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arXiv:quant-ph/0204042 · Journal of Physics A Mathematical and General

Simple diamagnetic monotonicities for Schr$ouml$dinger operators with inhomogeneous magnetic fields of constant direction

Hajo Leschke, Rainer Ruder, Simone Warzel

Published Jun 28, 2002• 2 citations
arXivHTMLPDF

Abstract

Under certain simplifying conditions we detect monotonicity properties of the ground-state energy and the canonical-equilibrium density matrix of a spinless charged particle in the Euclidean plane subject to a perpendicular, possibly inhomogeneous magnetic field and an additional scalar potential. Firstly, we point out a simple condition warranting that the ground-state energy does not decrease when the magnetic field and/or the potential is increased pointwise. Secondly, we consider the case in which both the magnetic field and the potential are constant along one direction in the plane and give a genuine path-integral argument for corresponding monotonicities of the density-matrix diagonal and the absolute value of certain off-diagonals. Our results complement to some degree results of Loss and Thaller (1997 Commun. Math. Phys. 186 95) and Erdös (1997 J. Math. Phys. 38 1289).

DOI
10.1088/0305-4470/35/27/311

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Simple diamagnetic monotonicities for Schroumloumloumldinger operators with inhomogeneous magnetic fields of constant direction
Under certain simplifying conditions we detect monotonicity properties of the ground-state energy and the canonical-equilibrium density matrix of a spinless charged particle in the Euclidean plane subject to a perpendicular, possibly inhomogeneous magnetic field and an additional scalar potential. Firstly, we point out a simple condition warranting that the ground-state energy does not decrease when the magnetic field and/or the potential is increased pointwise. Secondly, we consider the case in which both the magnetic field and the potential are constant along one direction in the plane and give a genuine path-integral argument for corresponding monotonicities of the density-matrix diagonal and the absolute value of certain off-diagonals. Our results complement to some degree results of Loss and Thaller (1997 Commun. Math. Phys. 186 95) and Erdös (1997 J. Math. Phys. 38 1289).
In studying the diamagnetism of free electrons in a uniform magnetic field it was found that reducing the field to zero in the wavefunction did not yield the experimentally indicated free particle plane wave wavefunction. However, solving the Schroedinger Equation resulting from setting the field equal to zero in the original equation did yield a plane wave wavefunction. This paradox was not found to be peculiar to the case of a charged particle in a uniform magnetic field but was found to occur in a number of other systems. In order to gain an understanding of this unexpected behavior, the following systems were analyzed: the one-dimensional square well potential; a charged, spinless particle in a Coulomb field and in a uniform electric field; a one-dimensional harmonic oscillator; and a charged, spinless particle in a uniform magnetic field. From these studies the following were obtained: conditions for determining the result of reducing the potential in a wavefunction; the condition under which the potential of a system may be switched off while maintaining the energy of the system constant; the relationship between the result of physically switching off a potential, the result of reducing it in the wavefunction, and the solution of the Schroedinger Equation obtained by decreasing the potential to zero in the original wave equation; and a general property of any wavefunction with respect to reducing any parameter within this wavefunction.
We consider a quantum charged particle moving in the xyxyxy plane under the action of a uniform perpendicular constant magnetic field, in the presence of a parabolic binding potential. The magnetic-moment operator has a continuous spectrum, and its eigenfunctions in the momentum representation are expressed in terms of modified Bessel functions. The probability distribution of the magnetic moment in the thermodynamic equilibrium state is calculated. At zero temperature, it has a simple exponential form in the diamagnetic region, with a sharp jump to zero at the origin. With an increase of temperature, a paramagnetic wing of the distribution becomes more and more pronounced. In the high-temperature regime, the diamagnetic and paramagnetic wings of the distribution have almost identical forms, described with a high precision by simple exponential functions with very large extensions. Therefore, diamagnetic and paramagnetic contributions almost cancel each other. The remaining non-zero diamagnetic value is due to a small asymmetry of the distribution nearby the origin, where some nonexponential fine structure is observed. The total width of the distribution function strongly depends on the strength of the binding potential. Strong fluctuations of the magnetic moment (described in terms of the variance) are discovered in all temperature regimes.
We consider a quantum charged particle moving in the xyxyxy plane under the action of a uniform perpendicular time-dependent magnetic field, in the presence of a parabolic binding potential. The time-dependent probability distribution of the magnetic moment is calculated analytically and numerically in the case of initial thermodynamic equilibrium state. In the high-temperature regime, the initial distribution is almost symmetric, resulting in a tiny mean diamagnetism. However, if the magnetic field eventually changes its sign, the fragile balance between the diamagnetic and paramagnetic ``wings'' of the probability distribution becomes broken, resulting in a giant mean diamagnetic moment, exceeding the initial one by several orders of magnitude. The final mean value of the magnetic moment is inversely proportional to the strength of the binding potential, and it does not depend on the Planck constant in the high-temperature regime. Strong fluctuations of the magnetic moment (described in terms of the variance) exist in all temperature regimes.
The total-current Hohenberg-Kohn conjecture asserts that the ground-state particle density and total current determine the scalar potential up to an additive constant and the vector potential up to a gauge transformation. We give three counterexamples for two spinless fermions in the physically relevant setting of three spatial dimensions. In each, a Hamiltonian Hv,AH^{v,\mathbf{A}}Hv,A has a nondegenerate ground state with identically zero total current but nonzero magnetic field. Complex conjugation then yields a second Hamiltonian Hv,−AH^{v,-\mathbf{A}}Hv,−A, with the same scalar potential and interaction, whose ground state has the same density and the same zero current, while the magnetic field is reversed. The first two examples are noninteracting. One is explicit, with a smooth bounded vector potential and a Gaussian density. The other has smooth bounded scalar and vector potentials in the standard potential spaces. The third example has Coulomb repulsion with unit coupling, an isotropic harmonic scalar potential, and Dirac monopole fields of magnetic charge ±1/2\pm 1/2±1/2. The monopole example uses vector potentials defined on separate spatial patches, since these fields are not curls of globally defined smooth vector potentials.