标题： 时间依赖的非对易背景中的量子谐振子 显示英文标题

标题： Quantum harmonic oscillator in a time dependent noncommutative background

摘要： 这项工作探讨了在时变背景中非交换谐波振子的行为，如之前在[1]中所研究的。 具体而言，我们检查了当用对易变量表示系统时的情况，利用了最近在[2]中引入的标准Bopp-shift关系的广义形式。 我们求解了时变系统，并使用Lewis不变量方法得到了本征函数的解析形式，该方法与Ermakov-Pinney方程相关，这是一个非线性微分方程。 然后，我们获得了Ermakov-Pinney方程的精确解析解集。 有了这些解，我们继续分析能量期望值的动力学并探索其图形表示，针对Ermakov-Pinney方程的不同解集以及特定的量子数选择。 最后，我们确定了对易和非对易情况下各算符之间的不确定性等式关系的广义形式。 可以预期，我们的研究与[1]中的发现一致，特别是在坐标映射关系退化为标准Bopp-shift关系的特定极限情况下。 显示英文摘要

摘要： This work explores the behaviour of a noncommutative harmonic oscillator in a time-dependent background, as previously investigated in [1]. Specifically, we examine the system when expressed in terms of commutative variables, utilizing a generalized form of the standard Bopp-shift relations recently introduced in [2]. We solved the time dependent system and obtained the analytical form of the eigenfunction using the method of Lewis invariants, which is associated with the Ermakov-Pinney equation, a non-linear differential equation. We then obtain exact analytical solution set for the Ermakov-Pinney equation. With these solutions in place, we move on to compute the dynamics of the energy expectation value analytically and explore their graphical representations for various solution sets of the Ermakov-Pinney equation, associated with a particular choice of quantum number. Finally, we determined the generalized form of the uncertainty equality relations among the operators for both commutative and noncommutative cases. Expectedly, our study is consistent with the findings in [1], specifically in a particular limit where the coordinate mapping relations reduce to the standard Bopp-shift relations.