摘要:
角动量理论是量子力学的重要内容。基于角动量分量同三维转动生成元具有同样的基本对易关系的考量，则由基本对易关系就得到了分立的本征值谱(J2,Jz)~(j(j+1),m)。又由于球坐标系下角动量平方J2与动能算符的角部分相同，角动量就这样被纳入了波力学方程，由此解得的定态波函数是(H,J2,Jz)的共同本征函数。然而，不同于波函数理论体系，在后来发展的算符—态矢理论体系中，态矢携带关于系统的全部信息。将角动量用必要的三组独立产生—湮灭算符表示，并在算符—态矢理论体系中考察角动量算符，会发现定态不必然还是(H,J2,Jz)的共同本征态。以在球坐标系和直角坐标系下皆可分离变量的严格可解三维各向同性谐振子为依据，作者详细研究了定态对应的态矢子空间中的(H,J2,Jz)本征值谱问题。在给定总粒子数n的情形下，即限制在特定的n所决定的子空间中，算符—态矢表示给出的角动量分量Jz具有分立的本征值而角动量的本征值却可以是连续变化的，而这正反映出角动量算符J=x × p的根本性质。当角动量分量本征值(以ħ为单位)接近总粒子数n时，基于态矢的计算与基于波函数的计算其结果是一致的，原因是n一定的定态被限制在态矢空间中的一个由等能面所定义的子空间中了。认识到既有的量子力学角动量理论的一些缺陷，则此前涉及轨道角动量之物理效应的相关表述都有修正的必要。
Abstract:
The angular momentum theory is an important constituent of quantum mechanics. The discrete eigenvalue spectrum of the angular momentum (J2, Jz)~ (j(j + 1), m) is determined through the three generators of rotation, as they share the same fundamental commutation relation. Moreover, since under spherical coordinate system J2 is identical to the angular part of the Laplacian operator, the angular momentum is thus incorporated into the wave equation, and the wavefunctions for stationary states are believed to be the common eigenfunctions of (H, J2, Jz). We notice that unlike wave functions, the state vectors in ket space are expected to carry out all information about the system concerned. When the angular momentum is represented with three sets of independent creation-annihilation operators as it should be, and its action upon the state vectors is scrutinized, it is found that the stationary states are not necessarily the common eigenvectors of (H, J2, Jz). By taking the advantage of being separable under both the spherical coordinates and the Cartesian coordinates, the eigenvalue spectrum of (H, J2, Jz) for the stationary states of the three dimensional isotropic harmonic oscillator is carefully investigated. For a given stationary state, i. e., with a fixed particle number n which also denotes the energy eigenvalue, while Jz has a discrete eigenvalue, the eigenvalue j (j + 1) can be continuous, thus revealing the true nature of the angular momentum operator J = x × p. For a fixed n , when the m values are sufficiently large, the resulting j (j + 1) is the same as that obtained in terms of eigenfunctions, resulting from the fact that now the stationary states are restricted to a subspace defined by an isoenergetic surface in ket space. Some formulations for angular momentum-related problems should be accordingly readdressed.
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