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International Journal of Quantum Chemistry

Linear integrability of wave functions

Toshikatsu Koga, Ajit J. Thakkar

Published Aug 1, 1988• 9 citations
Publisher

Abstract

Abstract Finiteness is usually imposed as a condition for physical admissibility of a wave function. Examination of this condition in both position‐ and momentum‐space shows that finiteness of the wave function at the origin in one space implies “linear” integrability in the reciprocal space, except in some pathological cases. Some implications of this result are discussed.

DOI
10.1002/qua.560340203

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Abstract Finiteness is usually imposed as a condition for physical admissibility of a wave function. Examination of this condition in both position‐ and momentum‐space shows that finiteness of the wave function at the origin in one space implies “linear” integrability in the reciprocal space, except in some pathological cases. Some implications of this result are discussed.
Abstract It is shown that the dispute about the conditions for physical admissibility of Schrödinger wave functions may be settled if the following two questions are clarified: (a) Are these condittions rooted in the foundations of quantum mechanics or invoked only for particular physical models? (b) Do we consider either the position‐space wave function or the momentum‐space wave function? Both the case of closed states and scattering states are considered. Some general remarks about these conditions are made in the Conclusion.
Based on the Fourier transform of the position–space Schrödinger equation in a finite domain, a set of relations is derived among position moments linearly averaged over the wave function in that domain. The present results add nontrivial information for the structure of the Schrödinger equation, including the known relations for an infinite domain. A special case provides a regional form of the zero momentum energy expression, which measures the accuracy of wave functions in some specific region.
Admissibility conditions for wave functions are derived from general quantum-mechanical principles. The conditions apply in general curvilinear coordinates to functions that describe general N-particle systems. They restrict the behavior of wave functions so severely as to suggest that the Hilbert space of ordinary quantum mechanics cannot be enlarged unless the fundamental principles of quantum mechanics are modified. One of the conditions implies that all particles must be either fermions or bosons, i.e., that no particles obeying intermediate statistics can exist.
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
We consider a Hamiltonian H^\hat HH^ with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of H^\hat HH^ in a finite basis of square-integrable functions. For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues, except for the eigenenergy associated with that pseudostate. This feature was observed for the Coulomb continuum represented in a Laguerre basis [M. McGovern et al., Phys. Rev. A 79, 042707 (2009)] and later explained using special properties of the Laguerre functions [I. B. Abdurakhmanov et al., J. Phys. B 44, 075204 (2011)]. We establish that a sufficient condition for the zero-overlap condition to occur is that the image space of the operator Q^H^P^\hat Q \hat H \hat PQ^​H^P^, where P^\hat PP^ is the projection operator onto the subspace spanned by the basis and Q^=1^−P^\hat Q = \hat 1 - \hat PQ^​=1^−P^ its complement, has dimension one. We show that the condition is met for the one-dimensional free-particle problem by a basis of harmonic oscillator eigenstates and by a minimal set of simple momentum-space wave functions, and for the Coulomb problem by a Laguerre basis, thus offering an alternative proof for the latter case. The zero-overlap condition ensures that in, e.g., an ionizing collision or laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded system wave function onto eigenstates of $ \hat H $ are asymptotically stable.
A simple example shows that a standard tacit hypothesis concerning quadratic integrability in momentum space is not in general legitimate, if only quadratic integrability in configuration space is assumed. Inasmuch as only the latter assumption is justified, there arises the need for criteria which prevent the occurrence of this difficulty. Such a criterion is developed in terms of the qualitative behavior of the potential. The conditions on the potential are of such a nature as to be satisfied in all cases which seem to be reasonable.