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American Journal of Physics

Continuum computer solutions of the Schrödinger equation

Rodney C. Greenhow, J.A.D. Matthew

Published Jul 1, 1992• 7 citations
Publisher

Abstract

Computer integration of Schrödinger’s time-independent wave equation with graphical display of solutions provides a valuable teaching aid, and is much used in the study of bound states. Much less attention has been paid to continuum solutions although these provide a richer set of phenomena. Programs are described that integrate the one-dimensional equation for potential steps and barriers, and model impedance matching, barrier penetration, and field emission. The radial equation for spherically symmetric potentials is also integrated and graphical demonstrations of scattering phase shifts, a simple model of the Ramsauer effect, and resonance scattering are provided. The programs have been used by undergraduates to investigate resonance scattering from spherical step and Woods–Saxon potentials, and a new power law relating resonance cross-section breadths and energies has been established.

DOI
10.1119/1.17121

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Computer integration of Schrödinger’s time-independent wave equation with graphical display of solutions provides a valuable teaching aid, and is much used in the study of bound states. Much less attention has been paid to continuum solutions although these provide a richer set of phenomena. Programs are described that integrate the one-dimensional equation for potential steps and barriers, and model impedance matching, barrier penetration, and field emission. The radial equation for spherically symmetric potentials is also integrated and graphical demonstrations of scattering phase shifts, a simple model of the Ramsauer effect, and resonance scattering are provided. The programs have been used by undergraduates to investigate resonance scattering from spherical step and Woods–Saxon potentials, and a new power law relating resonance cross-section breadths and energies has been established.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
An important technique for teaching introductory wave mechanics is provided by the numerical calculation of stationary state solutions of the Schrödinger equation. Realistic problems in wave mechanics that do not require sophisticated computer techniques are very valuable for the instruction of undergraduates. In the central-field model for atoms, the calculation of electron states reduces to the solution of a one-dimensional differential equation whose numerical solutions are no more difficult to obtain than those for a finite square well. A method used by our students is discussed in this paper for the calculation of the electron wavefunctions in a central-field potential. By working this problem, students obtain meaningful results that give them insight into the theory of atomic structure. Results of their experience are included, and possibilities of extending this example are discussed.
Abstract We discuss the procedure of solving the time-dependent Schrödinger equation via a separation ansatz that leads to the time-independent Schrödinger equation. The resulting equation is in the form of an eigenvalue problem for the Hamiltonian, whose solutions are the stationary states. We derive and discuss the bound states and corresponding energy levels for several one-dimensional potentials of interest, including the finite and infinite potential well, the Dirac-delta potential, and the square double-well modelling the spatial configuration of the ammonia molecule. We then turn to the scattered solutions and discuss reflection from and transmission through a finite potential barrier, thus encountering the tunnel effect, which we in turn formulate more generally using the Gamow factor. This leads us to the interesting phenomenon of tunnelling resonances