费米子球谐函数的性质
Properties of Fermion Spherical Harmonics
摘要：
费米子球谐函数[ $Y_\ell^{m}(θ,ϕ)$的半奇数$\ell$和$m$ -在以前的文件中提出]被证明具有相同的本征函数的属性作为著名的玻色子球谐函数[ $Y_\ell^{m}(θ,ϕ)$的整数$\ell$和$m$ ]。费米子函数与玻色子函数的区别在于阶梯算符$M_+$对于给定的$\ell$值，使谐波序列在$m$值上上升（下降）的两个或多个参数（$M_-$）不产生预期结果{仅在一种情况下为\em}：当$m$值从$\pm{1/2}$变为$\mp{1/2}$时;即当$m$改变符号时;在所有其他情况下，当梯形运算符试图将$m$取到范围$-\ell\le m\le +\ell$之外时，梯形运算符产生通常预期的结果，包括无效化。在一种情况下，意外的结果并没有使自旋角动量的标量坐标表示无效，因为本征函数性质对于有效的量子力学态是必不可少的，而将具有不同本征值的态联系起来的梯形算符不是必不可少的，并且实际上只对少数物理系统是已知的;自旋角动量的坐标表示与抽象的角动量理论在这方面不同，这只是一个有趣的好奇心。这种自旋角动量的新表示有望在物理系统的理论描述和实验中找到应用，其中粒子的自旋角动量（和相关的磁矩）在空间中取向，因为该取向可由球极角$θ$和$ϕ$指定。
英文摘要：
The Fermion Spherical harmonics [$Y_\ell^{m}(θ,ϕ)$ for half-odd-integer $\ell$ and $m$ - presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [$Y_\ell^{m}(θ,ϕ)$ for integer $\ell$ and $m$]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators $M_+$ ($M_-$) that ascend (descend) the sequence of harmonics over the values of $m$ for a given value of $\ell$, do not produce the expected result {\em in just one case}: when the value of $m$ changes from $\pm{1/2}$ to $\mp{1/2}$; i.e. when $m$ changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take $m$ outside the range: $-\ell\le m\le +\ell$. The unexpected result in the one case does not invalidate this scalar coordinate representation of spin angular momentum, because the eigenfunction property is essential for a valid quantum mechanical state, whereas ladder operators relating states with different eigenvalues are not essential, and are in fact known only for a few physical systems; that this coordinate representation of spin angular momentum differs from the abstract theory of angular momentum in this respect, is simply an interesting curiosity. This new representation of spin angular momentum is expected to find application in the theoretical description of physical systems and experiments in which the spin-angular momentum (and associated magnetic moment) of a particle is oriented in space, since the orientation is specifiable by the spherical polar angles, $θ$ and $ϕ$.
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