In this paper, we investigate the quantum dynamics of underlying two one-dimensional quadratic Linard type nonlinear oscillators which are classified under the category of maximal (eight parameter) Lie point symmetry group [Tiwari A K, Pandey S N, Senthilvelan M and Lakshmanan M 2013 J. Math. Phys. 54, 053 506]. Classically, both the systems were also shown to be linearizable as well as isochronic. In this work, we study the quantum dynamics of the nonlinear oscillators by considering a general ordered position dependent mass Hamiltonian. The ordering parameters of the mass term are treated to be arbitrary to start with. We observe that the quantum version of these nonlinear oscillators are exactly solvable provided that the ordering parameters of the mass term are subjected to certain constraints imposed on the arbitrariness of the ordering parameters. We obtain the eigenvalues and eigenfunctions associated with both the systems. We also consider briefly the quantum versions of other examples of quadratic Linard oscillators which are classically linearizable.

在本文中，我们研究了基本的两个一维二次 Linard 型非线性振荡器的量子动力学，这些振荡器被归类为极大（八参数）李点对称群 [Tiwari AK, Pandey SN, Senthilvelan M 和 Lakshmanan M 2013 J 。 数学。物理。 54, 053 506]。经典地，这两个系统也被证明是可线性化和等时的。在这项工作中，我们通过考虑一般有序位置相关的质量哈密顿量来研究非线性振荡器的量子动力学。质量项的排序参数一开始就被认为是任意的。我们观察到这些非线性振荡器的量子版本是完全可解的，前提是质量项的排序参数受到对排序参数任意性强加的某些约束。我们获得了与这两个系统相关的特征值和特征函数。我们还简要地考虑了经典线性化的二次 Linard 振荡器的其他例子的量子版本。