论文标题
位置依赖性质量和角频率schrödinger方程的精确解：具有量化限制参数的谐波振荡器模型
Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter
论文作者
论文摘要
我们提出了非相关量子谐波振荡器的限制模型的精确解，其中有效质量和角频率取决于位置。所提出模型的自由汉密尔顿人具有Bendaniel-Duke动能运算符的形式。质量和角频率的位置依赖性是使谐波振荡器常数$ k $的均匀性质，因此保留了常规的谐波振荡器电位。结果，观察到限制参数的量化。结果表明，具有依赖位置的质量和角频率的受限谐波振荡器的离散能谱是有限的，具有非均等形式，取决于限制参数。以位置依赖性质量和角频率的固定振荡器的固定状态的波函数以相关的legendre或gegenbauer多项式表示。在限制参数趋向于$ \ indty $的极限中，能量谱和波函数都会收敛到众所周知的等距能谱，以及以Hermite多项式表示的固定非相关谐波振荡器的固定非相关谐波振荡器的波函数。位置依赖性有效质量和角频率在此限制下也变得恒定。
We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel--Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant $k$ and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to $\infty$, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.
