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Both Maxwell's equations and quantum mechanics are used to describe the behavior of electrons in circuits.

I am confused on the interlinking between the two and the dividing line between when you use one vs the other. When must you abandon Maxwell's equations and move to quantum mechanical model when analyzing electron behavior in homogeneous media? At what scale do Maxwell's equations "fail"?

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2 Answers 2

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Quick clarification, based on some of the comments. This answer focuses on where Maxwell's equations are invalid (the title asks when Maxwell's equations are valid, but the end of the post asks when Maxwell's equations are invalid). Second, I usually consider Maxwell's equations to tell us about electromagnetic interactions, and so quantizing Maxwell's equations involves situations where photons are important. There are many situations where quantum mechanics is important in describing the motion of charged matter like electrons and ions, but where the electromagnetic field itself can be considered a classical field. For example, when describing conductivity in various kinds of materials, which is a major topic in condensed matter physics, the applied field generating a current can usually be considered a classical field, but quantum properties of electrons in a periodic potential are very important in determining the band structure, and therefore whether a material conducts or not.

Anyway, two important examples where Maxwell's equations are invalid would be:

(a) When the number of photons is small (for example, you can't use Maxwell's equations to explain the photoelectric effect for low intensities / small numbers of photons).

(b) When describing the interactions of a charged particle whose Compton wavelength is comparable to or larger than the distance scale (or, equivalently, energy scale) of the process you are considering. For example, you don't need QED to describe the interactions of a piece of wool with static charge (the wool has a lot of mass and a tiny Compton wavelength). You don't need QED to describe the repulsion of two electrons that are far apart from each other, compared to the localization of their wavefunction. You do need QED to describe high-energy scattering of electrons where the electrons have enough energy that their wavefunctions overlap, such as in Compton scattering.

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    $\begingroup$ @FourierFlux No, the structure of the material matters a lot more than the size. Anyway a good rule of thumb is that you need quantum mechanics when you are thinking about a interactions between few particles (eg a few electrons interacting with a few lattice sites, or a few photons). Materials can actually cause more complication that break this rule of thumb, but a lot of times you can wrap up the quantum mechanical parts into some parameters and treat the material classically using those parameters. $\endgroup$
    – Andrew
    Apr 2 at 22:35
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    $\begingroup$ @FourierFlux I suppose IC here means integrated circuit? I'm not an expert on ICs specifically. But I know there are transistors that are small enough to only contain a few tens of atoms, and then you need to treat the transistors quantum mechanically. Eg: en.wikipedia.org/wiki/QFET $\endgroup$
    – Andrew
    Apr 3 at 0:11
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    $\begingroup$ @FourierFlux As an IC designer, the best I can say is "it depends." I work on TSMC's 65nm mixed-signal node and (like all semiconductors) it takes some QM and condensed matter physics to derive the behaviors of semiconductors and transistors. The fab neatly packages that up into a model such as BSIM. From there, I don't even need maxwell's emag; lumped analysis and KCL/KVL are enough at the 2.4 GHz frequency I work with, simply because my chip is tiny. Sometimes my mmwave colleagues will apply Maxwell/emag on-device, but not really quantum.... $\endgroup$
    – nanofarad
    Apr 3 at 15:59
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    $\begingroup$ ... work at the bleeding edge nodes that actually become small enough for quantum effects may be different. I am not by any means an expert on QM, but I would still suspect that the quantum phenomena are confined to the transistor and its model, and you don't need to treat the entire circuit as a quantum system unless you've been carefully maintaining coherency in an attempt to specifically build a quantum computer. (this is mostly an informed guess, nobody in my lab/department works at that scale at the moment) $\endgroup$
    – nanofarad
    Apr 3 at 16:00
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    $\begingroup$ I'd suggest mentioning at the start those cases are ones where Maxwell's equations are invalid, since the OP question title asks when they're valid but the close of the question text asks when they're invalid. $\endgroup$
    – aschepler
    Apr 3 at 19:57
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Two of Maxwell's equations assume a "continuity" in electricity where there is in fact "quantization".

Gauss's Law

$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$

and Ampere's Circuital Law with Maxwell's correction

$$\nabla \times \vec{B} = \mu_0 \left(\vec{J} + \varepsilon_0\frac{\partial\vec{E}}{\partial t}\right)$$

refer to $\rho$ (charge density) and $\vec{J}$ current density.

On a atomic/microscopic scale, however, these terms are problematic, as electric charge and electric current are quantized. Therefore, on a microscopic level, quantum electro-dynamics must be used.

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