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<p>Supersymmetric quantum mechanics (SUSY-QM) provides a powerful framework for analyzing exactly solvable quantum systems. This study extends this formalism to a class of hyperbolic potentials within position-dependent mass (PDM) systems, which are crucial for modeling graded semiconductor heterostructures and other effective quantum theories. While the constant-mass case is confirmed to be exactly solvable, introducing a position-dependent mass presents a fundamental theoretical challenge. The nonlinear coupling between the spectral parameters and the superpotential generally breaks the standard shape-invariance condition, preventing a general analytical solution through the standard SUSY hierarchy. Our central result is the derivation, by strictly enforcing shape invariance as a constraint, of a unique and exactly solvable scenario. This process self-consistently identifies a specific, compatible pair of mass profile <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(M\left( x\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and potential<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ V\left( x\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="4pt" /> <mi>V</mi> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. This pair emerges not as an arbitrary choice but as the fundamental solution that preserves the underlying supersymmetric symmetry, highlighting that shape invariance itself acts as a selection rule dictating which mass profiles permit exact solvability for a given potential type. This solvable model establishes a critical analytical benchmark for future studies on more complex, non-shape-invariant PDM systems. It demonstrates a robust methodology for identifying exact solutions in generalized quantum contexts, consolidating the role of SUSY-QM in tackling the challenges of position-dependent mass.</p>

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Analytical Solutions of Rosen-Morse-Type Potentials in Quantum Systems with Position-Dependent Mass via Shape Invariance

  • 期刊论文

  • 发表日期：

    2025年10月4日

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  • Kh. Bengherabi,
  • N. Zaghou,
  • F. Benamira

摘要

Supersymmetric quantum mechanics (SUSY-QM) provides a powerful framework for analyzing exactly solvable quantum systems. This study extends this formalism to a class of hyperbolic potentials within position-dependent mass (PDM) systems, which are crucial for modeling graded semiconductor heterostructures and other effective quantum theories. While the constant-mass case is confirmed to be exactly solvable, introducing a position-dependent mass presents a fundamental theoretical challenge. The nonlinear coupling between the spectral parameters and the superpotential generally breaks the standard shape-invariance condition, preventing a general analytical solution through the standard SUSY hierarchy. Our central result is the derivation, by strictly enforcing shape invariance as a constraint, of a unique and exactly solvable scenario. This process self-consistently identifies a specific, compatible pair of mass profile \(M\left( x\right) \) M x and potential \(\ V\left( x\right) \) V x . This pair emerges not as an arbitrary choice but as the fundamental solution that preserves the underlying supersymmetric symmetry, highlighting that shape invariance itself acts as a selection rule dictating which mass profiles permit exact solvability for a given potential type. This solvable model establishes a critical analytical benchmark for future studies on more complex, non-shape-invariant PDM systems. It demonstrates a robust methodology for identifying exact solutions in generalized quantum contexts, consolidating the role of SUSY-QM in tackling the challenges of position-dependent mass.