Orbital angular momentum and group representations

Abstract

A new approach is presented to the question why in quantum mechanics the orbital angular momentum has integral eigenvalues only. The problem is formulated in terms of linear operators on the Hilbert space h of square-integrable functions of the angular variables ϕ and cos ϑ. The components of the orbital angular momentum are represented by self-adjoint operators L j ( j = x, y, z) on h which are constructed starting from the familiar differential operators in terms of ϑ and ϕ. The construction is possible in infinitely many ways. These are labeled by a continuous parameter δ, the choice δ = 0 corresponding to integral eigenvalues. For every particular δ, the operator ∫ j L j 2 has one and only one extension L 2 which is self-adjoint on h and such that, for each j, the domain of L 2 is contained in the domain of L j . This L 2 is the only operator which may represent the square of the orbital angular momentum. It is only in the case δ = 0 that L 2 commutes with each L j . Also, δ = 0 is the only case in which the spectra of L x , L y , and L z are the same. These are results which refer directly to the outcome of measurements. It is suggested that this explains why nonintegral eigenvalues for the orbital angular momentum are not found in nature. It is observed that orbital angular momenta with nonintegral eigenvalues do not fit in the framework of group theory. This is discussed from a general point of view. For a set of self-adjoint operators L j on some Hilbert space R to generate a unitary representation of a simply connected, compact, semisimple Lie group, it is shown to be necessary and sufficient that the operators L j satisfy suitable commutation relations and commute with a self-adjoint Casimir operator L 2. If, in addition, the group is of rank 1, it is necessary that the generators all have the same spectrum, which must be symmetric with respect to the origin. There is thus a general connection between the occurrence of group representations and the properties of observables.