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arXiv 2111.04065math-phmath.MPquant-ph
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关于非矩形阶跃谐波分布的精确可解半无限量子阱

On the exactly-solvable semi-infinite quantum well of the non-rectangular step-harmonic profile

E. I. Jafarov, S. M. Nagiyev

备注：
23 pages, 8 figures, final version
发表信息：
Quantum Studies: Mathematics and Foundations 9 (2022) 387-404
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摘要：

建立了有效质量随位置变化的非相对论谐振子的精确可解模型。该模型表现为一个非矩形剖面的半无限量子阱。由于有效质量与位置和半封闭参数$a$的某种分析依赖性，这种形式的轮廓看起来像阶跃谐波势。研究了离散谱和连续谱的两种状态。在离散谱的情况下，振荡器模型的波函数通过贝塞尔多项式表示。由于离散能谱依赖于参数$a$，因此离散能谱也是非等距的和有限的。在连续谱的情况下，振子模型的波函数由$_1F_1$超几何函数表示。在极限处，当参数$a$趋于无穷大时，两个波函数和所构造模型的能谱都正确地还原为具有恒定有效质量的通常非相对论谐振子的相应结果。即，离散谱的波函数恢复厄米多项式形式的波函数，而连续谱的波函数简单地消失。我们还提出了一个新的极限关系，减少贝塞尔多项式直接厄米多项式，并证明其正确性，使用数学归纳法。

英文摘要：

An exactly-solvable model of the non-relativistic harmonic oscillator with a position-dependent effective mass is constructed. The model behaves itself as a semi-infinite quantum well of the non-rectangular profile. Such a form of the profile looks like a step-harmonic potential as a consequence of the certain analytical dependence of the effective mass from the position and semiconfinement parameter $a$. Both states of the discrete and continuous spectrum are studied. In the case of the discrete spectrum, wavefunctions of the oscillator model are expressed through the Bessel polynomials. The discrete energy spectrum is non-equidistant and finite as a consequence of its dependence on parameter $a$, too. In the case of the continuous spectrum, wavefunctions of the oscillator model are expressed through the $_1F_1$ hypergeometric functions. At the limit, when the parameter $a$ goes to infinity, both wavefunctions, and the energy spectrum of the model under construction correctly reduce to corresponding results of the usual non-relativistic harmonic oscillator with a constant effective mass. Namely, wavefunctions of the discrete spectrum recover wavefunctions in terms of the Hermite polynomials, and wavefunctions of the continuous spectrum simply vanish. We also present a new limit relation that reduces Bessel polynomials directly to Hermite polynomials and prove its correctness using the mathematical induction technique.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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