Exactly-solvable confined model of the nonrelativistic quantum harmonic oscillator is proposed. Its position-dependent effective mass Hamiltonian is defined via the von Roos kinetic energy operator. The confinement effect to harmonic oscillator potential is included as a result of certain behaviour of the position-dependent effective mass. The corresponding Schrodinger equation in the canonical approach is exactly solved and it is shown that the discrete energy spectrum of the system under consideration depends on the confinement parameter a, von Roos parameters α, β, γ and has a non-equidistant form. Wave functions of the stationary states of the model are expressed through the Gegenbauer polynomials. The limit a → ∞ recovers both equidistant energy spectrum and wave functions of the stationary nonrelativistic harmonic oscillator expressed by Hermite polynomials.

中文翻译：

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具有冯·罗斯动能算子的受限谐波振荡器模型的量子力学显式解

提出了非相对论量子谐振子的精确可解约束模型。其与位置相关的有效质量哈密顿量是通过 von Roos 动能算子定义的。由于位置相关有效质量的某些行为，包括对谐振子电位的限制效应。正则方法中相应的薛定谔方程得到了精确求解，结果表明所考虑系统的离散能谱取决于限制参数 a，von Roos 参数 α、β、γ 并具有非等距形式。模型稳态的波函数通过 Gegenbauer 多项式表示。