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I always thought, an electron could not pass through a hole, with a diameter, which is smaller than its wavelength. Just like microwaves could not pass through a metal mesh with small holes, but visible light could. The other day I came across a video of a series of lectures on Quantum Transport, in which during a Q&A, it is mentioned, that if one shrinks the chamber in front of electrons to be smaller than the wavelength, quantum effects would be more sensible, and it is implied, that the electrons pass through. Here is the time-coded link to that video:

Quantum Transport, Lecture 1: Introduction

My question, which is an abstract one, is if an electron can really pass through a hole or a chamber, with a diameter smaller than the wavelength. If yes, how, and does it have to do with quantum tunneling? What would be the mathematics, explaining this phenomenon? I would be grateful, if you could provide a concrete explanation about my question.

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marked as duplicate by Ben Crowell, user191954, John Rennie quantum-mechanics Oct 15 '18 at 10:48

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Well basically the answer is probabilities. Most of the electrons will not pass through it. A very low ratio of electrons might pass through.

According to Bethe's theory of light (but analogically for electrons too) transmission through small holes, the transmission scales as (a/λ)^4, where a is the size of the whole and λ the wavelength. There is a structure called Quantum Point Contact.

The transmission is extremely small when the wavelength exceeds the QPC.

You have to distinguish between conducting and non-conducting materials. If it is a non-conducting material, then some of the electrons can pass through the hole, regardless of the size of the hole. The ratio of transmitted electrons is just the incident power per unit multiplied by the hole area.

If it is a conducting material, that changes how you need to analyze it. The incident electron (traveling as a wave) induces oscillations in the metal of the screen, then these oscillations re-radiate EM waves that interfere with the incident waves. This is what blocks the incident wave (traveling electron).

Conducting screens like what you mention are usually treated as Faraday cages, and doing the math which you are asking for is in this answer:

What is the relationship between Faraday cage mesh size and attenuation of cell phone reception signals?

For quantum tunneling, you do not even need a slit. Experiments show that when a wavepacket of electrons is directed at a potential barrier, some electrons will tunnel through the barrier, and this was classically not possible.

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One explanation is the Heisenberg Uncertainty principle, which implies that there are no solutions with the probability of exactly zero (or 1), though a solution may approach infinity if, for example the calculation for the elecron's position was taken for the probability of 1, so its speed's probability would have to be infinity. Hence, the probability of the electron being on the opposite side of the barrier is non-zero. So such electrons will appear on the other side of the barrier with a relative frequency proportional to its probability.

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