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Journal of Geometry and Symmetry in Physics

Geometric Quantization of a Particle in a Perpendicular Magnetic Field

Marin Drinov Academic Publishing House, Guillermo Capobianco, Walter Reartes

Published Jan 1, 2016• 0 citations
Publisher

Abstract

In this paper we develop the quantization of a particle in the plane under the influence of a perpendicular magnetic field using the geometric quantization with half--forms in Hilbert space of holomorphic functions. An original coordinate transformation is applied to convert the problem into a system of harmonic oscillators. Then, it is solved highlighting the relationship between different representations. We emphasize the isomorphism between the holomorphic representation and the Schrödinger representation.

DOI
10.7546/jgsp-41-2016-17-32

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In this paper we develop the quantization of a particle in the plane under the influence of a perpendicular magnetic field using the geometric quantization with half--forms in Hilbert space of holomorphic functions. An original coordinate transformation is applied to convert the problem into a system of harmonic oscillators. Then, it is solved highlighting the relationship between different representations. We emphasize the isomorphism between the holomorphic representation and the Schrödinger representation.
We establish a precise isomorphism between the Schrödinger representation and the holomorphic representation in linear and affine field theory. In the linear case, this isomorphism is induced by a one-to-one correspondence between complex structures and Schrödinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schrödinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any Kähler polarization.
By using appropriate gauge transformations one can develop quasiclassical wavefunctions for charged particles in a uniform magnetic field which have large amplitudes only in the neighborhood of classical Larmor orbits centered at arbitrary points in a plane perpendicular to the field. Pairs of such wavefunctions are not orthogonal to each other. Interesting geometrical relationships between these solutions and the degenerate orthonormal sets of wave-functions in a particular gauge can be found as a result of evaluating overlap integrals. This has been done for both the cylindrical and Landau gauges. In the cylindrical gauge certain of the orthonormal wavefunctions corresponding to different energies give identical probability distributions. Relationships between quantum and classical results, and the effect of the uncertainty principle are discussed.
The conditions that must be fulfilled by a certain physical system to apply geometric quantization prescription on it are investigated. These terms are sought as mathematical requirements, which can be traced in an analysis of integrable systems, from the perspective of both potential function and Hamiltonian vector field. The answer is found in momentum map critical points. Basically, a certain disposal of points that allow a momentum map C∗C^*C∗ isomorphism, of observables, with harmonic oscillator momentum map enforce geometric quantization rules. Following the general theory, two newly presented examples, which exhibits these properties, are quantified through geometric quantization prescription. The Lennard-Jones' type potential is one of the examples, it is known as describing molecules in interaction. We end with a third example that shows the local isomorphism of potentials do not induce C∗C^*C∗ isomorphism of observables.
The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.
The geometrical approach to phase-space quantization introduced by Klauder [KQ] is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted to a specific polarization; in this way the first Landau level acquires the expected degeneracy. This unification with GQ makes it clear how algebraic relations between classical observables are or are not preserved under quantization. From this point of view all quantum systems appear as the low energy sector of a generalized theory in which all classical observables have a uniquely assigned quantum counterpart such that Poisson bracket relations are isomorphic to the commutation relations.