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Spherically Symmetric Potentials

Owen L. de Lange, R. E. Raab

Published Nov 28, 1991• 2 citations
Publisher

Abstract

Abstract A number of quantum-mechanical problems involve a potential that is spherically symmetric, at least to a first approximation. For such potentials one can consider a ‘onedimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (see (5.26)), and in this chapter we use operator methods to treat certain important cases for which exact solutions are possible.

DOI
10.1093/oso/9780198539612.003.0005

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Abstract A number of quantum-mechanical problems involve a potential that is spherically symmetric, at least to a first approximation. For such potentials one can consider a ‘onedimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (see (5.26)), and in this chapter we use operator methods to treat certain important cases for which exact solutions are possible.
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).
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
Abstract In Chapter 4 we considered a ‘twodimensional’ problem in which L2 and Lz constitute a c.s.c.o. and we derived shift operators for the angular momentum quantum numbers l and m in the eigenkets |lm> of these observables. In Chapter 5 we discussed a ‘one-dimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (eqn (5.25)). Here we use results from Chapters 4 and 5 to construct shift operators for the eigenkets of the three dimensional problem (5.11)–(5.13).
Abstract In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.
The cylindrically and the spherically symmetric harmonic oscillators are considered by means of matrix techniques in a basis that is appropriate for each symmetry. For example, the spherically symmetric oscillator is described by the radial and angular momentum quantum numbers. The energy eigenvalues and the matrix elements of the promotion-demotion operators are found in each case. By expressing the operators in curvilinear coordinates, the promotion-demotion operators are used to deduce the form of the corresponding wave functions.