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Journal of Physics A Mathematical and General

Elliptic eigenstates for the quantum harmonic oscillator

J Pollett, O. Méplan, Claude Gignoux

Published Dec 21, 1995• 28 citations
Publisher

Abstract

A new family of stationary coherent states for the two-dimensional harmonic oscillator is presented. These states are coherent in the sense that they minimize an uncertainty relation for observables related to the orientation and the eccentricity of an ellipse. The wavefunction of these states is particularly simple and well localized on the corresponding classical elliptical trajectory. As the number of quanta increases, the localization on the classical invariant structure is more pronounced. These coherent states give a useful tool to compare classical and quantum mechanics and form a convenient basis to study weak perturbations.

DOI
10.1088/0305-4470/28/24/024

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A new family of stationary coherent states for the two-dimensional harmonic oscillator is presented. These states are coherent in the sense that they minimize an uncertainty relation for observables related to the orientation and the eccentricity of an ellipse. The wavefunction of these states is particularly simple and well localized on the corresponding classical elliptical trajectory. As the number of quanta increases, the localization on the classical invariant structure is more pronounced. These coherent states give a useful tool to compare classical and quantum mechanics and form a convenient basis to study weak perturbations.
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.
The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.
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)].
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
The quantum harmonic oscillator is a quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Coherent state is physically interesting because it is a minimum uncertainty state at which the Heisenberg uncertainty relation reaches its minimum. If a charged particle of charge moves in an electromagnetic field, there is a Lorentz force acting on the particle. The electric and magnetic are invariant under the gauge transformation. In the presence of an external vector potential, the Hamiltonian is changed, so is the probability current operator.