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November 22, 2024

Wave function vs classical orbits

- sage -

> A）核外电子绕核运动，是呈驻波状态，我记得驻波状态表示不动的状态，那如何理解电子又有轨道角动量呢？
> 在两个板子之间来回反射的粒子如果形成驻波状态，经典地看粒子始终在来回运动着，但波动地看，没有能量传
> 播，因为驻波是不移动的波...这个，我是不是哪儿理解错了？
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It is a standing wave. However, it is a wave-function (not a mechanical wave). the notion of 'moving' is not as intuitive as in classical physics. In a particular state of the electron circling the the nucleus (which is a classical picture and almost completely wrong in this case), there is no energy propagation in any sense. electron is in an eigenstate of Hamiltonian.

> B）在电磁场（V,A)中的带电荷e质量为m的粒子，不考虑自旋，有关系：
> （E-eV)^2=(p-eA)^2+m^2
> 调整（V,A)的大小，似乎原则上可以有（E-eV)^2小于(p-eA)^2，从而m^2小于零...这岂不是搞出虚数质量来？
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m^2 is a parameter in the lagrangian. if we switch off EM field, you could see it must be real for all stable particles. switch back on the EM field, what should change is the state, not the lagrangian parameters. Therefore, the correct interpretation should be, the presence of the background EM field changes the energy level of a charged particle moving in it. In another word, the expresssion should be interpretted as a relation enable you to solve for E.

> 回sage兄：如果核外电子在核外某能级上形成驻波并且没有能量在核外流动，如何理解核外电子有角动量呢？
> 再如，粒子在两个平行板之间来回反射运动着——这是经典物理图像，从量子力学角度看，粒子在在两个平行板
> 之间形成驻波，没有能量流动，不知道这如何跟来回反射运动对应起来理解？波幅对应粒子出现几率极大的地
> 方，波节对应粒子出现几率极小的地方。D.Bohm跟Einstein六封通信中讨论过这个问题，但最后没有讨论出结
> 来，D.Bohm最后说：伟人也未必总是对的。D.Bohm对量子力学钻牛角尖的结果是，发现了AB效应，让
> Einstein－ Bohr的争论由理想试验和哲学式的变为物理可行的，为后来的量子测量理论发展包括量子信息奠定了基
> 础——这些知识都是来自于星空兄，事实上许多基础都来自于他，顺便借此感谢一下。D.Bohm认为粒子在波节出停
> 留时间趋于零，速度应该趋于无穷大.我觉得波节是理想的一点，停留时间趋于零可以理解，即使速度在该处不趋于 > 无穷大，除非再考虑到不确定关系。
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Strictly speaking, the classical concept of some particle moving around do not have a direct correspondance in quantum mechanics. Your intuition is that wave-function is somehow moving when the particle is 'moving'. however, this is only roughly true in the situation where the particle (nearly classical) is described by a somewhat localized wave packet. then the wave packet is moving. In this case, the classical intuition is approximately true that momentum if the particle is related to the movement of the wave packet.

However, as you can see, this picture completely breaks down in the example of electron in an atom. there, there is no localized wave-packet which more or less define the particle. in stead, the electron wavefunction is spread out over the full space of that atom.

This example shows that in quantum mechanics, one has to be really careful about what one means by 'moving'. As long as we define what we mean, there is no contradiction. for example, one could define 'moving' by non-zero momentum. Since the electron wave function is a superposition of many momentum mode, when you measure the momentum of the electron (for example, by compton scattering), you will find the the electron is indeed 'moving'. Similarly, one could define angular momentum.

In quantum mechanics. it is easier to think of momentum just as quantum numbers characterizing states, rather than directly from some classical motion.

You can also choose to define if the electron is moving by if the wave-function is moving. it is totally fine. it is just not a very useful definition in this case, since it is not associated with the momentum of the electron which is a physically more useful quantity.

二零零五年六月十五日