We consider a quantum charged particle moving in the $xy$ 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 the 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 rest nonzero diamagnetic value is due to a small asymmetry of the distribution nearby the origin, where some non-exponential 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.

中文翻译：

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热力学平衡中带电粒子的磁矩概率分布

我们认为量子带电粒子在 $Xÿ$平面在均匀的垂直恒定磁场的作用下，存在抛物线型结合势。磁矩算子具有连续的频谱，其动量表示中的本征函数用修正的贝塞尔函数表示。计算了热力学平衡状态下磁矩的概率分布。在零温度下，它在抗磁性区域具有简单的指数形式，在原点急剧跃升至零。随着温度的升高，分布的顺磁翼变得越来越明显。在高温状态下，分布的反磁和顺磁翼具有几乎相同的形式，通过具有非常大扩展性的简单指数函数以高精度描述。因此，抗磁性和顺磁性几乎相互抵消。其余的非零抗磁值是由于原点附近分布的不对称性所致，在原点附近观察到一些非指数的精细结构。分布函数的总宽度在很大程度上取决于结合电位的强度。在所有温度范围内都发现了磁矩的强烈波动（用方差表示）。