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Journal of Physics A Mathematical and Theoretical

Coherent states of inverse oscillators and related problems

Vladislav G. Bagrov, Dmitri M Gitman, Edmilson dos Santos Macêdo, A. S. Pereira

Published Jul 25, 2013• 20 citations
Publisher

Abstract

We consider an example of a quantum system with a continuous and unbounded spectrum—a particle in the parabolic potential barrier. Conditionally, we call this system the inverse oscillator. Following the method used in our previous publication (Bagrov et al 2012 J. Phys. A: Math. Theor. 45 125306), we construct different families of generalized coherent states (GCS). We discuss properties of the constructed GCS and their relation to what is already known in the literature about semiclassical states for the inverse oscillator. Then in the same manner, we construct families of GCS for the normal oscillator. Using special variables, we succeeded in constructing GCS for the normal oscillator that admit different limiting cases: free particle GCS, the usual Schrödinger, Glauber CS and GCS of the inverse oscillator.

DOI
10.1088/1751-8113/46/32/325305

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We consider an example of a quantum system with a continuous and unbounded spectrum—a particle in the parabolic potential barrier. Conditionally, we call this system the inverse oscillator. Following the method used in our previous publication (Bagrov et al 2012 J. Phys. A: Math. Theor. 45 125306), we construct different families of generalized coherent states (GCS). We discuss properties of the constructed GCS and their relation to what is already known in the literature about semiclassical states for the inverse oscillator. Then in the same manner, we construct families of GCS for the normal oscillator. Using special variables, we succeeded in constructing GCS for the normal oscillator that admit different limiting cases: free particle GCS, the usual Schrödinger, Glauber CS and GCS of the inverse oscillator.
Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian.However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion.In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle.We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density.We describe the physical conditions under which free-particle CS can be considered semiclassical states.
Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states.
We discuss the construction of coherent states (CS) for systems with continuous spectra. First, we propose to adopt the Malkin–Manko approach, developed for systems with discrete spectra, to the case under consideration. Following this approach, we consider two examples, a free particle and a particle in a linear potential. Second, we generalize the approach of action-angle CS to systems with continuous spectra. In the first approach we start with a well-defined quantum formulation (canonical quantization) of a physical system and the construction of CS follows from such a quantization. In the second approach, the quantization procedure is inherent to the CS construction itself.
An unusual family of generalized coherent states of the Heisenberg algebra is constructed. It consists of two families of functionals on the Schwartz space. Each of them has the key properties of ordinary coherent states: it intertwines the representations of a given algebra, has the completeness property, and minimizes the product of variances in the Heisenberg relation. However, unlike the usual coherent states that are eigenstates for the annihilation operator, the constructed distributions belong to the continuous spectrum of Hermitian generators of the Heisenberg algebra. The inner product of functionals from different families has the main properties of the overlap function of ordinary coherent states: it is continuous in parameters, satisfies the reproducing identity, and has the corresponding geometric meaning. The application of unusual coherent states of the continuous spectrum is demonstrated by an example of solving the spectral problem for an inverted oscillator.
Despite the didactic importance of a free particle in quantum mechanics, its coherent state analysis has long been untouched. It is only recently that it has been noticed and studied in the semiclassical domain. While the previously known solutions, reported by Bagrov et al. for a free particle, are described using the linear non-Hermitian invariant operator, we show in this work that the general solution of the Schrödinger equation can also be naturally derived using a simpler method based on an Hermitian linear invariant operator. According to this, an exact Gaussian wave function that corresponds to a coherent state solution is obtained. An interpretation for such general quantum solution designated within the Lewis-Riesenfeld framework is provided and, further, quantum-classical correspondence principle for the system is reexamined.
Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well known Landau - Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.