Exact solution of Schrödinger equation for the complex Morse potential to investigate physical systems with position-dependent complex mass

TL;DR

The paper solves the position-dependent mass Schrödinger equation for a one-dimensional complex Morse potential with a complex mass in an extended complex phase space. By deriving exact ground-state eigenfunctions and energies and introducing a two-dimensional phase-space normalization, the authors show that real spectra can emerge under specific parameter constraints despite non-Hermiticity. They analyze a general PDCM form and several special mass-profile cases, providing closed-form expressions for eigenvalues, eigenfunctions, and normalization conditions, and they illustrate the spectra and probability densities with phase-space plots. The work highlights that spectrum reality can arise from underlying symmetries or parameter choices even when the Hamiltonian is non-Hermitian, and it discusses potential implications for high-energy/cosmological physics and analogies to PT-symmetric optical systems, including speculative links to dark matter.

Abstract

This paper presents the exact ground state solution for a diatomic particle system with position-dependent complex mass under action of a complex Morse potential in the quantum domain. By solving the position-dependent Schrödinger equation in extended complex phase space without assuming a specific mass profile, we derive both the eigenfunctions and corresponding eigenenergies using the analyticity conditions of the eigenfunctions. A key focus is placed on addressing the challenge of normalization inherent in non-Hermitian Hamiltonians. To overcome the limitations of conventional normalization methods in systems with complex potentials and spatially varying mass, we propose a modified normalization approach based on a two-dimensional integral over phase space. The results reveal that, under certain parameter constraints, real energy spectra can arise in non Hermitian settings, supported by normalized and physically meaningful eigenfunctions. Probability density plots validate the existence of stable, localized bound states, maintaining essential characteristics of the traditional Morse potential. Moreover, the model offers potential applications in high-energy and cosmological physics, particularly in the quantum description of exotic systems like dark matter.

Figures (13)

• Figure 1: The Normalization condition plot for a position-dependent complex mass system. Here, the purple shade defines the intersection region of both the normalization conditions, whereas white, blue and pink shades show the region where eigenfunction can’t be normalized.
• Figure 2: The eigenvalue plots of mass profile $m_r = cx_1-dp_2+e_1$ and $m_i = dx_1+cp_2+e_2$ in extended complex plane. (a) $E_i$ Vs $x_1$ and $p_2$ plot (b) $E_r$ Vs $x_1$ and $p_2$.
• Figure 3: The general case plots for mass profile $m_r = cx_1-dp_2+e_1$ and $m_i = dx_1+cp_2+e_2$ in extended complex plane. (a) Probability density (PCD) Plot (b) Contour plot of peak in figure \ref{['fig3a']} (c) Behaviour of Imaginary part of eigenvalue at peak of PCD (d) Behaviour of real part of eigenvalue at peak of PCD.
• Figure 7: The probablity density plots for, (a) Case I(a) (b) Case II(a) and contour plots of probability density of the peak for (c) Case I(a) (d) Case II(a)
• Figure 8: Normalization condition plot of reality of spectrum for mass profile $m_r = cx_1-dp_2+e_1$ and $m_i = dx_1+cp_2+e_2$ in extended complex plane associated with (a) first root of $\beta_3$ (b) second root of $\beta_3$.