Marcos Moshinsky

Linear Canonical Transformations and Their Unitary Representations

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.

Transformation brackets for harmonic oscillator functions

Abstract We define the transformation brackets connecting the wave functions for two particles in an harmonic oscillator common potential with the wave functions given in terms of the relative and centre of mass coordinates of the two particles. With the help of these brackets we show that all matrix elements for the interaction potentials in nuclear shell theory can be given directly in terms of Talmi integrals. We obtain recurrence relations and explicit algebraic expressions for the transformation brackets that will permit their numerical evaluation.

Operators that Lower or Raise the Irreducible Vector Spaces of Un−1 Contained in an Irreducible Vector Space of Un

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.

Bases for the Irreducible Representations of the Unitary Groups and Some Applications

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...

GROUP THEORY OF HARMONIC OSCILLATORS. II. THE INTEGRALS OF MOTION FOR THE QUADRUPOLE-QUADRUPOLE INTERACTION

Abstract In the present paper we obtain 3N integrals of motion associated with a system of N particles moving in a common harmonic oscillator potential and having a quadrupole-quadrupole interaction. We obtain also the eigenvalues of these 3N integrals of motion by applying the corresponding operators to the reduced canonical wave function defined in a previous paper. The reduced canonical wave function is constructed explicitly and it is shown to give a basis for all the irreducible representations of the three dimensional unitary unimodular group. One of the integrals of motion obtained is essentially the quadrupole moment projected along the direction of the angular momentum and its eigenvalues are discussed in detail.

Group theory of harmonic oscillators (III). States with permutational symmetry

Abstract This article continues the analysis of the problem of n particles in a common harmonic oscillator potential that was initiated in two previous papers under the same general title. The first objective of the paper is to give an analytic procedure for the explicit construction of the states in the U 3n ⊃ U 3 × U n , U 3 ⊃ R 3 ⊃ R 2 , Un ⊃ Un−1 ⊃ … ⊃ U1 chain o subgroups, where the 3n dimensional unitary group U3n is the symmetry group of the Hamiltonian while U 3 is the symmetry group of the harmonic oscillator, R 3 is the ordinary rotation group, and Un is the unitary group in n dimensions associated with the particle indices. The second and main objective of this paper is to construct states with definite permutational symmetry. After taking out the centre-of-mass motion the states given in terms of n−1 relative Jacobi vectors will be a basis for irreducible representations of the unitary group Un−1 and its orthogonal subgroup On−1. The characterization of the states is completed with the help of the irreducible representations of the symmetric group Sn, which, through its representations D[n−1, 1](Sn), is a subgroup of On−1. This implies that the states transform irreducibly under the groups in the chain Un ⊃ Un−1 ⊃ On−1 ⊃ Sn rather than under those in the chain Un ⊃ Un−1 ⊃ … ⊃ U1. The states classified in this way contain as particular cases, those of both the shell and the cluster model. Explicit expressions are given for two, three and four particles.

Noninvariance groups in the second-quantization picture and their applications

We investigate the existence of noninvariance groups in the second‐quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular‐momentum basis. The full set of 0‐, 1‐, 2‐, ⋯, (2j + 1)‐particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many‐body forces, the state‐labeling problem, and the quasiparticle picture. Finally, the generalization to more c...

Group theory of the collective model of the nucleus

In the present paper we extend the group theoretical analysis of a previous publication to obtain explicitly, as a polynomical in sinγ, cosγ, the function φλμlk(γ) required in the discussion of the quadrupole vibrations of the nucleus. The states appearing in the collective model 〈νλμLV〉=F1λ(β) ΣKφλμLK(γ) DL*MK(φi), l= (ν−λ)/2, are then defined, as Fλl(β), DL*MK(φi) are well known. All matrix elements required in the collective model of the nucleus are related then with the expression (λμL;λ′μ′L′;λ″μ″L″= ∂π0ΣKK′K″ (LL′L″KK′K″) φλμLK(γ) φλ′μ′L′K′ (γ) φλ″μ″L″K″ (γ)sin 3γdγ, which is a reduced 3j‐symbol in the O(5) O(3) chain of groups.

Canonical Transformations and the Radial Oscillator and Coulomb Problems

In a previous paper a discussion was given of linear canonical transformations and their unitary representation. We wish to extend this analysis to nonlinear canonical transformations, particularly those that are relevant to physically interesting many‐body problems. As a first step in this direction we discuss the nonlinear canonical transformations associated with the radial oscillator and Coulomb problems in which the corresponding Hamiltonian has a centrifugal force of arbitrary strength. By embedding the radial oscillator problem in a higher dimensional configuration space, we obtain its dynamical group of canonical transformations as well as its unitary representation, from the Sp(2) group of linear transformations and its representation in the higher‐dimensional space. The results of the Coulomb problem can be derived from those of the oscillator with the help of the well‐known canonical transformation that maps the first problem on the second in two‐dimensional configuration space. Finally, we mak...

U (5) ⊃ O (5 )⊃ o (3) and the exact solution for the problem of quadrupole vibrations of the nucleus

Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ν in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) ⊆O(3). We furthermore characterize the states by a definite IR λ of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label μ that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications.Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentum L could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentum L and definite number of quanta ..nu.. in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) is contained inO(3). We furthermore characterize the states by a definite IR lambda of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label ..mu.. that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications. (AIP)