精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量

Exact quantization rule and the invariant

We present and prove the exact quantization rules both for the one-dimensional Schrdinger equation and for the three-dimensional Schrdinger equation with a spherically symmetric potential. In the exact quantization rule, in addition to the usual term Nπ, there is an integral term, called the correction term. For the quantum systems with a so-called shape invariant potential in the supersymmetric quantum mechanics, we find that the correction term is an invariant,independent of the number of nodes in the wave functions. In those systems, the invariant can be determined with the help of the energy and the wave function of the ground state, and then, the energy levels of all the bound states can be easily calculated from the exact quantization rule. Conversely, the validity of the calculated energy levels shows that the correction term is an invariant in those quantum system with a shape invariant potential. The systems with a shape invariant potential we calculated in this paper are the one-dimensional systems with a finite square well, with the harmonic oscillator potential, with the Morse potential and its generalizations, with the Rosen-Morse potentials, with the Pschl-Teller potentials, with the Eckart potential, and with the Hulthen potential, and the three-dimensional systems of harmonic oscillators and the hydrogen atom.