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Journal of Physics A Mathematical and General

Characterization of two-dimensional Euclidean Landau states by coherent state transforms

Zouhaı̈r Mouayn

Published Apr 15, 2004• 9 citations
Publisher

Abstract

We construct a family of generalized coherent states attached to Landau levels of a charged particle moving in the two-dimensional Euclidean plane under a perpendicular uniform magnetic field. We prove that the ranges of the corresponding coherent state transforms coincide with spaces of bound states of the particle. This provides us with a new characterization of two-dimensional Euclidean Landau states.

DOI
10.1088/0305-4470/37/17/011

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We construct a family of generalized coherent states attached to Landau levels of a charged particle moving in the two-dimensional Euclidean plane under a perpendicular uniform magnetic field. We prove that the ranges of the corresponding coherent state transforms coincide with spaces of bound states of the particle. This provides us with a new characterization of two-dimensional Euclidean Landau states.
Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.
We study the problem of a charged particle in a uniform magnetic field with two different gauges, known as Landau and symmetric gauges. By using a similarity transformation in terms of the displacement operator we show that, for the Landau gauge, the eigenfunctions for this problem are the harmonic oscillator number coherent states. In the symmetric gauge, we calculate the SU(1,1)SU(1,1)SU(1,1) Perelomov number coherent states for this problem in cylindrical coordinates in a closed form. Finally, we show that these Perelomov number coherent states are related to the harmonic oscillator number coherent states by the contraction of the SU(1,1)SU(1,1)SU(1,1) group to the Heisenberg-Weyl group.