gauge transformation of electromagnetism field AB effect Gauge

gauge transformation of electromagnetism field, AB effect

Gauge transformation of electromagnetic field vacuum: dig holes on the point where electron charge and current exsits, and μ=μ 0，ε=ε 0 Reference: 《电动力学》郭硕鸿，《费曼物理学讲义第二卷》，《Modern Quantum Mechanics》J. J. Sakurai(中科大 高等量子力学教材)

Gauge transformation of electromagnetic field • The number of solution of ρ、A is infinite, but fit the relation called gauge transformation。ρ、A are all called electromagnetic potentional • Left: Maxwell Equation In Vacuum Right: Gauge Transformation of EM Field • Classical Electric Field: φ(x)→E(x) 其中ψ被定义为规范函数，在经典情况可任意 取值，使A、φ取值有无穷多种，但E、B取值根 据电磁场的唯一性定理，只有一种 • Electromagnetic Field: (φ(x), A(x))→(E(x), B(x))，其中x=(x, y, z)

AB(Aharonov-Bohm) Effect(1960's work) • Is electromagnetic potentional ρ、A useful？？？Consider AB Effect • gauge potentional A is more fundamental than E and B 密绕螺线圈，内部B=μ 0 n. I，外部电场 • 磁场均为零。给螺线圈通电流 phase shift: related to interference pattern • movements of electron's Young's • double silt experiment With current in spiral coil, the interference pattern moves 电子的杨氏双 缝干涉实验， 双缝后加密绕 螺线管 E=0，B=0，the electron cannot feel any force, why the interference pattern moves？？ Explain it completely will use theoretical mechanics and quantum mechanics, so we don't explain, only show the amazing formula

Electron in the electromagnetic field • How to explain it's effect? 此公式的证明将在电磁场理论/电动 力学中严格讲解 • Introduce cononcial(正则) momentum P=p-e. A, p=d(mv)/dt • The trajectory of electron movement • Under gauge transformation, We define: Cononcial momentum P=p-e. A Cononcial Potentional φ‘=φ-v·A

n Use AB effect to observe the magnetization of crystal Reference: (A book)A. Tonomura《Electron Holography》

观察磁化的介质 Use AB Effect to Probe the inner magnetic field in crystal