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首页> 外文期刊>European journal of physics: A journal of the European Physical Society >The coherent states of the two-dimensional isotropic harmonic oscillator and the classical limit of the Landau theory of a charged particle in a uniform magnetic field
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The coherent states of the two-dimensional isotropic harmonic oscillator and the classical limit of the Landau theory of a charged particle in a uniform magnetic field

机译：二维各向同性谐波振荡器的相干状态和均匀磁场中带电粒子的地兰理论的经典极限

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In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z-axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a(+) and a(-). We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency omega(c) = |q|B/m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a(+) in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a(-). The phase of the circular motion is the same as the phase of the complex eigen value of a(+). So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z-axis. We also show that as the magnitude of the eigen value of a(+) becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.
机译：在本文中，我们示出了当沿z轴引导均匀磁场时，当均匀磁场引导XY平面中的带电粒子的经典结果如何使用两者的相干状态来源于均匀的磁场。 - XY平面上的各向同性的各向同性谐波振荡器。在这种情况下，相干状态是湮灭运营商A（+）和（ - ）的同时征指载体。我们证明，表示时间相关的相干状态的时间依赖性坐标空间波分组在圆圈中移动，具有回旋频ω（c）= | q | b / m，并且具有由经典表达式给出的半径，但给出量子机械期望值的条款。颗粒的能量的期望值和其圆半径的平方的预期值与相干状态的（+）的尖端值的大小的平方成比例，其中作为X和Y坐标圆的中心与（ - ）的特征值的真实值和虚部成正比。圆形运动的相位与A（+）的复杂特征值的相相同。因此，对于颗粒的给定能量或对于圆形轨道的给定半径，存在无限数量的圆圈，其由中心的x和y坐标彼此不同以及圆周运动的相位。 LANDAU水平的无限退化是由于XY平面中的空间翻译和围绕Z轴的旋转的能量特征值的不变性。我们还表明，随着（+）的尖端值的大小变得大于一个，电能和圆形轨道的半径的相对不确定性或波动随着经典状态的预期而忽略于忽略的小。

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