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Chinese Physics Letters

Non-Commutative Fock-Darwin System and Magnetic Field Limits

余晓敏, 李康

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

在军士的一个 Fock-Darwin 系统变化的量力学被学习。由构造海森堡代数学，我们在军士上获得层次变化空间和军士变化分阶段执行空间，并且在平常的量力学把修正给结果。而且，在三个空格之中寻找差别，退化被二个方法分析，值(终止)在在工具条(c)和某些代数学实现(苏( 2 )和苏( 1,1 ))上的bar/(终止)上，并且在磁场限制的一些有趣的性质被展出，例如为在强壮的磁场的一个系统的给定的频率或团的完全不同的退化和魔术的数字分发。

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在军士的一个 Fock-Darwin 系统变化的量力学被学习。由构造海森堡代数学，我们在军士上获得层次变化空间和军士变化分阶段执行空间，并且在平常的量力学把修正给结果。而且，在三个空格之中寻找差别，退化被二个方法分析，值(终止)在在工具条(c)和某些代数学实现(苏( 2 )和苏( 1,1 ))上的bar/(终止)上，并且在磁场限制的一些有趣的性质被展出，例如为在强壮的磁场的一个系统的给定的频率或团的完全不同的退化和魔术的数字分发。
The Fock–Darwin system is studied in noncommutative quantum mechanics. We not only obtain its energy eigenvalues and eigenstates in noncommutative phase space, but also give an electron orbit description as well as the general expressions of the magnetization and the susceptibility in a noncommutative situation. Further, we discuss two particular cases of temperature and present some interesting results different from those obtained from usual quantum mechanics such as the susceptibility dependent on a magnetic field at high temperatures, the occurrence of the magnetization in a zero magnetic field and zero temperature limit, and so on.
The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, Rényi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.
We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.