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Journal of Mathematical Physics

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

Anthony J. Bracken, Howard I. Leemon

Published Apr 1, 1981• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.524964

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Journal of Mathematical Physics•Sep 1, 1983•2 citations
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.
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.
Coherent angular momentum states are defined for the two-dimensional isotropic harmonic oscillator. They share many attractive properties with the familiar (Cartesian) coherent states, but are in general distinct from those states. The probabilities of obtaining particular values for the radial and angular momentum quantum numbers follow independent Poisson distributions in the new states, but not in the old. In a quasiclassical description of the oscillator, corresponding to a given classical trajectory, the uncertainty in the angular momentum of the system is smaller if the new states are used rather than the old. The new states are the natural analogs of the coherent angular momentum states introduced for the three-dimensional oscillator by Bracken and Leemon [A. J. Bracken and H. I. Leemon, J. Math. Phys. 22, 719 (1981)].
The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)×O(N ) which implies a unique result for the energy spectrum of the system.Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter.This allows the construction of an infinite number of coherent states to be carried out.In the N = 1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case.Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.
The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)X O(N)which implies a unique result for the energy spectrum of the system. Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter. This allows the construction of an infinite number of coherent states to be carried out. In the N=1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case. Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.
Calculates expectation values and uncertainties of the radial coordinate and momentum, and the azimuthal angle and angular momentum components for a three-dimensional harmonic oscillator in a coherent state. The operators corresponding to powers of the angular variable are defined by their Fourier expansions. For large distances of the particle from the origin, where the oscillator becomes classical, radial and angular uncertainty products approach the value h(cross)/ square root 2, except for the point of discontinuity, where the angular uncertainty becomes infinite. For small distances, where the oscillator tends to its quantum ground state, the radial uncertainty stays finite and the angular uncertainty product tends to zero. It is also shown that the concept of a polar momentum operator is meaningless.
The forms of the operators ν°, ν, λ°, λ, which enable one to write the Hamiltonian of the two-dimensional isotropic harmonic oscillator in the form H=ℏω(2ν°ν+λ°⋅λ+1), are presented. Here ν° and ν are, respectively, the raising and lowering operators for ν°ν, the ‘‘radial’’ quantum number operator, while λ° and λ are, respectively, the raising and lowering operators for M, the magnitude of the angular momentum operator. Corresponding to this decomposition of H in the angular momentum basis are the energy eigenvalues Ekm=ℏω(2k+‖m‖+1) with k=0, 1, 2, ⋅⋅⋅ and m=0, ±1, ±2, ⋅⋅⋅. Here k is a ‘‘radial’’ quantum number, and m is a ‘‘magnetic’’ quantum number. The commutation relations satisfied by the operators ν°, ν, λ°, and λ are also presented.