关于位置和/或动量具有非零最小不确定性的谐振子的SUSYQM方法
More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum
摘要：
我们继续我们以前的应用超对称量子力学方法的本征值问题的背景下，一些变形的正则对易关系，导致非零的最小不确定性的位置和/或动量。本文首次用形变参数$α$，$β$确定了均匀电场中一维谐振子的谱和本征向量.我们建立，只要有一个非零的最小不确定性的动量，即，对于$α\ne 0$，由于电场对谐振子本征值的校正是依赖于电平的。在相反的情况下，即，对于$α= 0$，即使在位置存在非零最小不确定性时，我们也恢复了总能谱偏移的常规量子力学图像，即，对于$β\ne 0$。然后我们考虑的问题的$D$维谐振子的情况下，各向同性非零的最小不确定性的位置坐标，取决于两个参数$β$，$β'$。我们扩展我们的方法来处理相应的径向方程的动量表示和重新推导的频谱和动量径向波函数以前发现通过求解微分方程以一种简单的方式。这为解决新的$D$维问题开辟了道路。
英文摘要：
We continue our previous application of supersymmetric quantum mechanical methods to eigenvalue problems in the context of some deformed canonical commutation relations leading to nonzero minimal uncertainties in position and/or momentum. Here we determine for the first time the spectrum and the eigenvectors of a one-dimensional harmonic oscillator in the presence of a uniform electric field in terms of the deforming parameters $α$, $β$. We establish that whenever there is a nonzero minimal uncertainty in momentum, i.e., for $α\ne 0$, the correction to the harmonic oscillator eigenvalues due to the electric field is level dependent. In the opposite case, i.e., for $α= 0$, we recover the conventional quantum mechanical picture of an overall energy-spectrum shift even when there is a nonzero minimum uncertainty in position, i.e., for $β\ne 0$. Then we consider the problem of a $D$-dimensional harmonic oscillator in the case of isotropic nonzero minimal uncertainties in the position coordinates, depending on two parameters $β$, $β'$. We extend our methods to deal with the corresponding radial equation in the momentum representation and rederive in a simple way both the spectrum and the momentum radial wave functions previously found by solving the differential equation. This opens the way to solving new $D$-dimensional problems.
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