The Fractional Order Fourier Transform and its Application to Quantum Mechanics
The Fractional Order Fourier Transform and its Application to Quantum Mechanics
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DOI：
10.1093/imamat/25.3.241
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发表时间：
1980-03
影响因子：
1.2
通讯作者：
V. Namias
中科院分区：
文献类型：
--
作者：
V. Namias
我们引入分数阶傅里叶变换的概念，普通傅里叶变换是1阶变换。这种变换的积分表示可用于构建分数阶傅里叶变换表。我们发展了一种广义运算微积，它与普通变换所熟知的运算微积类似。它的应用为求解量子力学中由经典二次哈密顿量产生的某些类常微分方程和偏微分方程提供了一种便捷的技术。首先通过将该解法应用于自由的和受迫的量子力学谐振子来说明这种解法。相应的格林函数以封闭形式得到。然后将这种新技术扩展到三维问题，并应用于恒定磁场中电子运动的量子力学描述。通过系统应用广义运算微积的规则得到定态、能级以及初始波包的演化。最后，将该方法应用于具有时变系数的二阶偏微分方程，该方程描述了时变磁场中电子的量子力学动力学。
We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field.