Quantum Mechanics Model on K\"ahler conifold

We propose an exactly-solvable model of the quantum oscillator on the class of K\"ahler spaces (with conic singularities), connected with two-dimensional complex projective spaces. Its energy spectrum is nondegenerate in the orbital quantum number, when the space has non-constant curvature. We reduce the model to a three-dimensional system interacting with the Dirac monopole. Owing to noncommutativity of the reduction and quantization procedures, the Hamiltonian of the reduced system gets non-trivial quantum corrections. We transform the reduced system into a MIC-Kepler-like one and find that quantum corrections arise only in its energy and coupling constant. We present the exact spectrum of the generalized MIC-Kepler system. The one-(complex) dimensional analog of the suggested model is formulated on the Riemann surface over the complex projective plane and could be interpreted as a system with fractional spin.Comment: 5 pages, RevTeX format, some misprints heve been correcte

A new set of coherent states for the isotropic harmonic oscillator: Coherent angular momentum states

The Hamiltonian for the oscillator has earlier been written in the form H=ℏω(2v+v+λ+·λ+3/2), where v+ and v are raising and lowering operators for v+v, 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 denned by diagonalizing v 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 N 3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables v, v +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

The isotropic harmonic oscillator in an angular momentum basis: An algebraic formulation

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 harnomic oscillator, and L is its angular momentum vector. It is shown that H can be written in the form ℏω(2ν†ν + 놕λ + 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 놕λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form ℏω(2k + l + 3/2), previously obtained only by working in the coordinate or momentum representation. The common eigenvectors are constructed by applying 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