We explicitly construct a complete set of states, useful in three-body problems, both in a boson operator realization and in terms of coordinates which are of interest to microscopic collective models. The states carry the angular momentum quantum number L and, for the classification scheme mentioned in the title, our expressions generalize to arbitrary L the results previously available only for L=0 and 1.
We explicitly construct a complete set of states, useful in three-body problems, both in a boson operator realization and in terms of coordinates which are of interest to microscopic collective models. The states carry the angular momentum quantum number L and, for the classification scheme mentioned in the title, our expressions generalize to arbitrary L the results previously available only for L=0 and 1.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as basis states for representing their respective symmetry groups — O(2), O(1,1), O(3), and O(2,1). The goal of this study is to establish a correspondence between Hilbert space descriptions found by solving the Schrodinger equation in polar coordinates, and Fock space descriptions constructed by expressing the symmetry operators in terms of creation/annihilation operators. We obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one group generator whose eigenvalue is m + s , for integer m and real constant parameter s . For the three groups that contain O(2), the solutions split into two inequivalent representations, one associated with s = 0, from which we recover the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s > 0 (in three dimensions, solutions are found for s = 0 and s = 1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. The O(1,1) solutions are singlet states, restricted to zero eigenvalue of the symmetry operator, which represents the boost, not angular momentum. For O(2), a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1) while the latter exhibit infinite degeneracy. Creation operators that produce the allowed integer states by acting on the non-degenerate ground state are constructed as irreducible tensor products of the fundamental vector representation. However, the half-integer eigenstates are infinite-dimensional, as expected for the non-compact O(2,1), and no Fock space description is found for this case. For all s ≠ 0 solutions, the SU(N) symmetry of the harmonic oscillator Hamiltonian recently discovered by Bars is spontaneously broken by the ground state. The connection of this symmetry breaking to the non-separability into one-dimensional Cartesian solutions is demonstrated. We conclude with a discussion of the O(2,1) solutions, and the problem of timelike ghost excitations.
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.
Abstract In Chapter 4 we considered a ‘twodimensional’ problem in which L2 and Lz constitute a c.s.c.o. and we derived shift operators for the angular momentum quantum numbers l and m in the eigenkets |lm> of these observables. In Chapter 5 we discussed a ‘one-dimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (eqn (5.25)). Here we use results from Chapters 4 and 5 to construct shift operators for the eigenkets of the three dimensional problem (5.11)–(5.13).
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.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as represents of their respective symmetry groups: O(2), O(3), and O(2,1). Solving the Schrodinger equation by separating variables in polar coordinates, we obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one observable component of orbital angular momentum, with eigenvalue m+s, for integer m and real constant parameter s. In each symmetry group, s splits the solutions into two inequivalent representations, one associated with s=0, which recovers the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s>0 (in three dimensions, s=0,1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. In two dimensions, a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1), and a ladder representation can be constructed via irreducible tensor products of the vector creation operator multiplet, while the latter exhibit infinite degeneracy and the finite-dimensional ladder representation fails for these states. These results are closely connected to the breaking of a unitary symmetry of the harmonic oscillator Hamiltonian recently discussed by Bars.
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.