论文标题
电磁场的空间演化发生器
Generator of spatial evolution of the electromagnetic field
论文作者
论文摘要
从麦克斯韦的方程式开始，并定义傅立叶空间中的正常变量，我们编写了电磁场的时间演变方程，并在哈密顿和拉格朗日形式中使用源，使所有中间步骤在标准教科书中经常省略。然后，我们遵循相同的步骤来编写该领域的进化方程，并沿着哈密顿和拉格朗日形式的空间维度编写。通过这种方式，我们达到了带有来源的电磁场空间演变的显式形式，并表明它具有通过给定平面正交向传播方向传递的动量模量的物理含义。在自由场的特定情况下，该发电机与在传播方向上的全部动量的投影相吻合，带有负符号。在考虑其空间进化而非时间进化时，对于正确量化了田间的正确量化是必不可少的，而对于相等空间的换向关系的正确定义是必不可少的。
Starting with Maxwell's equations and defining normal variables in the Fourier space, we write the equations of temporal evolution of the electromagnetic field with sources in the Hamiltonian and Lagrangian forms, making explicit all intermediate steps often omitted in standard textbooks. Then, we follow the same steps to write the equations of evolution of this field along a spatial dimension in the Hamiltonian and Lagrangian forms. In this way, we arrive at the explicit form of the generator of spatial evolution of the electromagnetic field with sources and show that it has a physical meaning of the modulus of momentum transferred through a given plane orthogonal to the direction of propagation. In a particular case of free field this generator coincides with the projection of the full momentum of the field on the propagation direction, taken with a negative sign. The Hamiltonian and Lagrangian formulations of the spatial evolution are indispensable for a correct quantization of the field when considering its spatial rather than temporal evolution, in particular, for a correct definition of the equal-space commutation relations.
