Abstract In the framework of geometric quantization we extend the harmonic oscillator rules to rotation and vibration of molecules. First, the geometric quantization of the rigid rotor is introduced. The novelty is that we found an explicit relation giving the impulse along the third axes, which takes discrete values and found that it depends not only on the principal quantum number, but also on the orbital and magnetic quantum numbers. Geometric quantization is further extended to the class of vibrational (damped) systems. The theory is sought as mathematical requirement, which can be traced in the analysis of integrable systems. It is shown that properties of the critical points of the momentum map can be the key to the integrability. A C ⁎ isomorphism of the momentum map between the Lie algebra of physical observables and that of the harmonic oscillator induces the geometric quantization on former. We show how the method works, taking into account the Lennard-Jones' potential, which characterizes the interaction between molecules. A second example shows a physical system having the energy levels depending on the magnitude of the frequencies.

中文翻译：

---

同核双原子分子旋转振动运动的几何量化

摘要 在几何量子化的框架下，我们将谐振子规则扩展到分子的旋转和振动。首先介绍刚性转子的几何量化。新颖之处在于，我们发现了一个明确的关系，给出了沿第三轴的脉冲，该关系采用离散值，并发现它不仅取决于主量子数，还取决于轨道和磁量子数。几何量化进一步扩展到振动（阻尼）系统类。该理论被视为数学要求，可以在可积系统的分析中进行追溯。结果表明，动量图临界点的性质是可积性的关键。物理可观测量的李代数和谐振子的李代数之间动量映射的 AC ⁎ 同构引起了前者的几何量化。我们展示了该方法的工作原理，同时考虑到了表征分子间相互作用的 Lennard-Jones 电位。第二个例子显示了一个物理系统，其能量水平取决于频率的幅度。