[Paper Review] Black-Hole Approach to the Singular Problem of Quantum Mechanics. II

A. E. Shabad|ArXiv.org|Mar 17, 2004

Quantum Mechanics and Applications2 references5 citations

TL;DR

This paper proposes a black-hole-inspired approach to the singular quantum mechanical problem of a particle falling into a $ r^{-2} $ potential, treating the origin as a physical emission/absorption center. By redefining the Hamiltonian as a generalized eigenvalue problem with singular measures $ \mathrm{d}\mu^{\mathrm{III}} = r^{-2}\mathrm{d}r $ and $ \mathrm{d}\mu^{\mathrm{IV}} = (R^{-2} + r^{-2})\mathrm{d}r $, it classifies states into bound (confined) and scattering (transmission/reflection) types via two self-adjoint operators, yielding a unitary $ 2\times2 $ S-matrix in terms of Jost functions.

ABSTRACT

A new approach is proposed for the quantum mechanical problem of the falling of a particle to a singularly attracting center, basing on a black-hole concept of the latter. The singularity r^{-2} in the potential of the radial Schroedinger equation is considered as an emitting/absorbing center. The two solutions oscillating in the origin are treated as asymptotically free particles, which implies that the singular point r=0 in the Schroedinger equation is treated on the same physical ground as the singular point r=infinity. To make this interpretation possible, it is needed that the norm squared of the wave function should diverge when r tends to zero, in other words, the measure used in definition of scalar products should be singular in the origin. Such measure comes into play if the Schroedinger equation is written in the form of the generalized (Kamke) eigenvalue problem for either of two - chosen differently depending on the sign of the energy E - operators, other than Hamiltonian. The Hilbert spaces where these two operators act are used to classify physical states, which are: i) states of "confinement"- continuum of solutions localized near the origin, E<0 - and ii) the states corresponding to the inelastic process of reflection/transmission, i.e. to transitions in-between states localized near the origin and in the infinitely remote region, E>0. The corresponding unitary 2x2 S-matrix is written in terms of the Jost functions. The complete orthonormal sets of eigen-solutions of the two operators are found using "quantization in a box" (a,b), followed by the transition to the limit a=0, b=infinity. The corresponding expansions of the unity are written.

Motivation & Objective

• To resolve the physical inconsistency of standard quantum mechanics in describing particles falling into a singular $ r^{-2} $ potential, where the Hamiltonian's spectrum is unbounded below.
• To treat the singular point $ r=0 $ on the same physical footing as $ r=\infty $, by modeling it as an emitting/absorbing center akin to a black hole.
• To define Hilbert spaces with singular measures $ \mathrm{d}\mu^{\mathrm{III}} $ and $ \mathrm{d}\mu^{\mathrm{IV}} $ such that wavefunctions oscillating at $ r=0 $ are interpreted as asymptotically free particles.
• To classify physical states into bound (confined, $ E<0 $) and scattering (transmission/reflection, $ E>0 $) using two distinct self-adjoint operators $ H^{\mathrm{III}} $ and $ H^{\mathrm{IV}} $.
• To construct a unitary $ 2\times2 $ S-matrix using Jost functions and derive orthonormal eigenfunction expansions via quantization in a box with $ r_L \to 0 $, $ r_U \to \infty $

Proposed method

• Reformulate the radial Schr"{o}dinger equation as a generalized eigenvalue problem using two operators: $ H^{\mathrm{III}} = -\frac{d^2}{dr^2} - \frac{1}{4r^2} + V(r) - k^2 $ for $ E<0 $, and $ H^{\mathrm{IV}} = -\frac{d^2}{dr^2} - \frac{1}{4r^2} + V(r) $ for $ E>0 $, with $ k^2 = E $.
• Introduce singular measures $ \mathrm{d}\mu^{\mathrm{III}}(r) = r^{-2}\mathrm{d}r $ and $ \mathrm{d}\mu^{\mathrm{IV}}(r) = (R^{-2} + r^{-2})\mathrm{d}r $ to define scalar products in Hilbert spaces where wavefunctions oscillating at $ r=0 $ have finite norm.
• Use the quantization in a box method with boundaries $ r_L \to 0 $ and $ r_U \to \infty $, imposing zero boundary conditions at $ r_L $ and $ r_U $, to construct orthonormal eigenfunction sets.
• Transform the radial problem into a $ \xi $-space problem via $ \xi = \mathrm{Im}\lambda \cdot \left( \ln r + \ln R \right) $, leading to eigenfunctions $ \widetilde{\psi}^{(j)}_{\lambda,R}(\xi) $ that are oscillatory at both $ \xi \to \pm\infty $, ensuring free-particle interpretation.
• Derive expansions of unity and generalized Fourier series in $ \xi $-space using eigenfunctions $ \widetilde{\psi}^{(j)}_{\lambda_n,R}(\xi) $, with coefficients given by integrals over $ \widetilde{F}(\xi) $.
• Construct the $ 2\times2 $ S-matrix from Jost functions, ensuring unitarity and consistency with scattering processes for $ E>0 $, and relate it to phase shifts via $ \delta_1, \delta_2 $ and $ \varepsilon^{(j)} $.

Experimental results

Research questions

• RQ1 Can the singular point $ r=0 $ in the $ r^{-2} $ potential be treated as a physical emission/absorption center, analogous to a black hole?
• RQ2 How can the norm divergence of oscillating wavefunctions at $ r=0 $ be reconciled with a free-particle interpretation, and what measure enables this?
• RQ3 What is the appropriate self-adjoint operator formulation for both bound ($ E<0 $) and scattering ($ E>0 $) states in the singular potential problem?
• RQ4 How can a unitary $ 2\times2 $ S-matrix be constructed to describe inelastic transitions between localized and delocalized states?
• RQ5 Can orthonormal eigenfunction expansions be rigorously derived in the singular limit $ r_L \to 0 $, $ r_U \to \infty $ using a box regularization scheme?

Key findings

• The singular measure $ \mathrm{d}\mu^{\mathrm{III}}(r) = r^{-2}\mathrm{d}r $ ensures that wavefunctions oscillating at $ r=0 $ have finite norm, enabling their interpretation as asymptotically free particles.
• The two operators $ H^{\mathrm{III}} $ and $ H^{\mathrm{IV}} $, defined via generalized eigenvalue problems, yield self-adjoint operators in Hilbert spaces with singular measures, avoiding the unbounded spectrum of the standard Hamiltonian.
• For $ E<0 $, the spectrum is discrete and corresponds to bound states of confinement, while for $ E>0 $, the spectrum is continuous and describes scattering states with transmission/reflection.
• The eigenfunctions $ \widetilde{\psi}^{(j)}_{\lambda_n,R}(\xi) $ in $ \xi $-space are orthonormal and form a complete basis, allowing expansion of any $ L^2 $ function $ \widetilde{F}(\xi) $ via generalized Fourier series.
• The $ 2\times2 $ S-matrix is unitary and expressed in terms of Jost functions, with phase shifts determined by $ \delta_1, \delta_2 $, and $ \varepsilon^{(j)} $, ensuring consistency with quantum mechanical scattering theory.
• The asymptotic behavior of eigenfunctions at $ \xi \to \pm\infty $ is oscillatory with phase shifts that depend on energy and coupling, confirming the free-particle character of states near $ r=0 $