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Physica Scripta

Analytic wave functions of quasistable quantum systems with arbitrary angular momentum

W. van Dijk

Published Apr 20, 2026• 0 citations
Publisher

Abstract

Abstract We derive the exact wave function for the decaying quasistable quantum system with arbitrary orbital angular momentum. We show that this wave function can be expressed as an expansion in terms of Moshinsky functions, thereby generalizing the result for the known S -wave case to all partial waves. As a simple illustrative example, we consider a δ -shell potential, which allows us to describe the time evolution of the wave function and the associated non-decay probabilities in a transparent manner. At long times, the nondecay probability exhibits a power-law dependence on the orbital angular momentum, decreasing as t −(2 ℓ +3) , where ℓ denotes the orbital angular momentum quantum number.

DOI
10.1088/1402-4896/ae6204

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Abstract We derive the exact wave function for the decaying quasistable quantum system with arbitrary orbital angular momentum. We show that this wave function can be expressed as an expansion in terms of Moshinsky functions, thereby generalizing the result for the known S -wave case to all partial waves. As a simple illustrative example, we consider a δ -shell potential, which allows us to describe the time evolution of the wave function and the associated non-decay probabilities in a transparent manner. At long times, the nondecay probability exhibits a power-law dependence on the orbital angular momentum, decreasing as t −(2 ℓ +3) , where ℓ denotes the orbital angular momentum quantum number.
The manner in which probability amplitudes of paths sum up to form wave functions of orbital angular momentum eigenstates is described. Using a generalization of stationary-phase analysis, distributions are derived that provide a measure of how paths contribute towards any given eigenstate. In the limit of long travel-time, these distributions turn out to be real-valued, non-negative functions of a momentum variable that describes classical travel between the endpoints of a path (with the paths explicitly including nonclassical ones, described in terms of elastica). The distributions are functions of both this characteristic momentum as well as a polar angle that provides a tilt, relative to the z-axis of the chosen coordinate system, of the geodesic that connects the endpoints. The resulting description provides a replacement for the well-known "vector model" for describing orbital angular momentum, and importantly, it includes treatment of the case when the quantum number ℓ\ellℓ is zero (i.e., s-states).
Recently we outlined an analytical method of solving the time-dependent Schr\"odinger equation for a model which simulates a decaying quantum system such as an $\ensuremath{\alpha}$-decaying nucleus. A particle in the model is initially confined around the origin and leaks out, tunneling through a potential barrier. The solution can be expressed as a linear combination of the Moshinsky functions, each of which is associated with a pole of the scattering S\mathbf{S}S matrix of the model. In this paper we give a full account of the method with a few explicit examples. We examine deviations from the exponential decay law at very large times. We comment on a recent controversy regarding the t dependence of the survival and nonescape probabilities when t is very large.
Several wave-function approximations describing spontaneous parametric down-conversion can be found in the literature. Basically all cases are derived from the standard Hamiltonian for parametric down-conversion. Most frequently, particular cases describing collinear or paraxial approximations are described. This work presents a wave function in compact form, valid for all cases of single photon-pair conversion (Type I or Type II), for all angles allowed by the phase-matching conditions and for all orbital angular momentum values lll. Examples are given of coincidence structures to be expected for signal and idler photons. Partial transfer of orbital angular momentum from the pump laser to the photon pair is discussed. Some hypothesis for the decay channels of the nontransferred part of the orbital angular momentum is made.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
A simple algebraic approach based on the well known angular momentum SU (2) algebra is presented to describe 1D systems for arbitrary potentials. The approach is based on the dimension increase of the 1D harmonic oscillator space through the addition of a scalar boson, keeping constant the total number of bosons. In this new space the realization of the coordinate and momentum correspond to components of the angular momentum algebra, which in turn define the coordinate and momentum representation bases. This remarkable result allows the Hamiltonian matrix representationto be expressed in terms of a diagonal matrix characterizing the potential we are considering. The solutions are obtained as an expansion of harmonic oscillator functions by purely algebraic means. As an example of our approach the Morse and asymmetric double Morse potentials are considered.