That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.
That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.
We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of angular momentum could be only integer. We reexamine the arguments based on this requirement and alternate condition imposed by Pauli and show that they do not follow from the first principles and therefore these constraints can dropped. Consequently, we arrive to the same conclusion as in [1]: there exist regular, normalizable eigenfunctions with the non-integer eigenvalues thus a non-integer angular momentum is perfectly admissible from the theoretical viewpoint. The issue of the nature of eigenvalues forming the spectrum of the angular momentum remains open. What can be derived from the first principles is that to a fixed value of the angular momentum L corresponds a discrete spectrum of eigenvalues of the third component of the angular momentum, m, defined by the relation |m|=L-k, k=0,1,...,[L], where [L] is an integer part of L. As a mathematical byproduct of our analysis of eigenfunctions, we present an alternate definition of a power of a complex number allowing to retain initial translational invariance of a base.
This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
The recently established existence of spherical harmonic functions, Yℓm(θ,ϕ) for half-odd-integer values of ℓ and m, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles θs,ϕs, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, S2,Sz, are represented by the usual differential operators in spherical polar coordinates (commonly used for L2,Lz), and their electron-spin eigenfunctions are sinθsexp(±ϕs/2). This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, α,β, that ``integration over spin coordinates'' is a true integration (over the angles θs,ϕs). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.