Saturday, September 6, 2014

vampireking magnitization01 因为电磁势不是物理可测量，而E和B才是。对电磁势作变换（规范变换），物理量E和B保持不变。因此，用电磁势语言处理量子问题的时候，尤其是在量子场论中，要么消去额外的规范自由度，要么证明所研究的物理观测量是规范无关的




张永德作者的文章《电磁AB 效应及有关问题》



几点讨论


关于场强和势的问题


前面已说过, 电磁场的场强都是局域量


与此同时, 电磁场的势是整

体量由上述效应, 已

经知道、不是基本场, 单靠它们不能描述

全部的微观电磁现象,


或者说, 、所提供

的信息不足但是, 由于, 价是规范变换可


变的, 因此它们虽然能描述全部微观电磁现象,


但却提供了过多的信息就是说, , 甲中


也包括了非物理信息在规范条件约束下的势


, 职, 既能描述全部有关现象, 又很少提供


多余的非物理信息更准确些说, 电磁学正是






不可积相因子




这里, 为磁场强度这个等式可以

这样来检验取它的分量交的表达式并和上


面“二表达式相比较这个量子化的算符公式


表明, 量子的力公式可以直接由经典

的力公式经一次量子化而得到和经


典物理一样, 在方程右边的力的表达式中, 当

地的力只和当地的条件有关, 即只含物理量在


当地的数值, 这说明不存在超距作用力http://spe.sysu.edu.cn/spe/electrodynamics/web/pdf/ext/AB1.pdf




电子是费米

子, 自旋左只当空间旋转二, 它的波函数

沙中才完全复原旋转二时价中将出负






号

磁场定矢势之旋度，散度完全自由之。故矢势散度为常数，为任意标量场


vampireking 2013-09-20 19:13:22 1) 微分形式的方程可以写成积分形式，反之未必。因为微分形式是某一时空点的场满足的方程，如果场的相互作用，也就是说某个空间点的取值与其他空间点相关，那么似乎方程无法写成微分形式。（未仔细查对教材）
2) 用E和B表示的麦克斯韦方程不满足协变性，也就是说在不同的参考系描述同一个物理系统的方程形式不一样。为了满足协变性，人们引入了思维矢量电磁势A，E和B分别是电磁势A的导数。用电磁势语言写下的方程，满足协变性。因此，对于麦克斯韦方程，电磁势是更加基本的量。
但是，因为电磁势不是物理可测量，而E和B才是。对电磁势作变换（规范变换），物理量E和B保持不变。因此，用电磁势语言处理量子问题的时候，尤其是在量子场论中，要么消去额外的规范自由度，要么证明所研究的物理观测量是规范无关的。
3) 根据波粒二象性，单个电子=一团物质波波包，当然波包是自相干涉的。






由于电磁场的场强都是一些关于电磁势的


微分量, 因而是一些表征着电磁场局域性质的

量效应不能用电磁场的局域性质来描述


就意味着它可能根源于电磁场的空间整体性质


这是效应从一开始就向我们启示的
测量过程是仪器和粒子相互作用的过程，粒子当然满足薛定谔方程，只不过哈密顿算子现在应该包括仪器与粒子的相互作用能。
测量结果的不确定，是波函数的几率解释带来的。就像你抛一个硬币，得到向上和向下的结果是不能准确预言的。
不确定关系指的是两外一回事。如果两个算子不对易，那么对某个态测量他们的值，测量结果的涨落存在反相关： DA * DB >= 1.


由于电磁场的场强都是一些关于电磁势的


微分量, 因而是一些表征着电磁场局域性质的

量效应不能用电磁场的局域性质来描述


就意味着它可能根源于电磁场的空间整体性质


这是效应从一开始就向我们启示的






http://spe.sysu.edu.cn/spe/electrodynamics/web/pdf/ext/AB1.pdf


next up previous contents Next: The Lorentz Gauge Up: Potentials Previous: Potentials   Contents


Gauge Transformations

Now comes the tricky part. The following is very important to understand, because it is a common feature to nearly all differential formulations of any sort of potential-based field theory, quantum or classical. We know from our extensive study of elementary physics that there must be some freedom in the choice of $\phi$ and $\mbox{\boldmath$A$}$. The fields are physical and can be ``directly'' measured, we know that they are unique and cannot change. However, they are both defined in terms of derivatives of the potentials, so there is an infinite family of possible potentials that will all lead to the same fields. The trivial example of this, familiar from kiddie physics, is that the electrostatic potential is only defined with an arbitrary additive constant. No physics can depend on the choice of this constant, but some choices make problems more easily solvable than others. If you like, experimental physics depends on potential differences, not the absolute magnitude of the potential. So it is now in grown-up electrodynamics, but we have to learn a new term. This freedom to add a constant potential is called gauge freedom and the different potentials one can obtain that lead to the same physical field are generated by means of a gauge transformation. A gauge transformation can be broadly defined as any formal, systematic transformation of the potentials that leaves the fields invariant (although in quantum theory it can be perhaps a bit more subtle than that because of the additional degree of freedom represented by the quantum phase). As was often the case in elementary physics were we freely moved around the origin of our coordinate system (a gauge transformation, we now recognize) or decided to evaluate our potential (differences) from the inner shell of a spherical capacitor (another choice of gauge) we will choose a gauge in electrodynamics to make the solution to a problem as easy as possible or to build a solution with some desired characteristics that can be enforced by a ``gauge condition'' - a constraint on the final potentials obtained that one can show is within the range of possibilities permitted by gauge transformations. However, there's a price to pay. Gauge freedom in non-elemetary physics is a wee bit broader than ``just'' adding a constant, because gradients, divergences and curls in multivariate calculus are not simple derivatives. Consider $\mbox{\boldmath$B$}= \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}$. $\mbox{\boldmath$B$}$ must be unique, but many $\mbox{\boldmath$A$}$'s exist that correspond to any given $\mbox{\boldmath$B$}$. Suppose we have one such $\mbox{\boldmath$A$}$. We can obviously make a new $\mbox{\boldmath$A$}'$ that has the same curl by adding the gradient of any scalar function $\Lambda$. That is:
\begin{displaymath} \mbox{\boldmath$B$}= \mbox{\boldmath$\nabla$}\times \mbox{\... ...Lambda) = \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}' \end{displaymath}(8.36)

We see that:
\begin{displaymath} \mbox{\boldmath$A$}' = \mbox{\boldmath$A$}+ \mbox{\boldmath$\nabla$}\Lambda \end{displaymath}(8.37)

is a gauge transformation of the vector potential that leaves the field invariant.
Note that it probably isn't true that $\Lambda$ can be any scalar function - if this were a math class I'd add caveats about it being nonsingular, smoothly differentiable at least one time, and so on. Even if a physics class I might say a word or two about it, so I just did. The point being that before you propose a $\Lambda$ that isn't, you at least need to think about this sort of thing. However, great physicists (like Dirac) have subtracted out irrelevant infinities from potentials in the past and gotten away with it (he invented ``mass renormalization'' - basically a gauge transformation - when trying to derive a radiation reaction theory), so don't be too closed minded about this either. It is also worth noting that this only shows that this is a possible gauge transformation of $\mbox{\boldmath$A$}$, not that it is sufficiently general to encompass all possible gauge transformations of $\mbox{\boldmath$A$}$. There may well be tensor differential forms of higher rank that cannot be reduced to being a ``gradient of a scalar function'' that still preserve $\mbox{\boldmath$B$}$. However, we won't have the algebraic tools to think about this at least until we reformulate MEs in relativity theory and learn that $\mbox{\boldmath$E$}$ and $\mbox{\boldmath$B$}$ are not, in fact, vectors! They are components of a second rank tensor, where both $\phi$ and $\mbox{\boldmath$A$}$ combine to form a first rank tensor (vector) in four dimensions. This is quite startling for students to learn, as it means that there are many quantities that they might have thought are vectors that are not, in fact, vectors. And it matters - the tensor character of a physical quantity is closely related to the way it transforms when we e.g. change the underlying coordinate system. Don't worry about this quite yet, but it is something for us to think deeply about later. Of course, if we change $\mbox{\boldmath$A$}$ in arbitrary ways, $\mbox{\boldmath$E$}$ will change as well! Suppose we have an $\mbox{\boldmath$A$}$ and $\phi$ that leads to some particular $\mbox{\boldmath$E$}$ combination:
\begin{displaymath} \mbox{\boldmath$E$}= -\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t} \end{displaymath}(8.38)

If we transform $\mbox{\boldmath$A$}$ to $\mbox{\boldmath$A$}'$ by means of a gauge transformation (so $\mbox{\boldmath$B$}$ is preserved), we (in general) will still get a different $\mbox{\boldmath$E$}'$:
$\displaystyle \mbox{\boldmath$E$}'$$\textstyle =$$\displaystyle -\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}'}{\partial t}$ 
 $\textstyle =$$\displaystyle -\mbox{\boldmath$\nabla$}\phi - \frac{\partial }{\partial t}(\mbox{\boldmath$A$}+ \mbox{\boldmath$\nabla$}\Lambda)$ 
 $\textstyle =$$\displaystyle \mbox{\boldmath$E$}- \frac{\partial \mbox{\boldmath$\nabla$}\Lambda}{\partial t} \ne \mbox{\boldmath$E$}$(8.39)



as there is no reason to expect the gauge term to vanish. This is baaaaad. We want to get the same $\mbox{\boldmath$E$}$.
To accomplish this, as we shift $\mbox{\boldmath$A$}$ to $\mbox{\boldmath$A$}'$ we must also shift $\phi$ to $\phi'$. If we substitute an unknown $\phi'$ into the expression for $\mbox{\boldmath$E$}'$ we get:
$\displaystyle \mbox{\boldmath$E$}'$$\textstyle =$$\displaystyle -\mbox{\boldmath$\nabla$}\phi' - \frac{\partial }{\partial t}(\mbox{\boldmath$A$}+ \mbox{\boldmath$\nabla$}\Lambda)$ 
$\displaystyle \mbox{\boldmath$E$}'$$\textstyle =$$\displaystyle -\mbox{\boldmath$\nabla$}\phi' - \frac{\partial \mbox{\boldmath$A$}}{\partial t} - \mbox{\boldmath$\nabla$}\frac{\partial \Lambda}{\partial t}$(8.40)



We see that in order to make $\mbox{\boldmath$E$}' = \mbox{\boldmath$E$}$ (so it doesn't vary with the gauge transformation) we have to subtract a compensating piece to $\phi$ to form $\phi'$:
\begin{displaymath} \phi' = \phi - \frac{\partial \Lambda}{\partial t} \end{displaymath}(8.41)

so that:
$\displaystyle \mbox{\boldmath$E$}'$$\textstyle =$$\displaystyle -\mbox{\boldmath$\nabla$}\phi' - \frac{\partial \mbox{\boldmath$A... ...$A$}}{\partial t} - \mbox{\boldmath$\nabla$}\frac{\partial \Lambda}{\partial t}$ 
 $\textstyle =$$\displaystyle -\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t} = \mbox{\boldmath$E$}$(8.42)



In summary, we see that a fairly general gauge transformation that preserves both $\mbox{\boldmath$E$}$ and $\mbox{\boldmath$B$}$ is the following pair of simultaneous transformations of $\phi$ and $\mbox{\boldmath$A$}$. Given an arbitrary (but well-behaved) scalar function $\Lambda$:
$\displaystyle \phi'$$\textstyle =$$\displaystyle \phi - \frac{\partial \Lambda}{\partial t}$(8.43)
$\displaystyle \mbox{\boldmath$A$}'$$\textstyle =$$\displaystyle \mbox{\boldmath$A$}+ \mbox{\boldmath$\nabla$}\Lambda$(8.44)



will leave the derived fields invariant.
As noted at the beginning, we'd like to be able to use this gauge freedom in the potentials to choose potentials that are easy to evaluate or that have some desired formal property. There are two choices for gauge that are very common in electrodynamics, and you should be familiar with both of them.





张永德作者的文章《电磁AB 效应及有关问题》

该文比较全面的介绍了 AB 效应的原理。量子力学的ＡＢ效应证明电磁势Ａ

和ϕ有着直接的物理观测效应，同时电磁场的场强已不能有效地描述带电粒子的




量子行为。


余招贤老师推荐

第卷


年第期





大学


下


』


物理


电磁效应及有关问题


中国科技大学张永德


摘要本文简要总结了电碌效应, 较全面


地论述了与之有关的问题


一、一效应


在经典力学中, 描述电磁场和带电粒子运

动的方程和力公式, 都是

直接用场强表达的矢势和标势的引人只是数

学上的方便, 不具有物理意义只有规范变

换下不变的场强才有物理意义在量子力学


中, 根据最小电磁祸合原理, 电磁场虽然以其


矢势和标势进入方程, 但电磁场

经规范变换后仅导致波函数多一个相因子即


撇方程在规范变换下不变, 因此人





·


们一直认为在量子力学中, 也如同经典力学中


一样, 只有电磁场的场强才具有可观测的物理

效应但是, 年灿和提


出, 在量子力学中, 电磁场的势有直接的可观






测的物理效应


磁效应


年和提出了一种

奇异量子效应’ , 表明在某些电磁过程中, 电

磁场的场强已不能有效地描述带电粒子的量子

行为这种效应称之为一效

应由于电磁场的场强都是一些关于电磁势的


微分量, 因而是一些表征着电磁场局域性质的

量效应不能用电磁场的局域性质来描述


就意味着它可能根源于电磁场的空间整体性质


这是效应从一开始就向我们启示的可用


图的理想实验来说明效应在电子双缝

实验的缝屏后面两缝之间紧靠缝屏的地方放置

一个细螺线管通电后管内铸, 但管外


、并这个细螺线管产生一细束磁力

线, 称为磁弦理论分析表明, 相对于无螺线


管或没通电来说, 干涉花样在包络不变情况


下发生了移动极值位置移动电流若反号,


花样也反移现对此作理论分析


图双缝效应实验示意图


由于缝、是相干分解, 不失一般性,


可以假设缝。, 和上电子波有相同的相位,


从而将此实验简化为图所示

通电前


访夕望立





刁


中。,


一‘ , · 丫


一下硕二甲又, , ‘ 少

‘ 刀飞


毋。一泊,





爪


尸一一一一、


咬公


、、、一





了


图双缝效应等效图


点的合振幅为





通电后


。一一号,


访晋一女。一号, , , ‘


它改变了两束电子的相位差, 从而改变了双缝

干涉的强度分布这个内部相因子还可改写为

素乡二‘ 一。素丁丁‘, 二,“ 一。希。


为求解此方程, 令


中, 一中。着一等


其中七为待定函数, 代入中的

方程, 注意中。所满足的方程, 可得


一牛艺矛月


, 。【一手】告‘, , 。


这里中是回路积分时回路所圈住的磁通由


于这个相因子并不改变单缝衍射的强度分布,


所以在条纹移动时, 诸条纹极值的包络仍不变


这已为实验所证实





电效应


从和也预言了电通量的

效应, 设不存在电磁场标势作用时, 体系的哈


· 一号, ‘


密顿量为。


访二苦兴二中。





‘


若要此方程成立, 只需要求


一手” ‘一”


即‘一素‘

积分便得不计常数乘子, 它在下面甲归






一时除去


一。希一





总之可得


, 一。景叮· 。, ,一,


有标势甲存在时


一甲


相应的方程解为


, 一。。一专“ , 一丁‘ ,


这很容易用直接计算来证实


可设计如图的理想实验来检验电的

效应, 、为两个屏蔽金属元筒, 电子束


在处被相干分解, 在处相干叠加采用时


控开关将电子束分成一个个波包, 波包线度满






足


注意, 这里中。前面的相因子在铸的区域

与路径有关, 因而是不可积的只在的

区域里, 它才与路径无关这也说明了, 磁场

及其物理效应毕竟是一种物理的实在, 是不能


够用任何数学变换将之转化成相因子的

于是, 在通电情况下, 。点的合振幅为


。一。、仃, 一少, 。。、, 一儿。


谷“ , ,


飞万‘。,


一、· ,


大括号外的相因子是新增的外部相因子, 它在


这里没有可观测的物理效应, 可以略去, 大括


号内。前的相因子为新增的内部相因子,


图电效应实验示意图


筒长度》波包线度》电子波长

这里第二个“ 》” 符号要求电子波足够单色,


以便分析中用一个时延机制来控制, 使


得波包未进人时, 筒的电势为, 进人后,


切随时间上升, 且
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