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American Journal of Physics

Unusual commutation relations in physics

André Nauts, Fabien Gatti

Published Nov 18, 2010• 6 citations
Publisher

Abstract

The angular momentum operator L̂ of a particle obeys the usual commutation rule L̂×L̂=iℏL̂. In contrast, the total angular momentum operator Ĵ of a set of particles relative to moving axes (as, for instance, when mounted on a tumbling molecule) obeys the anomalous commutation rule Ĵ×Ĵ=−iℏĴ. We give a pedagogical and mathematically nonsophisticated description of intermediate cases for which partial angular momentum operators, relative to various moving frames, obey unusual commutation relations that are neither usual nor anomalous. Insight into the origin of these unusual commutation relations provides a means of guessing the type of commutation rule that will be obeyed. In particular, it gives a way to avoid, as far as possible, the unusual commutation relations, which lead to complicated and nonsystematic expressions. Some examples from molecular physics are presented.

DOI
10.1119/1.3482257

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The angular momentum operator L̂ of a particle obeys the usual commutation rule L̂×L̂=iℏL̂. In contrast, the total angular momentum operator Ĵ of a set of particles relative to moving axes (as, for instance, when mounted on a tumbling molecule) obeys the anomalous commutation rule Ĵ×Ĵ=−iℏĴ. We give a pedagogical and mathematically nonsophisticated description of intermediate cases for which partial angular momentum operators, relative to various moving frames, obey unusual commutation relations that are neither usual nor anomalous. Insight into the origin of these unusual commutation relations provides a means of guessing the type of commutation rule that will be obeyed. In particular, it gives a way to avoid, as far as possible, the unusual commutation relations, which lead to complicated and nonsystematic expressions. Some examples from molecular physics are presented.
Abstract In general some operators in quantum mechanics are not commutable, which means the measurement of two or more operators can’t be done simultaneously. Angular momentum operator and Hamiltonian are examples of operators that can only be measured using a mathematical approach in the form of a commutation relationship. Based on the fifth postulate of quantum mechanics about a constant of motion, if an operator commute with Hamiltonian operator then the operator is a constant of motion. This study was conducted to examine the commutation of the angular momentum operators to the Hamiltonian operator. The commutation of the angular momentum operators <?CDATA L^x{\widehat{L}}_{x}Lx​?> L ^ x , <?CDATA L^y{\widehat{L}}_{y}Ly​?> L ^ y , <?CDATA L^z{\widehat{L}}_{z}Lz​?> L ^ z , <?CDATA L^+{\widehat{L}}_{+}L+​?> L ^ + , and <?CDATA L^−{\widehat{L}}_{-}L−​?> L ^ − to the Hamiltonian operator shows that the operators are commute because the values are zero. Otherwise, the results of the component quadratic angular momentum operators commutator <?CDATA L^x2{\widehat{L}}_{x}{}^{2}Lx​2?> L ^ x 2 , <?CDATA L^y2{\widehat{L}}_{y}{}^{2}Ly​2?> L ^ y 2 , <?CDATA L^z2{\widehat{L}}_{z}{}^{2}Lz​2?> L ^ z 2 , <?CDATA L^2{\widehat{L}}^{2}L2?> L ^ 2 , <?CDATA L+2{L}_{+}{}^{2}L+​2?> L + 2 and <?CDATA L−2{L}_{-}{}^{2}L−​2?> L
A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude l=mr∧v\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}l=mr∧v---is given by l^=r∧(−iℏ∇−ecA)\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})l^=r∧(−iℏ∇−ce​A) in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation [l^i,l^j]=iℏεijkl^k[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\hat{l}}_k[l^i​,l^j​]=iℏεijk​l^k​, that are at the foundation of the calculus of angular momentum and of the theory of \emph{spin}---and Bohm's example of the EPR argument---are not valid in the presence of an electromagnetic field. The expected value of the operator l^=−iℏr∧∇\mathbf{\hat{l}}=-i\hbar\mathbf{r}\wedge\mathbf{\nabla}l^=−iℏr∧∇ is not gauge invariant, it depends on the calibration of the electrodynamic potentials.
The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.
When a classical particle moves in a central field the energy and the three components of angular momentum are all constants of the motion. The motion is an orbit whose orientation in space is fixed. To discuss the corresponding situation in quantum mechanics we must examine the commutation properties of the associated operators: If they all commute, then the quantities which they represent may all simultaneously be constants of the motion; if not, we must find as many mutually commuting operators as we can and obtain a maximal set of constants of the motion.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.