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Angular momentum and spherical symmetry

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.

DOI
10.1093/acprof:oso/9780198749349.003.0018

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Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
Angular momentum theory provides a mathematical foundation for the rotation of the eigenvectors of the angular momentum operator, the spherical harmonics functions. A useful property of spherical harmonics is that their rotation in 3D space corresponds to rotation in the virtual space of their azimuthal degrees of freedom. In addition to that, spherical harmonics functions that have different orbital indices do not mix. Given that spherical harmonics functions have known rotation properties, and that any arbitrary function can be expanded in a series of spherical harmonics, this property allows the rotation of an electrostatic potential when it is expanded in the series of spherical harmonics. In this chapter, the exponential form of the rotation operator, the rotation matrix for a 3D rotation of the coordinate system, and the principles of rotation of vectors and functions are described. The chapter provides a detailed self-contained introduction to the theory of angular momentum, derives the commutative properties of the angular momentum operator, and explains shift operators, while minimizing references to quantum mechanics and entirely avoiding group theory.
The addition theorem for spherical harmonics is just a special case of the law for the transformation of spherical harmonics under rotation of their arguments. Certain operators are scalars under rotation; this means that they commute with the total angular momentum and are unchanged by the transformation. The types of operators having simple transformation properties under rotation are known generally as tensor operators. The energy eigenstates of an atom, taking into account the spin-orbit interaction, are not usually eigenstates of the z component of the total orbital angular momentum, but are eigenstates of the total angular momentum, spin plus orbital. Sehwinger has looked at the theory of angular momentum from a point of view that enables one to carry out very elegant calculations of the various quantities occurring in the theory of angular momentum. The quanta of some different oscillators are simply the quarks that compose the strongly interacting particles.
As in classical mechanics, rotation in quantum mechanics is a transformation which deals with angular momentum. The difference with classical mechanics comes from the fact that angular momentum is a vector operator and not a usual vector and its components do not commute. As for any transformation in quantum mechanics, to each rotation we can associate an operator which acts in state space. The expression of this operator depends on whether the rotation is passive, that is we do a rotation of the coordinate axes and the physical system is left unchanged, or active, in which case the coordinate axes are unchanged and the rotation is performed on the physical system. In the first part (Chaps. 1 and 2) of this book, details concerning both aspects are given. Following the definition of the geometrical transformation associated with the most general rotation, we give the expression of the rotation operator for specific cases. Transformation laws for scalar fields, vector fields and spinor fields are given as well as transformation laws for scalar operators, vector operators and more generally, for operators of any rank. The second part (Chaps. 3 and 4) deals with angular momentum algebra. We define the coupling coefficients of 2, 3 and 4 angular momenta as well as the recoupling coefficients. The definition of the irreductible tensor operator, which is a generalisation of scalar and vector operators, is given as well as the Wigner-Eckart theorem. The application of this theorem to more complex cases is studied.
The chapter begins with the introduction of the two-particle Schrödinger wave equation (SWE) and the solution of this equation for the hydrogen atom. The orbital angular momentum of the electron results from the SWE solution. The Pauli spinors are introduced, and the SWE wavefunctions are modified to account for the spin of the electron. The structure of multielectron atoms is then discussed. The discussion is focused on low-Z atoms for which Russell–Saunders or LS coupling is appropriate. Alternate coupling schemes are briefly discussed. Angular momentum coupling algebra, the Clebsch–Gordan coefficients, and 3j symbols are then introduced. The Wigner–Eckart theorem is discussed, and the use of irreducible spherical tensors for evaluation of quantum mechanical matrix elements is discussed in detail.