The European Physical Journal Plus ( IF 3.4 ) Pub Date : 2021-07-19 , DOI: 10.1140/epjp/s13360-021-01742-z E. I. Jafarov 1 , J. Van der Jeugt 2
We present a new model of a one-dimensional nonrelativistic canonical quantum harmonic oscillator which is semiconfined. Semiconfinement is achieved by replacing the constant effective mass with a mass that varies with position. The problem is exactly solvable and the analytic expression of the wavefunctions of the stationary states is expressed by means of generalized Laguerre polynomials. Surprisingly, the energy spectrum completely overlaps with the energy spectrum of the standard nonrelativistic canonical quantum harmonic oscillator. In the limit when the semiconfinement parameter a goes to infinity, the wavefunctions also tend to the wavefunction of the standard oscillator in terms of Hermite polynomials.
中文翻译:
具有位置相关有效质量的半受限谐振子模型的精确解
我们提出了一种新的半约束的一维非相对论正则量子谐振子模型。半约束是通过用随位置变化的质量代替恒定的有效质量来实现的。这个问题是完全可以解决的,并且稳态波函数的解析表达式是通过广义拉盖尔多项式来表达的。令人惊讶的是,能谱与标准非相对论正则量子谐振子的能谱完全重叠。在半限制参数a趋于无穷大的极限下,波函数也趋向于标准振荡器的波函数,即 Hermite 多项式。