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Permutational Structure of Nuclear States
Abstract
One of the fundamental properties of nuclear systems is that they are composed out of fermions. The variety of nuclear phenomena related to this property can hardly be overestimated. In the present section we elaborate some tools for a more detailed analysis of this property.
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Quantum mechanical problems are simplified and usefully structured through the use of group theory. Applications occur when a Hamiltonian commutes with the elements of a given group. In such a case, the group transformation properties of the Hamiltonian eigenkets are determined, quantum numbers and selection rules are supplied, and the construction of eigenkets and the evaluation of matrix elements can be simplified. The zero-order group provides approximate transformation properties, quantum numbers, and selection rules for the perturbed eigenkets.
This chapter presents the definition and the algebraic properties of certain square symbols appearing in the finite irreducible representations of the group U(n). This concept is linked to properties of the symmetric group S(N) and its representations. The recent developments of the technique of S(N) for many-body physics are closely related to the properties of these square symbols. The chapter discusses high-dimensional IR of S(N) for which new techniques are required. The concept of double cosets is very important as it is directly linked to the exchange properties. The properties known for S(N) can serve to bring out new features of the IR of U(n), such as Regge symmetry.
All analyses of interactions in molecular and solid-state systems, however diverse they may be in method of approach and degree of sophistication, must conform to basic principles imposed by what is called the quantum-mechanical “symmetry” of the system. The concept of symmetry of a system characterized by a Hamilton operator H is embodied by the group of Hilbert-space operators which have the following two important properties: a) they preserve the absolute value of the scalar product, |(f, g)|, between any two vectors f, g in the Hilbert space. Let A be such an operator. Then, either (Af, Ag) = (f, g) or (Af, Ag) = (g, f) = (f, g)*. From this it follows that A must be unitary,or an anti-unitary operator, respectively. Further, b) any A must commute with H: AHf = HAf, for all f. We call them “symmetry operators” for the system. It can indeed be easily verified, and it is left as an exercise to the Reader, from the properties a) and b), that these operators form a group in the abstract sense of the word, the symmetry group of the system under consideration. We denote this group by GH.
This chapter discusses the Heisenberg–Weyl ring in quantum mechanics. A program has been carried out that deals with quantum mechanics on a compact one-dimensional space. The system was defined as a quantum system whose momentum operator has a discrete, infinite spectrum of equally spaced eigenvalues. The importance of canonical transformations in quantum mechanics was recognized within a year of its original formulation. Point transformations have been used as transformation groups, the recent applications of which include many-body and scattering problems, while the role of linear canonical transformations has recently been appreciated. The transformation of a given physical system to a mathematically simpler one is a common technique in classical mechanics, usually by taking one of the new canonically conjugate observables to be a constant of motion as the Hamiltonian or the angular momentum. Linear transformations—in a higher dimensional space—become nonlinear when the radial part is isolated in a differential operator realization or when the requirement of the conservation of the commutator bracket is demanded between well-chosen states on a particular basis.
A study of the transition between the H2+ molecule and the corresponding three-body nucleus shows (1) that it is useful to regard the system as composed of two parts (heavy particle; heavy and light particles) between which acts an effective potential; (2) that this potential depends more and more on the relative velocity of the two parts as the masses of the light and heavy particles approach equality; (3) that the wave equation and the potential for the relative motion are obtained in a consistent manner by requiring that a certain form of approximate wave function give the best possible representation of the motion of the three particles, in the sense of the variation principle. This wave function represents a state in which the system resonates between the groupings atom-ion and ion-atom. It is adapted to the treatment of the scattering of neutrons in deuterium and is also used in the text to calculate the binding energy of H3. Application of the same type of approximate wave function to the description of nuclear structure in general, gives rise to the concept of resonating group structure. This picture regards the constituent neutrons and protons as divided into various groups (such as alpha-particles) which are continually being broken and reformed in new ways. Group theory gives information as to which groupings are most important in describing a particular state of a given nucleus. The interchange of neutrons and protons between the groups is rapid. It is largely responsible for the intergroup forces, but also prevents one from attributing any well-defined individuality to the groups except as follows: If the time required for a particle to diffuse between two parts of the nucleus vibrating in opposite phase (in the language of the liquid droplet model) is large in comparison with the period of the vibration, then the particles of the nucleus may be divided into groups which preserve their identity long enough to make possible a simple description of the nuclear motion in terms of the relative displacements of these clusters. Arguments are given to show that the diffusion condition is satisfied for low excitation energies. When the nuclear energy is higher, the groups have significance only in providing a suitable mathematical scheme to treat the nuclear motion (see following paper). Allowed types of motion and energies for low states of Be8, C12, and O16 are calculated in terms of the relative motion of alpha-particle groups, using the methods familiar in molecular structure. The modes of vibration are closely related to those given by the liquid model of Bohr and Kalckar, but many low levels are excluded on symmetry grounds. The general methods outlined here for the description of nuclear structure are to a large extent independent of the nature of the forces between elementary particles. A discussion of the possible existence of many-body forces is given (i.e., forces which cannot be described by a potential that is a sum of potentials involving two particles at a time). The observed variation of nuclear binding energy with atomic number is found not to give sufficient evidence from which to draw any general conclusion. Electron positron theory indicates that a part of the nuclear forces consists of many-body interactions.
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.
No group is of greater importance than the symmetric group. After all, any group can be embedded as a subgroup of a symmetric group. In this chapter, we construct the irreducible representations of the symmetric group S
n
. The character theory of the symmetric group is a rich and important theory filled with important connections to combinatorics. One can find whole books dedicated to this subject, cf. [16, 17, 19, 12]. Moreover, there are important applications to such diverse areas as voting and card shuffling [7, 8, 3].
Nuclear levels in the p-shell are analyzed with respect to supermultiplet quantum numbers. It is shown that effective interactions favour the lowest supermultiplets in both ground and excited states.
The semigroup of all operators T such that (Tx,Tx)⩾(x,x), for all elements of x of a finite-dimensional complex vector space with ( , ) a given, possibly indefinite Hermitian form on that space, is the object under study. It is shown that this semigroup is closed under the operation of taking adjoints with respect to the given form and that every semigroup element may be written as a product of a unitary and a self-adjoint operator (polar form), with the unitarity and self-adjointness defined with respect to the given form. It is further shown that the semigroup is generated by the group of isometries of the given form and the union of a finite family of semigroups, each of which is of an elementary nature. The intersection of the above semigroup with the complex symplectic group is also considered; all the above results have analogues in this case.
Representations of the symmetric group SN symmetry adapted to subgroup sequences ⊗SNj↗ [SE pointing arrow] SN ⊗SiNj[SE pointing arrow] ↗⊗SiN are considered using double-coset decomposition. The matrix elements of the double-coset representatives are given and their group theoretical properties are discussed. The matrix elements are identified with the recoupling transformations of the unitary group by considering the tensor representations of the latter. The orthogonality and completeness relations of the symmetric group expressed in terms of double-coset representative matrix elements are used to establish general relations that must be satisfied by the coupling coefficients of the unitary group.
In this paper we show that sets of polynomials in the components of (2j + 1)-dimensional vectors, solutions of certain invariant partial differential equations, form bases for all the irreducible representations of the unitary group U2j+1. These polynomials will play, for the group U2j+1, the same role that the solid spherical harmonics (themselves polynomials in the components of a three-dimensional vector) play for the rotation group R3. With the help of these polynomials we define and determine the reduced Wigner coefficients for the unitary groups, which we then use to derive the Wigner coefficients of U2j+1 by a factorization procedure. An ambiguity remains in the explicit expression for the Wigner coefficients as the Kronecker product of two irreducible representations of U2j+1 is not, in general, multiplicity-free. We show how to eliminate this ambiguity with the help of operators that serve to characterize completely the rows of representations of unitary groups for a particular chain of subgroups. The procedure developed to determine the polynomial bases of U2j+1 seems, in principle, generalizable to arbitrary semisimple compact Lie groups.
This chapter presents results concerning irreducible representations of the symmetric group, which appears to be unfamiliar to or unappreciated by most chemists and physicists are set forth without proof. These results are closely associated with the familiar methods expounded by Kotani for constructing symmetry-adapted spin functions. Young's diagrams provides explicit rule for obtaining the matrices of the irreducible representations. In order to describe a representation, names are needed for a set of basis vectors. Young tableaux associated with a fixed Young diagram are used to name a set of basis vectors for the corresponding representation. There are two other equivalent sets of symbols that could be used for the same purpose and which are explained, in passing, since they are useful: lattice permutations and Yamanouchi symbols. The topic of lattice permutations is treated by MacMahon and is discussed in the chapter. Yamanouchi symbol is the lattice permutation written backwards.
The theory of tensor operators in the unitary groups is developed in a form which parallels the familiar Racah-Wigner angular momentum calculus. The presentation is principally an attempt to synthesize published results into a comprehensive whole.
DOI:https://doi.org/10.1103/PhysRev.51.51
The fractional parentage coefficients are treated by using the Young operator of the symmetric group and general expressions for the coefficients are obtained. From these expressions one can evaluate the fractional parentage coefficients which separate any numbers of particles from the others for pure or mixed configurations. In order to calculate the fractional parentage coefficients which split off more than one particles, non-standard representations of the symmetric group are made use of. The representations are the outer products of two representations of the symmetric group and the transformation coefficients with the standard representations are calculated.
The problem of separating the spin and isospin dependence from the equation of motion of a system consisting of a small number of nucleons is considered. Certain coefficients, analogous to those used in the theory of angular momentum, are introduced and it is demonstrated that with their use the equation of motion may be reduced to a system of coupled differential equations involving the position coordinates only. Some of the properties of these coefficients and their connection with the permutation group are discussed. Tables of coefficients for three-nucleon and four-nucleon problems are also included.
A new method for antisymmetrizing many nucleon systems and its application to the elastic alpha-16O-scattering is discussed.
Asymptotic wave functions for nuclear reactions with two fragments are expanded into supermultiplets. The expansion coefficients turn out to be essentially coefficients of fractional parentage for the spin-isospin part.
This chapter discusses the supermultiplet expansion and oscillator cluster parentage of light nuclei and the translational-invariant oscillator cluster parent states containing k sets of internally unexcited oscillator clusters. The state of the relative motion carries NS quanta of excitation. Antisymmetrization is affected by projection of an orbital partition and coupling with spin-isospin states. The matrix elements for orbital operators of tensor rank χ between projected cluster parent states can be written in terms of two-body matrix elements. The chapter also discusses 3-cluster configurations that cover nuclear states up to 12C and describes the general features of a k-cluster configuration. The chapter presents the evidence that for light nuclei and low energy, only the few-cluster configurations corresponding to the lowest supermultiplets are involved.
Configurations of two clusters are analysed and adapted to orbital symmetry. Using the concept of double coset generators, basic integrals for normalization and for the application of two-body operators are determined and classified according to their exchange type. The normalization and the two-body matrix elements for symmetry-adapted states are written as linear combinations of basic integrals. The coefficients in these linear combinations are given explicitly.
The matrix elements of two-body interactions between the most general states of n nucleons characterized by the supermultiplet and L-S coupling scheme are considered. The matrix elements can be expressed in terms of reduced matrix elements with respect to the symmetric and rotation group. Reduced matrix elements with respect to the symmetric group result from the separation of the last pair or from the use of tensor operators. These two types of matrix elements are derived and analysed.
Using general properties of the representations of unitary groups and their relations to representations of symmetric groups, the 3j symbol of the unitary unimodular group LU(2) is written in terms of a 9j symbol of the unitary unimodular group LU(J) withJ being the sum of the threej's. The result yields the Regge symmetry of the 3j symbol as a consequence of new relations between Wigner coefficients and special invariants of unitary groups on one hand and the association symmetry of the symmetric group on the other.
The translationally invariant shell model in which the nucleons oscillate harmonically with respect to the nuclear centre of mass is considered. In this model the wave function of the nucleus depends on the 3(A-1) Jacobi coordinates. Thus, the problem of the spurious states is absent in this model. The transformation from one set of Jacobi coordinates into another for the oscillator wave functions is discussed. The method for calculating the fractional parentage coefficients is set out.
The wave function for the composite nucleus is written as a properly antisymmetrized combination of partial wave functions, corresponding to various possible ways of distributing the neutrons and protons into various groups, such as alpha-particles, di-neutrons, etc. The dependence of the total wave function on the intergroup separations is determined by the variation principle. The analysis is carried out in detail for the case that the configurations considered contain only two groups. Integral equations are derived for the functions of separation. The associated Fredholm determinant completely determines the stable energy values of the system (Eq. (33)), Eq. (48) connects the asymptotic behavior of an arbitrary particular solution with that of solutions possessing a standard asymptotic form. With its help, the Fredholm determinant also determines all scattering and disintegration cross sections (Eqs. (50)···(54) and (57)), without the necessity of actually obtaining the intergroup wave functions. The expressions (43) and (60) obtained for the cross sections, taking account of spin effects, have general validity. Details of the application of the method of resonating group structure to actual problems are discussed.
It is shown that the supermultiplet approximation remains valid for
large values of A. The energy differences between states belonging to different
supermultiplets were analyzend by comparing the ground-state energies of three
isobars. (R.E.U.)
Topics covered include: elements of group theory; vector spaces; representations; finite representations of the linear groups GL(n,C) and of their principal subgroups; generalities on Lie groups; rotation group SO(3); the Lorentz group; brief survey of the Poincare group; symmetries of elementary particles; and the unitary symmetry and its generalizations.
In a recent article (V. Bargmann, Comm. Pure Appl. Math., 14:
187(1981)) a family of Hilbert spaces were studied whose elements are entire
analytic functions of n complex variables. The methods developed therein appear
appropriate for a fairly effortless treatment of the representation theory of the
rotation group. Further applications of these methods are considered. (W.D.M.)
The purpose of this monograph is to describe a microscopic nuclear theory which can be used to consider all low-energy nuclear phenomena from a unified viewpoint. In this theory, the Pauli principle is completely taken into account and translationally invariant wave functions are always employed. It can be utilized to study reactions initiated not only by nucleons but also by arbitrary composite particles.
Three structurally distinct and explicit expressions are developed for the boson polynomials. The relationship of these polynomials to the representations of the general linear group and the Gel'fand‐Graev generalized beta functions is noted. A by‐product of these results is a new, closed expression for the irreducible representations of the symmetric group. Some similarities, as well as dissimilarities, between the boson polynomial forms and the canonical tensor operator forms are presented and discussed, the origin of these properties being traced to the similarities and distinctions between Wigner coefficients and Racah coefficients. One of the boson polynomial expressions is used to prove an important new relation in the Racah‐Wigner calculus: the identity of the set of extended projective coefficients to a subset of Racah coefficients. This relationship becomes one‐to‐one for SU(2) and establishes a pattern calculus for the Racah coefficients of angular momentum theory.
The structure of the totally symmetric unit tensor operators (and their conjugates) in U(n) is examined from the viewpoint of the pattern calculus and the factorization lemma. The geometrical properties of the arrow patterns of the fundamental projective (tensor) operators are demonstrated to be the origin of the existence of simple structural expressions for a class of reduced matrix elements of the totally symmetric unit projective operators. An extension of the pattern calculus rules is given whereby these matrix elements can be written out directly. This class of reduced matrix elements is sufficient to permit the construction of the general totally symmetric unit tensor operator. The canonical splitting of the multiplicity in U(3) is similarly shown to be implied uniquely by the geometrical properties of the arrow patterns of the fundamental projective operators and their conjugates. This fact is used to construct explicitly the class of U(3) unit tensor operators having maximal null space. Explicit expressions for a large class of Racah coefficients are also given, and the implications of their limit properties discussed.
The canonical splitting of the multiplicities of the unit tensor (Wigner) operators in U(3) was used in I to determine explicitly one Wigner operator in each (arbitrary) multiplicity set. The denominator function whose zeroes define the null space of this Wigner operator is presented in a new form from which the complete identification of the null space is made. Using the properties of the intertwining number of U(3), the null spaces of all the U(3) Wigner operators are determined, and it is demonstrated that the null spaces of the operators belonging to a multiplicity set are simply ordered by inclusion. The Wigner operator previously obtained from the canonical splitting is shown to be the one having the maximal null space for its multiplicity set.
We define operators that lower or raise the irreducible vector spaces of a semisimple subgroup of a semisimple Lie group contained in an irreducible vector space of the group. We determine the lowering and raising operators for the canonical subgroup Un−1 of the unitary group Un. With the help of these operators, which are polynomial functions of the generators of Un, and the corresponding operators for the subgroups in the canonical chain Un⊃ Un−1⊃ …⊃ U2⊃ U1 we can obtain, in this chain, the full set of normalized basis vectors of an irreducible vector space of Un from any given normalized basis vector of the vector space. In particular we can obtain, using only the lowering operators, the set of basis vectors from the basis vector of highest weight of the vector space. This result is of importance in applications to many‐body problems and in the determination of the Wigner coefficients of Un. In future papers we plan to determine the lowering and raising operators for the orthogonal and symplectic groups.
If An denotes the Abelian group of n × n unitary diagonal matrices and Sn the symmetric group represented by n × n permutation matrices, the set of elements ap with a ∊ An and p ∊ Sn form a group Kn which is the semidirect product of An and Sn. Irreducible representations of Kn and the chains Kn⊃Sn and Un⊃Kn, with Un being the unitary group in n dimensions, are discussed with applications in the harmonic‐oscillator shell model.
The analog of the SU2 (1 − j) symbol is defined and discussed in detail for SUn. An appropriate generalization to SUn of the Condon‐Shortley phase convention is explicitly given.
We show that the group of linear canonical transformations in a 2N‐dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well‐known dynamical group of the N‐dimensional harmonic oscillator. Finally, we study the case of n particles in a q‐dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.
We discuss the calculational procedures of the kernels of GCM, RGM and OCM and some properties of them related to their calculation.
The GCM kernels for various types of systems are treated and methods are discussed on the analytical evaluation and on the
decomposition in terms of the number of nucleons exchanged between clusters. The RGM kernels are evaluated by the integral
transformation of GCM kernels. Various formulas of this transformation are presented including those for the systems of clusters
with unequal oscillator widths. The problems related to the RGM norm kernel (RGM-NK) are discussed; firstly on the solution
of the eigen-value problem of RGM-NK for various kinds of systems, secondly on the evaluation of kernels or physical quantities
obtainable from the knowledge of RGM-NK and finally on the cluster model space for whose character the solution of the eigen-value
problem of RGM-NK gives an indispensable information. The projection operator onto the Pauli-allowed states in OCM is obtained
directly from the solution of the eigenvalue problem of RGM-NK. IN this paper we also present another method of construction
of this operator of OCM which needs not to solve the eigen-value problem of RGM-NK which is tedious for complex systems even
with our present calculational techniques.