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Journal of Mathematical Physics

Coherent states on horospheric three-dimensional Lobachevsky space

Yu. A. Kurochkin, I. Yu. Rybak, D. V. Shoukavy

Published Aug 1, 2016• 6 citations
Publisher

Abstract

In the paper it is shown that due to separation of variables in the Laplace-Beltrami operator (Hamiltonian of a free quantum particle) in horospheric and quasi-Cartesian coordinates of three dimensional Lobachevsky space, it is possible to introduce standard (“conventional” according to Perelomov [Generalized Coherent States and Their Applications (Springer-Verlag, 1986), p. 320]) coherent states. Some problems (oscillator on horosphere, charged particle in analogy of constant uniform magnetic field) where coherent states are suitable for treating were considered.

DOI
10.1063/1.4960474

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КОГЕРЕНТНЫЕ СОСТОЯНИЯ НА ОРИСФЕРЕ ТРЕХМЕРНОГО ПРОСТРАНСТВА ЛОБАЧЕВСКОГО

Y. A. Kurochkin, І. Ю. Рибак, Д. В. Шёлковый

The separation of variables in the Laplace–Beltrami operator (Hamiltonian for a free quantum mechanical particle) of three-dimensional Lobachevsky space in the horospherical and quasi Cartesian coordinates provides an opportunity to introduce standard (conventional) coherent states. We consider problems, (oscillator, quantum mechanical particle in the uniform magnetic field) for which introduced coherent states are the adequate method of investigation.

Doklady of the National Academy of Sciences of Belarus•Jun 10, 2016•0 citations

Relativistic coherent states and charge structure of the coordinate and momentum operators

Bohdan I. Lev, А. А. Семенов, Constantin V. Usenko +1 more

We consider relativistic coherent states for a spin-0 charged particle that satisfy the next additional requirements: (i) the expected values of the standard coordinate and momentum operators are uniquely related to the real and imaginary parts of the coherent state parameter $\ensuremath{\alpha};$ (ii) these states contain only one charge component. Three cases are considered: free particle, relativistic rotator, and particle in a constant homogeneous magnetic field. For the rotational motion of the two latter cases, such a description leads to the appearance of the so-called nonlinear coherent states.

arXiv:quant-ph/0206180•Physical Review A•Aug 27, 2002•16 citations

A generalized Fourier transform of the P-quasi distribution function

Dušan Popov

In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).

arXiv:2212.04345•Dec 6, 2022•0 citations

Introduction to Coherent States

Didier Robert, Monique Combescure

In this Chapter we study the Weyl–Heisenberg group in the Schrödinger representation in arbitrary dimension n. One shows that it operates in the Hilbert space of quantum states (and on quantum operators) as a phase-space translation. Then applying it to a Schwartz class state of arbitrary profile we get a set of generalized coherent states. When we apply the Weyl–Heisenberg translation operator to the ground state of the n-dimensional Harmonic Oscillator, one gets the standard coherent states introduced by Schrödinger [233] (Naturwissenshaften 14:664–666, 1926) in the early days of quantum mechanics (1926). Later the coherent states have been extensively studied by Glauber [118, 119] (Phys. Rev. 131:2766–2788, 1963; Phys. Rev. 130:2529–2539, 1963) for the purpose of quantum optics and it seems that their name comes back to this work. The standard coherent states have been generalized by Perelomov [208] (Generalised Coherent States and Their Applications, 1986) to more general Lie groups than the Weyl–Heisenberg group. We also introduce the usual creation and annihilation operators in dimension n which are very convenient for the study of coherent states. We show that coherent states constitute a non-orthogonal over-complete system which yields a resolution of the identity operator in the Hilbert space and which allows a computation of the Hilbert–Schmidt norm and of the trace of respectively Hilbert–Schmidt class and trace-class operators. We study their time-evolution for the quantum Harmonic Oscillator hamiltonian and show that a time evolved coherent state located around phase-space point z is up to a phase a coherent state located around the phase-space point \(z_t\), where \(z_t\) is the phase-space point of the classical flow governed by the Harmonic Oscillator. This property was described by Schrödinger as the non-spreading of the time evolution of coherent states under the quantum Harmonic Oscillator dynamics. We also show how to go from the Schrödinger to the Fock–Bargmann representation using the standard coherent states.

Theoretical and mathematical physics•Jan 1, 2021•0 citations

Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

Yu. A. Kurochkin

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

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Coherent states of a charged particle in a uniform magnetic field

K. Kowalski, Jakub Rembieliński

The coherent states are constructed for a charged particle in a uniform magnetic field based on coherent states for the circular motion which have recently been introduced by the authors.

arXiv:quant-ph/0509108•Journal of Physics A Mathematical and General•Sep 7, 2005•33 citations
In the paper it is shown that due to separation of variables in the Laplace-Beltrami operator (Hamiltonian of a free quantum particle) in horospheric and quasi-Cartesian coordinates of three dimensional Lobachevsky space, it is possible to introduce standard (“conventional” according to Perelomov [Generalized Coherent States and Their Applications (Springer-Verlag, 1986), p. 320]) coherent states. Some problems (oscillator on horosphere, charged particle in analogy of constant uniform magnetic field) where coherent states are suitable for treating were considered.
We consider relativistic coherent states for a spin-0 charged particle that satisfy the next additional requirements: (i) the expected values of the standard coordinate and momentum operators are uniquely related to the real and imaginary parts of the coherent state parameter $\ensuremath{\alpha};$ (ii) these states contain only one charge component. Three cases are considered: free particle, relativistic rotator, and particle in a constant homogeneous magnetic field. For the rotational motion of the two latter cases, such a description leads to the appearance of the so-called nonlinear coherent states.
In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).
In this Chapter we study the Weyl–Heisenberg group in the Schrödinger representation in arbitrary dimension n. One shows that it operates in the Hilbert space of quantum states (and on quantum operators) as a phase-space translation. Then applying it to a Schwartz class state of arbitrary profile we get a set of generalized coherent states. When we apply the Weyl–Heisenberg translation operator to the ground state of the n-dimensional Harmonic Oscillator, one gets the standard coherent states introduced by Schrödinger [233] (Naturwissenshaften 14:664–666, 1926) in the early days of quantum mechanics (1926). Later the coherent states have been extensively studied by Glauber [118, 119] (Phys. Rev. 131:2766–2788, 1963; Phys. Rev. 130:2529–2539, 1963) for the purpose of quantum optics and it seems that their name comes back to this work. The standard coherent states have been generalized by Perelomov [208] (Generalised Coherent States and Their Applications, 1986) to more general Lie groups than the Weyl–Heisenberg group. We also introduce the usual creation and annihilation operators in dimension n which are very convenient for the study of coherent states. We show that coherent states constitute a non-orthogonal over-complete system which yields a resolution of the identity operator in the Hilbert space and which allows a computation of the Hilbert–Schmidt norm and of the trace of respectively Hilbert–Schmidt class and trace-class operators. We study their time-evolution for the quantum Harmonic Oscillator hamiltonian and show that a time evolved coherent state located around phase-space point z is up to a phase a coherent state located around the phase-space point ztz_tzt​, where ztz_tzt​ is the phase-space point of the classical flow governed by the Harmonic Oscillator. This property was described by Schrödinger as the non-spreading of the time evolution of coherent states under the quantum Harmonic Oscillator dynamics. We also show how to go from the Schrödinger to the Fock–Bargmann representation using the standard coherent states.
The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.