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American Journal of Physics

One-dimensional hydrogen atom in an infinite square well

I. Richard Lapidus

Published Jun 1, 1982• 10 citations
Publisher

Abstract

The effect of boundary conditions on solutions to the Schrödinger Equation is demonstrated.
DOI
10.1119/1.12805

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We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)
It is shown that the condition of finiteness at the origin, usually imposed on solutions to the nonrelativistic Schrödinger equation for the hydrogen atom (and some other systems) is not necessary. The quadratically integrable l = 0 “anomalous” solution which is usually excluded in this fashion is not a solution to the Schrödinger equation at the origin, and consequently does not need to be excluded by a boundary or other type condition. The relativistic generalization through the Dirac equation of this anomalous solution is not quadratically integrable. The solution generalized according to the relativistic Schrödinger equation is quadratically integrable and obeys the equation at the origin. Consequently, a boundary or other type condition may still be needed in a general relativistic theory.