三维约束：再论WKB
Three Dimensional Confinement : WKB Revisited
摘要：
在WKB近似下，给出了一种确定量子力学系统能量本征值的方法，该系统被限制在半径为$r_0$的刚性不可穿透球盒中.这里采用的方法不是考虑离心项的兰格修正，而是Hainz和Grabert的方法：离心项被微扰展开（以$\hbar$的幂），将其分解为2项-经典离心势和量子修正。Hainz和Grabert发现，这种方法再现了氢原子的确切能量，在$\hbar$中达到一阶，所有高阶修正都消失了。在本研究中，这种形式主义的径向势硬壁限制下的情况下，检查是否同样的论点也适用于这样的限制系统。作为明确的例子，3个广为人知的势被研究，这是相当重要的各种原子现象的理论处理涉及原子跃迁，即，三维谐振子、氢原子和Hulthen势。
英文摘要：
An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert : The centrifugal term is expanded perturbatively (in powers of $\hbar$), decomposing it into 2 terms -- the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energies of the hydrogen atom, to the first order in $\hbar$, with all higher order corrections vanishing. In the present study, this formalism is extended to the case of radial potentials under hard wall confinement, to check whether the same argument holds good for such confined systems as well. As explicit examples, 3 widely known potentials are studied, which are of considerable importance in the theoretical treatment of various atomic phenomena involving atomic transitions, viz., the 3-dimensional harmonic oscillator, the hydrogen atom, and the Hulthen potential.
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