# How the wave nature of electron manifests the information that it is charged?

In the introductory courses and texts it is mentioned that there is a , $$\textrm{wave}-\textrm{particle}$$ duality in our nature, i.e. the matter interacts like a particle but propagates like a wave. By propagates like a wave I mean that we can observe diffraction and inference phenomena from matter (these are characteristics of waves) and by interaction I mean that matter can be affected by gravity (this confirms that matter have mass and hence a characteristic of a particle) or matter can be absorbed or emitted in lumps.

But when we describe the electron waves how do we confirm that the electron waves are charged? How does the electron waves diffract or interfere differently than other matter waves to manifest that they are charged? My question is what is the difference between the wave nature of neutral matter and a charged matter, a charged matter's wave nature should behave a little differently because it has the duty to spread the electric and magnetic field no matter whether it has to interact or not it has to produce the electric field (that's what the theory of electromagnetism claims).

Thank you. Any help will be much appreciated.

• Why do you think the charge should manifest in "diffracting or interfering differently"? How do you classically see that the electron is charged, and why do you think that doesn't work for its quantum version? – ACuriousMind Jan 11 '20 at 12:00
• I think so because the classical electron gets affected by the electromagnetic field, so their diffraction pattern also gets affected if we put some field but then the question arises “were waves charged?” Because diffraction shows that wave was involved and shift of the pattern tells us that charge was involved somehow. – user240696 Jan 11 '20 at 12:44
• For me you need to clarify the question. I would have thought that your comment above would be an answer. The wave of a charged particle does behave differently than that of an uncharged particle. For example, as you point out, a diffraction pattern will change in the presence of an external charge. – garyp Jan 11 '20 at 14:06
• @garyp I found it very unintuitive to think of a charged wave and therefore I inferred that interaction with EM field was not of wave characteristic but showed the particle nature, but I read somewhere that vector potential is a real thing in QM and that was my confusion. Vector potential is created by a moving charge but moving of an electron happens like a wave so it means that wave is charged , isn’t it? – user240696 Jan 12 '20 at 3:46

I can give you an answer at a level of quantum mechanics, not quantum field theory though, assuming non-relativistic interaction of a matter and an electromagnetic field. I will try to make a discussion based on master equation sense, using Schrodinger equation and interpret meaning of each term in the equation. In a relativistic case, all you need to do is to replace the Schrodinger equation into Dirac equation. If you want to consider electromagnetic wave emitted by an accelerating electron or a dipole, then you can solve Dirac equation (or Schrodinger equation for non-relativistic case) and Maxwell equation at the same time.

Let us limit our physical system of interest on a single electron interacting with an external electromagnetic field, to make the discussion simple. Schrodinger equation of an electron interacting with the external electromagnetic wave can be written as $$i\hbar\frac{\partial}{\partial t}\psi_e(\vec{x},t) = \frac{1}{2m_e}\left(\vec{p}-e\vec{A}(\vec{x},t)\right)^2\psi_e(\vec{x},t) + V(\vec{x},t)\psi_e(\vec{x},t),$$ where $$\hbar$$ is the Planck constant divided by $$2\pi$$, $$m_e$$ is the mass of an electron in vacuum, $$e$$ is the magnitude of the charge of the electron, $$\psi$$ is the wave function of the electron, $$\vec{A}$$ is the vector potential of the external electromagnetic field, and $$V$$ is the external electrostatic potential energy, charge of the electron contained in $$V$$.

Let us imagine a matter that does not exist in the observable universe; the matter has the same mass as the electron but has zero charge. Then Schrodinger equation for such a particle under the external electromagnetic field can be written as $$i\hbar\frac{\partial}{\partial t}\psi_n(\vec{x},t) = \frac{p^2}{2m_e}\psi_n(\vec{x},t),$$ where $$\psi_n$$ is the wave function of our imaginary particle. Note that our imaginary matter propagates in space without any interaction of electromagnetic wave even when there is external electromagnetic fields. In a non-relativistic case, the term proportional to the $$p^2$$ describes propagation of the wave and any wave-like behavior. Both the electron and the imaginary matter have the $$p^2$$ term and they behave like a wave.

The main difference between the imaginary matter and the electron in terms of governing equation (in this case, Schrodinger equations) is that the electron wave has additional terms related to the momentum, $$-e\vec{A}$$, and related to the spatial profile of the wave, $$V$$. This means that the electron wave at each different position locally interact with the external electromagnetic fields, at each different time. The vector potential can boost (if the vector potential changes over time, or if the electrostatic field make a slope, maybe locally, maybe globally) for the electron wave to gain momentum in its propagation direction locally, and can change the direction of the electron (if the vector potential changes over space, in a way that it can generate magnetic fields, or if the electrostatic potential make a slope in different directions locally at different positions). Or the electron wave can change the phase while overall dynamics did not change at all. Actually, local phase change of the electron wave can make change of the wave one or more of the followings.

1. Acceleration
2. Change of interference phase (causing change in interference pattern)
3. Any wave-optics-like phenomenon (for example, Coulomb focusing of an electron wave incident to an attractive Coulomb potential)

But in the case of our imaginary neutral matter, none of the above three can happen because it does not interact with the external electromagnetic field. That is, all these are due to charge (electron has, while the imaginary particle does not have).

Alright. The electron wave interact with the external electromagnetic field locally, at each different point, at the same time. That is the way the charge of the electron wave interact with the external electromagnetic field. In the case of emission of electromagnetic field from an electron wave can be understood in the same wave, local acceleration of dipole moment at each different position of the electron wave can generate local electromagnetic wave.