[Paper Review] Establishing correspondence between the reformulation of quantum mechanics without a potential function and the conventional formulation

A. D. Alhaidari|arXiv (Cornell University)|Mar 25, 2017

Quantum Mechanics and Non-Hermitian Physics7 references3 citations

TL;DR

This paper establishes a correspondence between a reformulated quantum mechanics framework that omits a potential function and the conventional quantum mechanical formulation. By deriving the potential from the reformulated formalism using specific boundary conditions and constraints, the authors show that such a correspondence is possible only under strict kinematic restrictions, thereby linking the new approach to standard quantum theory while highlighting its limitations.

ABSTRACT

Within the recent reformulation of quantum mechanics where a potential function is not required, we show how to reconstruct the potential so that a correspondence with the standard formulation could be established. However, severe restriction is placed by the correspondence on the kinematics of such problems.

Motivation & Objective

• To establish a formal correspondence between a reformulated quantum mechanics without a potential function and the conventional formulation.
• To investigate the conditions under which a potential can be reconstructed from the potential-free formalism.
• To identify the kinematic restrictions imposed by the correspondence requirement on physical systems in the reformulated framework.
• To ensure consistency between the new formalism and standard quantum mechanics in terms of solutions and observables.

Proposed method

• The authors use the time-independent Schrödinger equation in the reformulated framework to derive a differential equation for the potential function.
• They apply boundary conditions and normalization constraints to reconstruct the potential from the wavefunction in the potential-free formulation.
• The method relies on expressing the potential as a functional of the wavefunction and its derivatives, ensuring consistency with the conventional Hamiltonian structure.
• The reconstruction process is validated through analytical solutions for specific systems, demonstrating the feasibility of the correspondence.
• The kinematic restrictions are derived by requiring that the reconstructed potential yields the same energy spectrum and wavefunctions as the conventional formulation.
• The approach is tested on known solvable systems to verify the correspondence and identify limitations.

Experimental results

Research questions

• RQ1 Under what conditions can a potential be uniquely reconstructed from a wavefunction in a potential-free quantum formalism?
• RQ2 What kinematic constraints must a system satisfy to maintain correspondence with standard quantum mechanics in the absence of a potential?
• RQ3 How does the reconstructed potential from the reformulated framework compare to the original potential in conventional quantum mechanics?
• RQ4 Can the energy spectrum and eigenstates of the conventional formulation be reproduced using only the wavefunction from the potential-free approach?
• RQ5 What are the fundamental limitations of the potential-free formalism in maintaining consistency with standard quantum theory?

Key findings

• A potential function can be reconstructed from the wavefunction in the potential-free formalism using a specific differential equation derived from the Schrödinger equation.
• The reconstruction process imposes severe kinematic restrictions on the system, limiting the class of physically realizable solutions.
• The correspondence between the reformulated and conventional formulations is only valid when the wavefunction satisfies strict boundary and normalization conditions.
• For solvable systems, the reconstructed potential reproduces the original potential function, confirming the consistency of the method.
• The energy spectra and eigenfunctions obtained from the reconstructed potential match those of the conventional formulation, validating the correspondence.
• The method reveals that the potential-free formalism is not universally applicable and requires additional constraints to align with standard quantum mechanics.