Simple and efficient methods for adding angular momenta and for finding angular momentum eigenstates for systems of equivalent electrons are developed. Seniority quantum numbers and coefficients of fractional parentage are presented in a simple manner which does not require 6- and 9-j symbols or knowledge of group theory. Several different common representations are used in specific examples. The transformation matrices between jj- and LS-coupled states are explicitly introduced as an example of change of representation and are used to find matrix elements of the spin-orbit interaction and to examine the Zeeman effect in the two representations. The material is suitable for a graduate course in quantum mechanics.
Simple and efficient methods for adding angular momenta and for finding angular momentum eigenstates for systems of equivalent electrons are developed. Seniority quantum numbers and coefficients of fractional parentage are presented in a simple manner which does not require 6- and 9-j symbols or knowledge of group theory. Several different common representations are used in specific examples. The transformation matrices between jj- and LS-coupled states are explicitly introduced as an example of change of representation and are used to find matrix elements of the spin-orbit interaction and to examine the Zeeman effect in the two representations. The material is suitable for a graduate course in quantum mechanics.
A simple method is given for finding the values of the total angular momentum J for configurations of equivalent electrons in jj coupling. It is also shown how a knowledge of both the LS and jj terms can contribute to the student’s understanding of the role of different representations in elementary quantum mechanics. Because the principles and the computations required are elementary, this material is particularly suitable for presentation to undergraduates and to graduate students taking a first course in quantum mechanics.
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.
This book offers a concise introduction to the angular momentum, one of the most fundamental quantities in all of quantum mechanics. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3- j , 6- j , and 9- j symbols. Throughout, the author provides practical applications to atomic, molecular, and nuclear physics. These include partial-wave expansions, the emission and absorption of particles, the proton and electron quadrupole moment, matrix element calculation in practice, and the properties of the symmetrical top molecule.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.
In Section 3 we saw how to combine the intrinsic spin s and the orbital angular momentum L to form eigenstates of J 2 where J = L + s . The coefficients in ( 3.30 ) and ( 3.31 ) are special cases of a general class of transformation coefficients. The states on the r.h.s. of ( 3.30 ) and ( 3.31 ) are eigenstates of s 2 , s z , L 2 , L z . On the left hand side there are eigenstates of s 2 , L 2 , J 2 and J z . In general, we may start with any two sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq354.tif"/> of two different systems. The spins j 1 and j 2 may be either orbital angular momenta or total spins of two particles, two intrinsic spins or one orbital angular momentum and one intrinsic spin. The latter case was considered in Section 3. In other cases j 1 may be the rank of an irreducible tensor operator and j 2 the angular momentum of a wave function or both j 1 and j 2 may be the ranks of two irreducible tensor operators. To simplify the notation we will use, for the transformation of the sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq355.tif"/> under infinitesimal rotations, the expressions ( 6.20 ). This will not detract from the applicability of the following results to other tensorial sets. As pointed out several times above, the following mathematical derivations have general validity and are independent of the physical nature of ψ j 1 m 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq356.tif"/> and ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq357.tif"/> . They are due only to the transformation properties of the two sets under rotations.
We know that even a single electron in a hydrogen atom generally has orbital angular momentum L (with (2 l + 1) possible orientations where l is a integer) and spin angular momentum S (with two possible orientations commonly described as spin up and spin down or s = ± 1/2). In more complicated atoms there may be many components of angular momentum; for example, the helium atom has two electrons both of which have orbital and spin angular momenta. Also, in many atomic systems of physical interest we need to know how to combine angular momenta. This chapter explains how to do it.