[Paper Review] On a relativistic scalar particle subject to the Klein-Gordon oscillator, the Coulomb potential and a linear scalar potential

R. L. L. Vitória, C. Furtado+1 more|arXiv (Cornell University)|Nov 16, 2015

Quantum Mechanics and Non-Hermitian Physics69 references43 citations

TL;DR

This paper investigates the relativistic quantum dynamics of a scalar particle under the Klein-Gordon oscillator, Coulomb potential, and a linear scalar potential in (2+1) dimensions. It derives bound state solutions by solving the Klein-Gordon equation with a biconfluent Heun equation, revealing that the oscillator's angular frequency is quantized and depends on quantum numbers n and l, with the ground state energy and frequency determined by a cubic algebraic equation.

ABSTRACT

The relativistic quantum dynamics of an electrically charged particle subject to the Klein-Gordon oscillator and the Coulomb potential is investigated. By searching for relativistic bound states, a particular quantum effect can be observed: a dependence of the angular frequency of the Klein-Gordon oscillator on the quantum numbers of the system. The meaning of this behaviour of the angular frequency is that only some specific values of the angular frequency of the Klein-Gordon oscillator are permitted in order to obtain bound state solutions. As an example, we obtain both the angular frequency and the energy level associated with the ground state of the relativistic system. Further, we analyse the behaviour of an electrically charged particle subject to the Klein-Gordon oscillator, the Coulomb potential and a linear scalar potential.

Motivation & Objective

• To investigate the relativistic quantum dynamics of a scalar particle under the Klein-Gordon oscillator and Coulomb potential in (2+1) dimensions.
• To determine how the Coulomb potential modifies the energy spectrum and angular frequency of the Klein-Gordon oscillator.
• To analyze the effect of adding a linear scalar potential on the bound state solutions and quantization of the oscillator frequency.
• To establish that only specific values of the angular frequency are allowed, constrained by the quantum numbers n and l.
• To derive the energy levels and angular frequency for the ground state via polynomial solutions to the biconfluent Heun equation.

Proposed method

• Formalism based on the Klein-Gordon equation with minimal coupling to the electromagnetic potential (Coulomb term) and modification of the mass term via scalar potentials.
• Use of the Klein-Gordon oscillator via a vector-like coupling: $\hat{p}_\mu \to \hat{p}_\mu - i m \omega \rho \hat{\rho}$, introducing harmonic confinement.
• Transformation of the radial equation into a biconfluent Heun equation through variable substitution and series expansion.
• Imposition of polynomial solutions via the condition $a_{n+1} = 0$, leading to quantization of the angular frequency and energy levels.
• Solution of the recurrence relations for coefficients $a_j$, leading to algebraic constraints on $\theta_{n,l} = \sqrt{m^2 \omega_{n,l}^2 + \nu^2}$.
• Derivation of a cubic equation for $\theta_{1,l}$ in the ground state ($n=1$) to determine allowed values of $\omega_{1,l}$ and $\mathcal{E}_{1,l}$.

Experimental results

Research questions

• RQ1 How does the presence of a Coulomb potential affect the relativistic bound state spectrum of the Klein-Gordon oscillator?
• RQ2 What constraints does the system impose on the angular frequency $\omega$ of the Klein-Gordon oscillator, and how are these related to quantum numbers $n$ and $l$?
• RQ3 Can polynomial solutions be obtained for the biconfluent Heun equation in the presence of both Coulomb and linear scalar potentials?
• RQ4 How does the inclusion of a linear scalar potential modify the energy levels and allowed values of $\omega$ compared to the pure Coulomb case?
• RQ5 What is the analytical form of the ground state energy and angular frequency when all three potentials are present?

Key findings

• The angular frequency $\omega_{n,l}$ of the Klein-Gordon oscillator is not arbitrary but quantized and depends on the quantum numbers $n$ and $l$, with only specific values allowing bound state solutions.
• For the ground state ($n=1$), the allowed values of $\theta_{1,l} = \sqrt{m^2 \omega_{1,l}^2 + \nu^2}$ are determined by a third-degree algebraic equation, implying a non-trivial dependence of $\omega_{1,l}$ on system parameters.
• The energy level of the ground state $\mathcal{E}_{1,l}$ is determined by solving the cubic equation for $\theta_{1,l}$, with the exact expression being complex and not explicitly written due to length.
• The ground state is defined by $n=1$ rather than $n=0$, indicating a shift in the quantum number labeling due to the interplay of the Coulomb and oscillator potentials.
• The inclusion of a linear scalar potential modifies the energy spectrum and further constrains the allowed values of $\omega_{n,l}$, which remain dependent on $n$ and $l$.
• The system yields a polynomial solution to the biconfluent Heun equation only when the parameters satisfy specific algebraic conditions derived from the $a_{n+1} = 0$ criterion.