Quantum Harmonic Oscillator

A diatomic molecule vibrates somewhat like two masses on a spring with a potential energy that depends upon the square of the displacement from equilibrium. But the energy levels are quantized at equally spaced values.

The energy levels of the quantum harmonic oscillator are

量子谐振子的能量本征值为

and for a diatomic molecule the natural frequency is of the form

并且对于双原子分子来说，自然频率的形式为

where the reduced mass is given by

其中约化质量由下式给出

This form of the frequency is the same as that for the classical simple harmonic oscillator. The most surprising difference for the quantum case is the so-called "zero-point vibration" of the n=0 ground state. This implies that molecules are not completely at rest, even at absolute zero temperature.

这种频率形式与经典简谐振子的相同。量子情况中最令人惊讶的不同之处在于所谓的“零点振动”，即n=0的基态。这说明即使在绝对零度时，分子也不完全静止。

The quantum harmonic oscillator has implications far beyond the simple diatomic molecule. It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. In real systems, energy spacings are equal only for the lowest levels where the potential is a good approximation of the "mass on a spring" type harmonic potential. The anharmonic terms which appear in the potential for a diatomic molecule are useful for mapping the detailed potential of such systems.

量子谐振子的影响远不止于简单的双原子分子。它是理解较大分子中复杂振动模式、固体晶格中原子运动、热容量理论等的基础。在实际系统中，只有最低能级的能量间隔相等，此时势能可以很好地近似为“弹簧上的质量”类型的谐振势能。双原子分子中出现的非谐振项对于映射此类系统的详细势能曲线是有用的。

双原子分子的振动类似于两个质量在弹簧上的振动，其势能取决于偏离平衡位置的位移的平方。但能量级是量子化的，且等间距分布。