[Paper Review] Scattering and Bound States of a Deformed Quantum Mechanics

Chee-Leong Ching, Rajesh R. Parwani|arXiv (Cornell University)|Jul 6, 2012

Noncommutative and Quantum Gravity Theories5 references3 citations

TL;DR

This paper constructs the exact position representation for a deformed quantum mechanics with a maximum momentum, derived from a modified commutation relation (MCR) that introduces a fundamental momentum scale. Using this formalism, it analyzes scattering and bound states—showing that bound states in a finite well form later than in standard quantum mechanics due to the maximum momentum. The work resolves two literature puzzles by demonstrating that low-momentum expansions fail to capture non-perturbative effects and that Hermiticity requires additional boundary conditions beyond Dirichlet conditions.

ABSTRACT

We construct the exact position representation of a deformed quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and scattering from a step potential, among others. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well. In the process we also highlight some limitations and pit-falls of low-momentum or perturbative treatments and thus resolve two puzzles occurring in the literature.

Motivation & Objective

• To construct the exact position representation for a deformed quantum mechanics with a maximum momentum, derived from a modified commutation relation (MCR) with f(P) = 1 − 2αP + qα²P².
• To analyze scattering and bound states—specifically the infinite square well and step potential—using the exact position representation, avoiding approximations from low-momentum expansions.
• To resolve two puzzles in the literature: (1) a discrepancy between perturbative and exact results for the infinite well spectrum, and (2) an erroneous claim of length quantization in the infinite well under the MCR.
• To show that the Hamiltonian in the infinite well with Dirichlet conditions is not Hermitian, despite having real eigenvalues, and to identify the correct boundary conditions for Hermiticity.

Proposed method

• Derives the exact position representation by solving the MCR (X, P) = if(P) in the x-basis, determining P(p) as a function of momentum p.
• Uses the momentum representation (P = p, X = if(p)∂/∂p) from prior work to derive the position-space form of the momentum operator P(x) via inverse transformation.
• Applies the exact formalism to solve the free particle and infinite well problems, obtaining energy spectra and wavefunctions without low-momentum truncations.
• Compares results with semi-classical quantization (Sommerfeld-Wilson rule) and shows exact agreement, validating the formalism.
• Identifies that standard perturbative methods fail due to non-Hermiticity of the perturbative Hamiltonian and derives a more fundamental formula for consistent perturbation theory.
• Demonstrates that Dirichlet boundary conditions alone are insufficient for Hermiticity; additional constraints on wavefunction derivatives or total probability current are required.

Experimental results

Research questions

• RQ1 How does the presence of a maximum momentum in a deformed quantum mechanics affect the formation of bound states in a finite potential well compared to standard quantum mechanics?
• RQ2 Why do perturbative calculations of the infinite well spectrum using low-momentum expansions disagree with exact results, and what is the correct perturbative approach?
• RQ3 What are the implications of the non-Hermiticity of the Hamiltonian in the infinite well with Dirichlet boundary conditions, and what boundary conditions restore Hermiticity?
• RQ4 Does the MCR with maximum momentum lead to quantization of the well size, as previously claimed in the literature?
• RQ5 How can the probability current be consistently defined in deformed quantum mechanics, and what role does it play in ensuring probability conservation?

Key findings

• The bound state spectrum in a finite potential well terminates at finite energy when q ≤ 1, due to the intrinsic maximum momentum, delaying bound state formation compared to standard quantum mechanics.
• The infinite well energy spectrum derived from the exact position representation shows a finite upper bound for q ≤ 1, consistent with the semi-classical analysis and contrasting with the unbounded spectrum in the standard case.
• Perturbative calculations using low-momentum expansions yield incorrect results because the resulting Hamiltonian is not Hermitian; a more fundamental formula is required for consistency.
• The standard textbook perturbative formula fails due to non-Hermiticity; the correct perturbative approach requires ensuring the Hamiltonian remains Hermitian under deformation.
• Dirichlet boundary conditions alone do not ensure Hermiticity; the vanishing of the total probability current (J₀ + J₁) at the boundaries is necessary, requiring additional constraints on wavefunction derivatives.
• The claim of length quantization in the infinite well under the MCR is unfounded; no such quantization is found in exact, semi-classical, or consistent low-momentum treatments.