Quantum Mechanics Simulations

• John Hiller of University of Minnesota at Duluth
• Ian Johnston, University of Sydney
• Daniel Styer, Oberlin College

• BOUND1D ( One Dimensional Bound States ) written by Ian Johnston is a tool which allows you to explore energy eigenfunctions for an electron in various potential wells, which can be square, parabolic, ramped, asymmetric, double or Coulombic. The first part of the program deals with finding the eigenvalues and eigenfunctions of different wells. You may find them yourself, using a "hunt and shoot" method, or else the program will compute the eigenvalues automatically, by counting the number of nodes to determine where theeigenvalues occur. The second part of the program looks at properties of eigenfunctions normalization, orthogonality and the evaluation of many kinds of overlap integrals. The third part examines time development of general states made up of a superposition of over. bound state eigenfunctions. Facility is provided for you to incorporate your own procedures to specify different potential wells or different overlap integrals.

• SCATTR1D ( Stationary Scattering States in One Dimension ) written by Jihn Hiller solves the time-independent Schrodinger equation for stationary scattering states in one-dimensional potentials. The wave function is displayed in a variety of ways, and the transmission and reflection probabilities are computed. The probabilities may be displayed as functions of energy. The computations are done by numerically integrating the Schrodinger equation from the region of the transmitted wave, where the wave function is known up to some overal normalization and phase, to the region of the incident wave. There the reflected and incident waves are separated. The potential is assumed to be zero in the incident region and constant in the transmitted region.

• QMTIME ( Quantum Mechanical Time Development ) written by Dan Styer simulates quantal time development in one dimension. A variety of initial wave packets (Gaussian, Lorentzian, etc.) can evolve in time under the influence of a variety of potential energyfunctions (step, ramp, square well, harmonic oscillator, etc.) with or without an external driving force. A novel visualization technique simultaneously displays the magnitude and phase of complex-valued wavefunctions. Either position-space or momentum-space wave functions, or both, can be shown. The program is particularly effective in demonstrating the classical limit of quantum mechanics.

• LATCE1D ( Electron States in One -Dimensional Lattice) written by Ian Johnston is a tool which allows you to explore energy eigenfunctions for an electron in a lattice made up of a number of simple potential wells (up to twelve), which can be square, parabolic or Coulombic. You may find the eigenvalues yourself, using a "hunt and shoot" method, or allow the program to compute them automatically. You can firstly explore regular lattices, where all wells are the same and spaced at regular intervals. These will demonstrate many of the properties of regular crystals, particularly the existence of energy bands. Secondly you can change the width, depth or spacing of any of the wells, which will mimic the effect of impurities or other irregularities in a crystal. Lastly you can apply an external electric across the lattice. Facility is provided for you to incorporate your own procedures to calculate wells, lattice arrangements or external fields of their own choosing.

• BOUND3D (Bound States in Three Dimensionsis) wriiten by Ian Johnston a tool which allows you to explore energy eigenfunctions for an particle in a spherically symmetric potential well, which can be square, parabolic, Coulombic, or several other shapes of importance in molecular or nuclear applications. The first part of the program deals with finding the eigenvalues and eigenfunctions of different wells, assuming that the angular part of the wave functions are spherical harmonics. You may find them yourself for a given angular momentum quantum number using a "hunt and shoot" method, or else the program will compute the eigenvalues automatically, by counting the number of nodes to determine where the eigenvalues occur. The second part of the program looks at properties of eigenfunctions normalization, orthogonality and the evaluation of many kinds of overlap integrals. Facility is provided for you to incorporate your own procedures to specify different potential wells or different overlap integrals.

• IDENT: ( Identical Particles in Quantum Mechanics) written by Dan Styer shows the probability density associated with the symmetrized, antisymmetrized, or nonsymmetrized wave functions of two noninteracting particles moving in a one-dimensional infinite square well. It is particularly valuable for demonstrating the effective interaction of noninteracting identical particles due to interchange symmetry requirements.

• SCATTR3D ( Scattering in Three Dimensions ) wriiten by John Hiller performs a partial-wave analysis of scattering from a spherically symmetric potential. Radial and three-dimensional wave functions are displayed, as are phase shifts, and differential and total cross sections. The analysis employs an expansion in the natural angular momentum basis for the scattering wave function. The radial wave functions are computed numerically; outside the region where the potential is important they reduce to a linear combination of Bessel functions which asymptotically differs from the free radial wave function by only a phase. Knowledge of these phase shifts for the dominant values of angular momentum is used to approximate the cross sections.

• CYLSYM (Cyllindrically Symmetric Potentials) wriitten by John Hiller,solves the time-independent Schrodinger equation Hu=Eu in the case of a cylindrically symmetric potential for the lowest state of a chosen parity and magnetic quantum number. The method of solution is based on evolution in imaginary time, which converges to the state of the lowest energy that has the symmetry of the initial guess. The Alternating Direction Implicit method is used to solve a diffusion equation given by HU=-hbar dU/dt, where H is the Hamiltonian that appears in the Schrodinger equation. At large times, U is nearly proportional to the lowest eigenfunction of H, and the expectation value <H>=<U|H|U>/<U|U> is an estimate for the associated eigenenergy.