Advanced Chemistry Collection

Schroedinger.m enables the user of Mathematica, a software system available from Wolfram Research, Inc. (1), to define a potential and determine bound-state energy eigenvalues for a one-dimensional Hamiltonian. Students new to both quantum mechanics and Mathematica find Schroedinger.m easy to use. It can be used in a junior-level physical chemistry course to provide examples or to provide the means for students in such a course to carry out numerical experiments. The commands defined in the package require only elementary knowledge of Mathematica. However, the more Mathematica one knows, the further one can go with the applications. Knowing even a very few basic Mathematica functions enables a student to use solutions from this package to do some fairly sophisticated computations with one-dimensional wave functions. Schroedinger.m defines commands in Mathematica to set the potential, solve the Schrödinger equation to obtain both energies and wave functions, display the solutions graphically, and calculate certain integrals involving the wave functions. It is limited to one-dimensional bound-state problems with the boundary conditions (a) = (b) = 0; it cannot be applied to other one-dimensional problems with periodic boundary conditions. It is well suited to solving the one-dimensional R-dependent Schrödinger equation that describes most diatomic molecular potentials. A student does not need to know the details of Schroedinger.m in order to use it. However, it helps to be able to understand how the computer represents the solutions, in order to move beyond the simplest applications or to understand the limitations of the package. The numerical method used is described in detail in the program documentation. The user has direct control over parameters that define the extent of the grid, its spacing, and the range of solutions. In many cases, however, it is not necessary to worry about these numerical parameters. Simply using built-in defaults gives an adequate answer. However, to apply the method to an unusual or complex problem, it may be necessary to carefully consider how to set these parameters. Fortunately, the methods for doing so are fairly straightforward. This screen from the first example notebook shows the n=4 and n=10 wave functions calculated by Schrödinger.m. Hardware and Software Requirements Programs in Series D of JCE: Software require Microsoft Windows version 3.1 or later and associated hardware as defined by Microsoft. We recommend an IBM-PC or compatible computer with a 80386 or higher processor, a minimum of 4 MB of memory, a hard disk, one floppy disk drive to install the software, a mouse, and a Windows-compatible graphics card, such as an IBM VGA or Super VGA adapter with a compatible color monitor. DOS 5.0 or later is highly recommended. The issue is supplied on 3.5-in. disks. Additional requirements are noted below: Schroedinger.m requires a version of the Mathematica program with notebook-type interface. Literature Cited Mathematica, Wolfram Research, Inc., P.O. Box 6059, Champaign, IL 61820-6059.


News \| Issues \| CD-ROM / Video \| Find It! \| Technical Support \| For Authors
JCE Online \| Journal \| Software \| Internet \| Happenings \| About JCE \| Contact JCE

Last Updated: July 19, 2001 Created: December 3, 1996	Created by: J. L. Holmes Comments to: jceonline@chem.wisc.edu

© 1997 Division of Chemical Education, Inc., American Chemical Society. All rights reserved.