This page intentionally left blank www.com ANGULAR MOMENTUM IN QUANTUM MECHANICS BY A. EDMONDS PRINCETON, NEW JERSEY PRIKCETON UNIVERSITY PRESS 1957 www.com ANGULAR MOMENTUM IN QUANTUM MECHANICS www.com INVESTIGATIONS IN PHYSICS Edited by EUGENE W IGNER and ROBERT HOFSTADTER 1. Mathematical Foundations of Quantum Mechanics by JOHN VON NEUMANN 3. Shell Theory of the Nucleus by EUGENE FEENBERG 4.
Angular Momentum in Quantum Mechanics by A.com Copyright ® 1957, by Princeton University Press London: Oxford University Press L. Card 57-5446 Printed in the United States of America www.com PREFACE Th1 concepts of angular momentum and rotational invariance play an important part in the analysis of physical systems. They have a special significance in quantum mechanics, for here we find that calcu- lations may be divided in a natural way into two parts, namely (i) the computation of quantities which are invariant under rotations (for example the Slater integrals of atomic spectroscopy) and (li) the evalua- tion of expressions which depend only on the rotational properties of the various operators and state vectors involved. It is remarkable that the structure of an expression of this latter kind is prima'rily & function of the complexity of the system being studied (e.
the number of angular momenta in the coupling scheme) and is relatively independent of its precise physical nature. This fact has made it possible to develop & very general theory of angular momentum algebra, from which can be derived computational methods applicable to problems in such fields as atomic, molecular and nuclear spectroscopy, nuclear reactions, and the angular correlation of successive radiations from nuclei. It has been my aim not only to give an account of this theory, but also to provide a practical manual for the physicist who wishes to use" the associated computational methods. To this end I have paid attention to questions of notation and phase convention a.nd have included tables of formulas a.nd references to numerical compilations, so as to facilitate the evaluation of the various coefficients defined in the text.
The reader is assumed to have a general knowledge of quantum mechanics; an acquaintance with the theory of group represerfations should not be necessary. /' The text is based upon the notes of lecture courses given during the last few years in the Universities of Birmingham, Manchester, Paris, Copenhagen, and Uppsala. The greater part of the writing was done while I was a member of the CERN Theoretical Study Division in Copenhagen. I am grateful to Professor Niels Bohr for the privilege of working during that time in the friendly and stimulating atmosphere of his Institute.
A number of colleagues have contributed by discussions and criticisms. In particular I should like to thank K.com TABLE OF CONTENTS CHAPTER 1. Group Theoretical Preliminaries 3 1. Elementary Theory of Groups 5 1.
The Euler Angles. The Quantization of Angular Momentum 10 2. Definition of Angular Momentum in Quantum Mechanics 10 2. Angular Momentum of a System of Particles.
Representation of the Angular Momentum Operators. The Physical Significance of the Quantization of Angular Momentum. The Eigenvectors of the Angular Momentum Operators J2 and J. The Spin Eigenvectors.
Angular Momentum Eigenfunctions in the Case of Large l 27 2. Time Reversal and the Angular Momentum Operators 29 CHAPTER 3. The Coupling of Angular Momentum Vectors. The Addition of Angular Momenta.
Commutation Relations between Components of Jl, J2' and J. Selection Rules for the Matrix Elements of Jl and 12 35 3. The Choice of the Phases '"'~ the States W('YiJ2im) 36 3. The Vector Coupling Coefficients.
Computation of the Vector Coupling Coefficients. The Wigner 3-j Symbol. Tabulation of Formulas and Numen,J.l Values for Vector- Coupling Coefficients. Time Reversal and the Eigenvectors Resulting from Vector Coupling.
The Representations of Finite Rotations. The Transformations of the Angular Momentum Eigen- vectors under Finite Rotations. The Symmetries of the ~:!}m. Products of the ~":'m(af3'Y).
Recursion Relations for the d;,t?m({~) 61 4. Computation of the d;:,lm(/3). Integrals Involving the ~;':-')m(a{3'Y). The ~;':-')m(W) as Angular Momentum Eigenfunctions 64 4.
The Symmetric Top .com viii TABLE OF CONTENTS CHAPTER 5. Spherical Tensors and Tensor Operators 68 5. The Tensor Operators in Quantum Mechanics. Factorization of the Matrix Elements of Tensor Operators (Wigner-Eckart Theorem).
The Reduced Matrix Elements of a Tensor Operator 75 5. Hermitian Adjoint of Tensor Operators. Electric Quadrupole Moment of Proton or Electron. The Gradient Formu]a.
Expansion of a Plane Wave in Spherical Waves 80 5. Vector Spherical Harmonics. Spin Spherical Harmonics. Emission and Absorption of Particles.
The Construction of Invariants from the Vector- Coupling Coefficients. The Recoupling of Three Angular Momenta 90 6. The Properties of the 6-j Symbol. Numerical Evaluation of the 6-j Symbol.
The Evaluation of Matrix Elements in Actual Problems. Matrix Elements of the Tensor Product of Two Tensor Operators. Selected Examples from Atomic, Molecular and Nuclear Physics < • • • • • • • • • • • 113 ApPENDIX 1. Theorems Used in Chapter 3.
Approximate Expressions for Vector-Coupling Co- efficients and 6-j Symbols 122 Tables 1-5. 124 Cited References and Bibliography 133 Index .com ANGULAR MOMENTUM IN QUANTUM MECHANICS www.com This page intentionally left blank www.com CHAPTER 1 Group Theoretical Prelirninaries 1. Introduction The subject of this book is the detailed development of the uses of the principle of conservation of angular momentum in the analysis of physical systems. While this principle is by no means trivial in classical mechanics, it is of fundamental importance in the quantum mechanics of many-particle systems.
Such systems include the more complex atoms, the atomic nuclei treated from the point of view of the inde- pendent particle model, and experiments in which particles are emitted from or absorbed by nuclei. We shall first discuss the relevance of conservation of the angular momentum of a system in classical mechanics, and see how it is related to the symmetry of the Hamiltonian of the system with re~pect to rotations of the frame of reference. Thus even in a classical a.ualysis we find that 'the theory of the group of rotations in three dimensions is bound up with the idea of angular momentum.l l THE SYMMETRY OF THE HAMILTONIAN. A constant of the motion is a function of the canonical variables which does not change with time; and in the classical mechanics a knowledge of all the constantsof the motion of a system amounts to a solution of the equations of motion.
Now for any function u of the canonical variables which does not depend explicitly on the time the Poisson bracket of the function with the Hamiltonian is zero; for du dt = [u, H] = O. An infinitesimal contact transformation may be defined as a contact tran.s- formation which changes the canonical variables q;) Pi (i = i, 2,. , n) by an infinitesimal amount: The gonerating function F of the infinitesiInal transformation differs IFor a more detailed treatment see any advaneed textbook on classical mechanics, e. GROUP THEORETICAL PRELIMINARIES only infinitesimally from the generating function of the identity trans- formation, which is L, qiP~.
We may write it therefore as F = L qiP: i + e G(q, p') where e is an infinitesimal parameter. It is customary to call G(q, p') the generating function of the infinitesimal transformation, in spite of the fact that this is also the name of the quantity F. It may be shown that the change in a function u of the canonical variables due to this transformation is ~ = s[u, G]. Hence replacing u by the Hamiltonian H, we have oH = e[H, G].
Thus we deduce that the constants of the motion are the generating functions of those infinitesimal contact transformations which leave the Hamiltonian invariant. We find in particular that the angular momentum components are the generating functions of the infinitesimal rotations about the cor- responding axes of the frame of reference. Thus if the angular momentum is a constant of the motion, then the Hamiltonian of the system is symmetric with respect to rotation of the frame of reference about the origin. We say that the group of the Hamiltonian, i.
the group of trans- formations which leave the Hamiltonian invariant, contains the group 80(3) of rotations in three-dimensional space. This fact is of importance in quantum mechanics, for the theorem Qf Wigner-Eckart states 2 that if T is an element of the group GH of the Hamiltonian H, and if u is an eigenvector of H, then Tu is also an eigenvector of H with the same eigenvalue. This implies that all eigenvectors of H belonging to a given irreducible representation of GR have the same eigenvalue, i. are degenerate in energy; however this statement contains group theoretic~l terminology which has not yet been explained.
In the case of a system with rotational symmetry, the theorem implies that, as is well known, the angular momentum eigenvectors are eigenvectors of the energy and that the set of states with the same total angular momentum and different values of the z-component is degenerate. ELEMENTARY THEORY OF GROUPS 5 1. Elementary Theory of Groups3 The concept of group is a generalization of the properties of a large number of systems of mathematical interest; such systems as the set of all permutations of n objects, the set of all rotations of a rigid body, the set of all nonsingular linear transformations on a given vector space. An ab8tract group is defined without reference to any particular physical or mathemaiicalsystem.Itis in fact a set of elements among which a law of composition is defined such that the composition of any two elements a and b of the group taken in this order and denoted by ba is an element of the set." We must add to this property the following conditions: 1.
The associative law c(ba) = (cb)a. There exists a unit element 1, which leaves any element a unaltered on composition with it: la = al = a. To each element a corresponds an inverse a-I which gives on com- position with a the unit element: The number of elements in a group, its order, may be finite, or denumer- ably or nondenumerably infinite. Among finite groups are the symmetry groups of the regular solids and the permutation groups on a finite number of objects.
The positive and negative integers form a group of denumerably infinite order with respect to addition. The simplest group with a nondenumerable set of elements is the set of real numbers with respect to addition, or equivalently the set of all translations of a point on a line. A subgroup h of a group g is a set of elements of g which itself fulfills the group conditions. The unit element must thus belong to h, and if a and b both belong to h, then so do a-I and ba.
The groups we shall be concerned with are those with a nondenumer- able infinity of elements. Let us consider first the set of all nonsingular linear homogeneous transformations on an n-dimensional vector space; we suppose the transformation matrices to have complex coefficients. This set clearly forms a group with respect to composition of the trans- formations (i. to matrix multiplication); it is known as the full linear group GL(n).
Restriction of these transformations to unitary trans- B'The reader is referred for a more detailed treatment of the applications of the theory of groups to quantum mechanics to the well-known works of Wey! (1931), Wigner (1931), Eckart (1930), van der Waerden (1931), and Bauer (1933). "Note that in general ab y&. GROUP THEORETICAL PRELIMINAQ1ES formatipl1s giv~ us the unitary group U(n), which' is a subgroup of GLr",.this relation being symbolized by U(n) C GL(n) We~may make the further restriction that t)le unitary matrices have + deterIDinant. 'rhe resulting group is called the special unitary group SU(n).
The group of all real linear homogeneous .