The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.
The global rotational degrees of freedom in the Schr\"odinger equation for an N-body system are completely separated from the internal ones. After removing the motion of the center of mass, we find a complete set of
(2l+1) independent base functions with angular momentum
l. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.
In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum
j. In contrast, here we show that a spatial wavefunction,
jm(ϕ,θ,χ)= eimϕδ(θ−θm) ei(j+1/2)χ, which treats
j in the
∣jm> state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators;
ϕ,
θ,
χ are the Euler angles, and
cosθm=(m/∣j∣) is the Vector-Model polar angle. The
jm(ϕ,θ,χ) gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in
j and
m) which predicts the effective wavepacket angular uncertainty relations for
ΔmΔϕ,
ΔjΔχ, and
ΔϕΔθ, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive
j-rotation measurements. We also use the
jm(ϕ,θ,χ) to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles,
g=2, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to
j=1/2, these expressions are either exact (the last two) or excellent approximations (the first two), showing that
jm(ϕ,θ,χ) gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.
Abstract The spin-independent properties of an electronic system are completely determined by spinless density functions, P1(1; 1') and P 2( 1,2; 1',2 '), derived from the 1-and 2-electron density matrices, p1(1; 1') and p2(1,2 ; 1',2'). These functions, and their physical inter-pretation, have been dealt with in part I: the present paper contains the generalizations necessary when the Hamiltonian contains spin terms (as in discussions of electron and nuclear spin resonance). The density matrices, and more generally the transition density matrices p1(kλ|1; 1') and p2(kλ|l,2; 1',2') associated with states and Φkand Φλ are resolved into spinless components associated with different spin situations: p1 has four such components (one giving the density of a-spin electrons, a second the density of β-spin electrons, and two with less direct physical significance), while p2 has sixteen components. It is shown that all components, for states k=(S, M),λ= (S',M') of given S, S', are determined by three 2-electron functions.For k=λ these reduce to p2, the spinless pair function, from which the electron density p1 may be derived by integration; Q2a ‘conditional' spin density, from which the (1-electron) spin density Q1 may be derived (as in the case of p1 and P2); Q c. a., a ‘coupling anisotropy function', which describes the anisotropy of spin-spin coupling associated with quantization of the spin z-component. It is also useful to introduce (though it is not an independent function). Qc, a ‘coupling density function', which measures the coupling between the spins of two electrons at different points in space. All spin-dependent effects, including the usual spin-spin and spin-orbit couplings and also (indirect) nuclear and electron-nuclear spin-spin couplings, may be related to the many- electron wave function through the above density functions. All the densities have a physical interpretation, independent of the precise nature of the wave functions (which may, for example, be the exact eigenfunctions of a spinless Hamiltonian or merely approximations of orbital form). The derivation of the density functions from approximate wave functions is considered in some detail. For functions describing weakly coupled electron groups (e. g. inner-shell and valence electrons) the densities may be expressed in terms of those of the separate groups. This reduction may be followed by an orbital description of each group; and subsequently, by a linear-combination-of-atomic-orbitals (l. c. a. o.) approximation. In the simpler approximations, all the coupling effects discussed are determined by electron and spin ‘populations'. Preliminary calculations suggest that such approximations may in many cases be adequate.