量子力学中的角动量本征值问题
通讯作者:
曹则贤,email:zxcao@iphy.ac.cn
- 1 中国科学技术大学近代物理系 合肥 230026;
- 2 中国科学院物理研究所 北京 100190
摘要: 在量子力学中，作用于希尔伯特空间中的矢量ψ上满足对易关系J × J=iħJ的算符即可称为角动量。角动量与SU(2)群的生成元有相同的对易关系，由后者得到的作为{J2，Jz}共同本征态的表示|jm>常常会被当作量子力学的角动量表示。然而，这样得到的表示是纯代数的，是可能的数学表示之一。轨道角动量J=r × p是物理量，J2似无明确的物理意义，轨道角动量的表示问题应该从物理的角度出发加以考察。本文将轨道角动量算符用升降算符ak+，ak，k=1，2，3，表示，基于与空间参照系相结合的占据态空间表示{n1，n2，n3 ；nk=0，1，2，…}探讨J2，Jz的本征值问题。利用Jz与守恒量nb=n1+n2的对易性，J2与守恒量nt=n1+n2+n3的对易性，可以轻松得到算符J2和Jz各自的本征态/本征值表达式，其表现出与算符J2和Jz的固有对称性相关联的结构。只有偶发的(J2，Jz)共同本征态，这真实地反映了J2与Jz之间的角色关系。我们的理论可为涉及角动量的量子力学问题，如朗道能级、自旋—轨道耦合等，提供简单有效的求解路径，并有助于发掘本征值/本征态的内在结构。
Eigenvalue problem for angular momentum in quantum mechanics
Corresponding author:
CAO Ze-Xian, zxcao@iphy.ac.cn
- 1 中国科学技术大学近代物理系 合肥 230026;
- 2 中国科学院物理研究所 北京 100190
- Received Date:
2026-03-01
Abstract: In quantum mechanics, an operator acting on vectors ψ in a Hilbert space satisfying the commutation relation J × J = iħJ is generally called angular momentum. The angular momentum and the generators for rotation group share the same fundamental commutation relation, thus the representation |jm> for the generators of SU(2) group, with an eigenvalue spectrum (J2, Jz)~(j (j + 1), m), is often taken as the representation for angular momentum. However, such a representation is purely algebraic, while the representation for orbital angular momentum should be formulated on the physical footing since the orbital angular momentum J = r × p is a physical quantity. In the current work, we represent the orbital angular momentum with raising and lowering operators ak+, ak, k = 1, 2, 3, and tackle the eigenvalue problem for (J2, Jz) on the basis of the occupation number notation {n1, n2, n3; nk = 0, 1, 2,…} associated with a given spatial reference system. The eigenvalue problem for (J2, Jz) can be easily solved by making advantage of the commutativity between Jz and nb = n1 + n2 and that between J2 and nt = n1 + n2 + n3. It is found that the eigenvalues/eigenvectors may reveal a structure regarding to the inherent symmetry of J2 or Jz. Our theory can provide a simple and effective methodology for handling angular momentum in quantum mechanics such as in the Landau level problem, spin-orbital coupling, etc., and can be helpful to revealing the inherent structure underlying eigenvalues/eigenstates.
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