[Paper Review] Singular Centre in Quantum Mechanics as a Black Hole

A. E. Shabad|ArXiv.org|Aug 19, 2002

Crystallography and Radiation Phenomena5 references3 citations

TL;DR

This paper proposes a physical interpretation of the continuum of states arising from a singular $1/r^2$ potential in quantum mechanics, showing that these states correspond to particles asymptotically free near the origin. By introducing a singular measure $\mathrm{d}r/r^2$, the authors establish orthonormality and derive a unitary $2\times2$ S-matrix for two-channel scattering, drawing a deep analogy to black hole physics, including event horizon-like behavior and momentum-like quantum numbers tied to the singularity's strength.

ABSTRACT

We consider the radial Schroedinger equation with an attractive potential singular in the origin. The additional continuum of states caused by the singularity, that usually remain nontreatable, are shown to correspond to particles, asymptotically free near the singularity (in the inner channel). Depending on kinematics, they are either confined by the centre or may escape to infinity (to the outer channel). The orthonormality within the continuum of confined states is established and the scattering phase of the particle emitted by the centre and then reflected back to it is found. For the deconfinement case a unitary 2x2 S-matrix is found in terms of the Jost functions, and describes transitions within and between the two channels. The volume elements in the two channels are different. The two-channel situation is analogous to the known behaviour of radiation in the black hole metrics. We discuss the black hole essence of singularly attracting centre for classical motion and the relativity of time inherent to this problem.

Motivation & Objective

• To provide a physical interpretation of the non-normalizable continuum of states induced by a singular $1/r^2$ potential in radial Schrödinger equations.
• To resolve the long-standing issue of orthonormality and normalization for wave functions diverging at the origin by introducing a singular measure $\mathrm{d}r/r^2$.
• To model the singular center as analogous to a black hole, with particle emission, absorption, and reflection, akin to Hawking radiation and event horizon dynamics.
• To establish a two-channel scattering framework with distinct volume elements, enabling unitary evolution between inner (near-origin) and outer (infinite) regions.
• To explore the relativity of time and the role of the parameter $r_0$ as a fundamental scale separating inner and outer dynamics.

Proposed method

• Transform the radial Schrödinger equation via $r_* = r_0 \ln(r/r_0)$, mapping the origin to $-\infty$, turning the singularity into a confining potential.
• Introduce a singular measure $\mathrm{d}r/r^2$ in the inner region to ensure linear divergence of the norm, enabling interpretation of oscillating wave functions as free particles near the origin.
• Define two channels: an inner channel (near $r=0$) with wave functions $\sim r^{\pm i\mathrm{Im}\lambda + 1/2}$, and an outer channel (at infinity) with standard asymptotic behavior.
• Construct a unitary $2\times2$ S-matrix using Jost functions, describing transitions between inner and outer channels, with different volume elements in each.
• Use the eigenvalue problem formulation $(-\mathrm{d}^2/\mathrm{d}r^2 + V(r))\psi = \kappa(r)\psi$, with $\kappa(r) = k^2 - (\lambda^2 - 1/4)/r^2$, to ensure self-adjointness and completeness of the state set.
• Establish orthonormality of confined states via the singular measure, ensuring unitarity and consistency with quantum mechanical probability conservation.

Experimental results

Research questions

• RQ1 How can the continuum of states induced by a singular $1/r^2$ potential be physically interpreted, especially when standard normalization fails?
• RQ2 What measure is required to make the norm of oscillating wave functions near the origin finite and physically meaningful?
• RQ3 In what way does the singular center behave like a black hole, particularly in terms of particle emission, absorption, and reflection?
• RQ4 How can a unitary S-matrix be constructed for a system with two distinct channels—inner (near-origin) and outer (infinite)—with different volume elements?
• RQ5 What is the physical significance of the parameter $r_0$ in the transformation $r_* = r_0 \ln(r/r_0)$, and how does it relate to the size or scale of the system?

Key findings

• The singular measure $\mathrm{d}r/r^2$ ensures linear divergence of the norm for oscillating wave functions near $r=0$, enabling their interpretation as free particles in the inner region.
• Orthonormality of the confined states in the inner channel is established using the singular measure, validating the completeness of the state set in the confinement regime.
• For $k^2 < 0$, the system exhibits a purely elastic, confining process where all emitted particles are reflected back to the center, with no net emission or absorption.
• For $k^2 > 0$, a two-channel regime emerges: particles can escape to infinity or be absorbed from infinity, with transitions described by a unitary $2\times2$ S-matrix.
• The scattering phase for particles emitted and reflected back by the center is explicitly computed, showing a dependence on $\mathrm{Im}\lambda$ and $r_0$.
• The system exhibits a deep analogy to black hole physics, including horizon-like behavior, event-like absorption, and a relativity of time, with $\mathrm{Im}\lambda/r_0$ playing the role of momentum in the inner world.