Dodonov Viktor V, Horovits Matheus B
Institute of Physics, University of Brasilia, P.O. Box 04455, Brasília 70919-970, Brazil.
International Center for Physics, University of Brasilia, P.O. Box 04455, Brasília 70919-970, Brazil.
Entropy (Basel). 2021 Nov 26;23(12):1579. doi: 10.3390/e23121579.
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential A=B(t)-y(1+α),x(1-α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.
我们考虑一个无自旋的非相对论性带电粒子，它在随时间变化的磁场作用下在xy平面内运动，该磁场由线性矢量势(A = \frac{B(t) - y(1 + \alpha), x(1 - \alpha)}{2})描述，规范参数(\alpha)有两个固定值：(\alpha = 0)（圆规范）和(\alpha = 1)（朗道规范）。虽然在所有情况下磁场是相同的，但对于非平稳磁场，由于感应电场结构不同，具有不同规范参数值的系统并不等效，其电力线对于(\alpha = 0)是圆，对于(\alpha = 1)是直线。我们推导了对于任意函数(B(t))，能量和磁矩的时间相关平均值及其方差的一般公式。它们用经典运动方程(\ddot{\varepsilon} + \omega_{\alpha}^2(t)\varepsilon = 0)的解来表示，其中(\omega_1 = 2\omega_0)。在磁场突然跳跃、参数共振、绝热演化以及几个特定函数(B(t))的情况下找到了明确结果，此时解可以用初等函数或超几何函数表示。这些例子表明，如果演化不是绝热的，对于这两种规范，上述平均值的演化可能会有很大不同。似乎当磁场趋于零时绝热近似失效。此外，在这种情况下突然跳跃近似也可能失效。磁场缓慢变化并改变符号的情况似乎特别有趣。在所有情况下，磁矩的涨落都非常强烈，经常超过平均值的平方。
