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College Physics

Views on the angular momentum in quantum mechanics

Xi Zhou

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

It is pointed out that the angular momentum operator J ^ does not have complete set of eigenvalues and eigenvectors,the dynamical quantity described by angular momentum operator J ^ is non observable,it is not appropriate to treate angular momentum J as a whole.

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Discusses angular momentum,an important but seemingly difficult concept in physics.An analysis of angular momentum direction in fix axis movement finds the direction different from the fix axis in normal condition while a probe into the angular momentum in atom movement leads to the conclusion that the atomic orbit angular momentum operator are not complete set of mechanical eigenvalues and eigenvectors and the orbit angular momentum operator is non-observable.Points out a quantum physical nature in photon spin angular momentum,which is not corresponding to classical physics.
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.
We propose an algebraic diagonalization procedure for the angular momentum matrix Jx which is based on the introduction of a suitable generating function. The consideration of the analytic properties of the generating function leads to the eigenvalues of the problem. The coefficients of the power series expansion of this function generate the components of the eigenvectors of Jx. This procedure provides the beginning student of quantum mechanics with an alternative derivation of the eigenvalue spectrum of the angular momentum.
The recent two papers have successively proposed the concepts and more precise expressions of strict conservation or statistical conservation of physical quantities, which are different from the expression of a conserved quantity in quantum mechanics. The Hamiltonian operator of the system discussed in this paper are independent of time, so it is commutative with the operators of an angular momentum itself, its component and its square, so that all these quantities are conserved quantities. In an isolated composite system, one of the angular momentum components of the total system are strictly conserved, while the angular momentum components of the subsystems can only be conserved statistically, both of which can be expressed in terms of eigenvalues and eigenfunctions. Generally, angular momentum itself does not have eigenvalues and eigenfunctions, so it cannot be expressed as strict conservation in terms of these concepts. Although there are eigenvalues and eigenfunctions for the square of angular momentum, they cannot express strict conservation either. And they can only be expressed as statistical conservation.
Eigenfunctions and eigenvalues of the operator of the square of the angular momentum are studied. It is shown that neither from the requirement for the eigenfunctions be normalizable nor from the commutation relations it is possible to prove that the eigenvalues spectrum is a set of only integer numbers (in units ℏ=1\hbar=1ℏ=1). We present regular, normalizable eigenfunctions with the non-integer eigenvalues thus demonstrating that a non-integer angular momentum is admissible from the theoretical viewpoint.
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.