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arXiv:2108.01981

Fall of Quantum Particle to the Center: Exact results

Michael I. Tribelsky

Published Aug 4, 2021• 0 citations
arXivHTMLPDF

Abstract

The fall of a particle to the center of a singular potential U(r) is one of a few fundamental problems of quantum mechanics. Nonetheless, its solution is not complete yet. The known results just indicate that if U(r) decays fast enough at r tends to zero, the spectrum of the Schrodinger equation is not bounded from below. However, the wave functions of the problem are singular at r = 0 and do not admit the limiting transition to the wave function of the ground state. Therefore, the unboundedness of the spectrum is only the necessary condition. To prove that a quantum particle indeed can fall to the center, a wave function describing the fall should be obtained explicitly. This is done in the present paper. Specifically, an exact solution of the time-dependent Schrodinger equation describing the fall to the center is obtained and analyzed. The law describing the collapse to a single point of the region of the wave function localization is obtained explicitly, as well as the temporal dependences of the average kinetic and potential energy of the falling particle and its momentum. It is shown that the known necessary conditions for the particle to fall simultaneously are sufficient.

DOI
10.48550/arxiv.2108.01981

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The fall of a particle to the center of a singular potential U(r) is one of a few fundamental problems of quantum mechanics. Nonetheless, its solution is not complete yet. The known results just indicate that if U(r) decays fast enough at r tends to zero, the spectrum of the Schrodinger equation is not bounded from below. However, the wave functions of the problem are singular at r = 0 and do not admit the limiting transition to the wave function of the ground state. Therefore, the unboundedness of the spectrum is only the necessary condition. To prove that a quantum particle indeed can fall to the center, a wave function describing the fall should be obtained explicitly. This is done in the present paper. Specifically, an exact solution of the time-dependent Schrodinger equation describing the fall to the center is obtained and analyzed. The law describing the collapse to a single point of the region of the wave function localization is obtained explicitly, as well as the temporal dependences of the average kinetic and potential energy of the falling particle and its momentum. It is shown that the known necessary conditions for the particle to fall simultaneously are sufficient.
Exact solutions describing a fall of a particle to the centre of a non-regularized singular potential in classical and quantum cases are obtained and compared. We inspect the quantum problem with the help of the conventional Schrödinger’s equation. During the fall, the wave function spatial localization area contracts into a single zero-dimensional point. For the fall-admitting potentials, the Hamiltonian is non-Hermitian. Because of that, the wave function norm occurs time-dependent. It demands an extension to this case of the continuity equation and rules for mean value calculations. Surprisingly, the quantum and classical solutions exhibit striking similarities. In particular, both are self-similar as the particle energy equals zero. The characteristic spatial scales of the quantum and classical self-similar solutions obey the same temporal dependence. We present arguments indicating that these self-similar solutions are attractors to a broader class of solutions, describing the fall at finite energy of the particle.
Quantum collapse (QC) occurs under the action of the inverse-power potential U(r)=-U_{0}/(2r^{n}) in the framework of the Schroedinger equation. The corresponding stationary wave functions feature an infinite number of spatial oscillations at r→0r\rightarrow 0r→0. The ground state (GS) is missing in the presence of the QC, but the available bound states at finite energies may be physically relevant, as they are normalizable. This work aims to produce singular stationary wave functions. The analytical findings represent an asymptotic form of the stationary wave functions at r→0r\rightarrow 0r→0, which fill a continuous spectrum, and some exact solutions for the wave functions, obtained under appropriate conditions. As a regularized version, we also consider the potential truncated at a finite difference from the center. The regularized wave functions feature a finite number of spatial oscillations, the missing GS reappears, and the spectrum of the bound states is discrete.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.
Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.