Two new closely related boson realizations of the Heisenberg–Weyl algebra are derived that are directly equivalent to the introduction of spherical coordinates for the isotropic three-dimensional harmonic oscillator. This provides another way to obtain matrix elements of all physical operators connecting angular-momentum eigenstates, which are represented here by simple products of boson operators acting on the vacuum. Two of these bosons carry the angular-momentum quantum numbers while the third is a novel ‘‘radial boson.’’ In this approach, one begins by finding the nonunitary Dyson and then the unitarized Holstein–Primakoff realizations for the spectrum-generating SU(1,1)×SO(3) algebra of the oscillator. The corresponding mappings of the Heisenberg–Weyl generators are then obtained from their tensor properties under SU(1,1)×SO(3). The Dyson map of the Heisenberg–Weyl generators is not uniquely determined in this way, but the Holstein–Primakoff map is nevertheless unique. The possibility exists for generalization to isotropic oscillators of higher dimension, which are of interest in connection with nuclear collective motion.
Two new closely related boson realizations of the Heisenberg–Weyl algebra are derived that are directly equivalent to the introduction of spherical coordinates for the isotropic three-dimensional harmonic oscillator. This provides another way to obtain matrix elements of all physical operators connecting angular-momentum eigenstates, which are represented here by simple products of boson operators acting on the vacuum. Two of these bosons carry the angular-momentum quantum numbers while the third is a novel ‘‘radial boson.’’ In this approach, one begins by finding the nonunitary Dyson and then the unitarized Holstein–Primakoff realizations for the spectrum-generating SU(1,1)×SO(3) algebra of the oscillator. The corresponding mappings of the Heisenberg–Weyl generators are then obtained from their tensor properties under SU(1,1)×SO(3). The Dyson map of the Heisenberg–Weyl generators is not uniquely determined in this way, but the Holstein–Primakoff map is nevertheless unique. The possibility exists for generalization to isotropic oscillators of higher dimension, which are of interest in connection with nuclear collective motion.
The factorization of four vector operators, D±(ω) and D±(−ω), which occur in a representation-independent, spectrum-generating algebra for the three-dimensional, isotropic harmonic oscillator in an angular momentum basis, is considered (ω is the angular frequency of the oscillator). The D±(ω) are quantum-mechanical analogs of the classical vectors (1∓iL̂×)Fc (ω), where Fc(ω)=−Mωr×L+pL is constant in a frame rotating with angular velocity ωL̂. It is shown that these four vector operators can be factorized in two different ways to yield operators that, apart from their dependence on a constant of the motion (L2), are linear in either p or r. In this way 20 abstract operators are obtained. The properties of these operators are discussed: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖lm〉 of L2 and Lz. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the corresponding operators in the momentum representation. (ii) The remaining eight operators factorize linear combinations of the Hamiltonian and the dimension operator. In linearized, differential form four of these operators are ladder operators for energy and angular momentum in the radial part of the coordinate-space wave functions, while the other four are the corresponding operators in the momentum representation.
Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.
The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.
A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harmonic oscillator, and L is its angular momemtum vector. It is shown that H can be written in the form h/ω(2ν°ν+λ°⋅λ+3/2), where ν° and ν are raising and lowering (boson) operators for ν°ν, which has nonnegative integer eigenvalues k; and λ° and λ are raising and lowering operators for λ°⋅λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form h/ω(2k+l+3/2), previously obtained only by working in the coordinating the operators ν° and λ° to a ’’vacuum’’ vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν,ν°, λ, and λ°, in order to clarify their physical interpretation.
The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.