We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schrödinger picture. Generally in the most papers in the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of ωby iω, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti- $\mathcal{P}\mathcal{T}$-symmetric harmonic oscillator, and we obtain the standard quasi-Hermiticity relation which would ensure the time invariance of the eigenfunction's norm. We give a complete description for the eigenproblem. We show that the wave functions for this system are normalized in the sense of the pseudo-scalar product. A Gaussian wave packet of the inverted oscillator is investigated by using the ladder operators method. This wave packet is found to be associated with the generalized coherent state that can be crucially utilized for investigating the mean values of the space and momentum operators. We find that these mean values reproduce the classical motion.

我们处理谐振子的量子动力学及其在薛定谔图像中的倒置对应物。通常在文献中的大多数论文中，通过将ω替换为i ω，从谐振子形式上获得倒谐振子，这导致无界特征向量。这明确地表明，在谐振子反演中重新定义变量涉及一些不清楚的地方。为了补救这种情况，我们引入了一个缩放算子（戴森变换），方法是将倒谐振子连接到一个反$\mathcal{P}\mathcal{吨}$-对称谐振子，我们得到了标准的准Hermiticity关系，这将确保本征函数范数的时间不变性。我们给出了本征问题的完整描述。我们表明，该系统的波函​​数在伪标量积的意义上进行了归一化。利用梯形算子方法研究了倒置振荡器的高斯波包。发现该波包与广义相干态相关联，可以关键地用于研究空间和动量算子的平均值。我们发现这些平均值再现了经典运动。