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Background

Quantum mechanics considers only electromagnetic interactions. This fact might be a jut obscured in the introductory texts, but all the potentials in discussions of scattering, tunneling, etc. are the electric potentials interacting with a charged particle.

Problem

Let us now consider an ion incident on a dielectric surface (I chose ion rather than an electrin, so that we could ignore the exchange effects). Dielectric contains positive and negative charges, but, since overall it is electrically neutral, it creates no potential that could interact with the ion, i.e. the wall is invisible! 

Our naive intuition tells us that the ion is "mechanically" prevented from passing through the wall, however quantum mechanical description of this process seems non-trivial. Obviously, it has to do with the inner structure of the dielectric, Casimir and van der Waals forces are probably a part of it. 

Question  

Admittedly, this is a problem from solid state/surface physics rather than pure QM. I am seeking an explanation or references, preferably supplemented by a Hamiltonian.

Comment
A possible direction to look in is the scanning tunneling microscopy, where tunneling through a dielectric is an essential phenomenon.

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    $\begingroup$ it is not that simple, do not forget that the quantum mechanical desription of solids is given by the band theory of solids. To have scatters on that.... see this review for example researchgate.net/profile/Andrey_Borissov/publication/… $\endgroup$
    – anna v
    Mar 28, 2020 at 9:02
  • $\begingroup$ There is also Pauli exclusion, which is not electromagnetic in origin. Boundary conditions depend on it. Ions are blocked by it. Also , for an ion you cannot neglect exchange. $\endgroup$
    – my2cts
    Mar 28, 2020 at 9:12
  • $\begingroup$ @my2cts Indeed, although it is not usually included among the known forces/interactions. This is why I suggested an ion rather than an electron in my formulation. $\endgroup$
    – Roger V.
    Mar 28, 2020 at 9:14
  • $\begingroup$ @annav Thank you for this helpful reference! Could you also take a look at my other question (which was actually my departing point for thinking about these issues) physics.stackexchange.com/questions/538603/… ? $\endgroup$
    – Roger V.
    Mar 28, 2020 at 9:16

3 Answers 3

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since overall it is electrically neutral, it creates no potential that could interact with the ion, i.e. the wall is invisible

This is not true. The overall charge of the wall is zero, but the same thing is true of an electric dipole and it certainly does produce a field. Charge neutrality implies that the ion will not experience an interaction with the wall if it is sufficiently far away. This obviously does not apply if the ion tries to pass through the wall. In this case it will experience the electric field from the non-uniform distribution of charge inside the wall.

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  • $\begingroup$ Indeed, as I pointed in teh question itself: "it has to do with the inner structure of the dielectric, Casimir and van der Waals forces are probably a part of it". Do you have a reference for a particle scattered from a dipole? - that could be a good start. $\endgroup$
    – Roger V.
    Mar 28, 2020 at 13:07
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The walls are impenetrable to ions because of the Pauli principle. Note that you cannot neglect exchange in the case of an ion so you might as well stick to electrons. Pauli exclusion also dictates the shell structure of atoms and the Fermi sea structure of metals. It is as important to electronic structure as electromagnetism.

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  • $\begingroup$ So, if an electron is incident on a dielectric with an energy corresponding to the conduction band, it will travel through the material almost, as if it were an empty space? $\endgroup$
    – Roger V.
    Mar 28, 2020 at 9:26
  • $\begingroup$ No it will be scattered by each and every core electron in the dielectric. $\endgroup$
    – my2cts
    Mar 28, 2020 at 9:45
  • $\begingroup$ There are many experimental techniques to probe materials based on electron scattering. Look into electron diffraction, electron microscopy, EELS, etc. Most of these are surface techniques because electrons do not penetrate into the bulk. $\endgroup$
    – my2cts
    Mar 28, 2020 at 9:51
  • $\begingroup$ I agree that the electrons do not penetrate in bulk - this is an experimental fact. What I need is the mathematical description (I think QM should be sufficient for that). $\endgroup$
    – Roger V.
    Mar 28, 2020 at 10:02
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In fact in the case of an electron, the problem is well studied under the name of metal-insulator junction or metal-semiconductor junction.

Indeed, the region outside the insulator can be viewed as a metal with a very broad band, in which electron is close to the bottom (and in many practical realizations of this system this will be indeed a metal or an n-doped semiconductor). The rest is well-known: the Fermi level is the same in the metal and in the insulator, but while in the metal it lies in the conduction band, in the insulator it is in the middle of the gap between the conduction and the valence bands. Thus, en electron incident onto the insulator has to change its kinetic energy as energy $$\epsilon_k = \frac{\hbar^2k^2}{2m} \longrightarrow \frac{E_g}{2} + \frac{\hbar^2k^2}{2m^*},$$ i.e. it is facing an effective potential barrier of height $E_g/2$. (Hint - it helps thinking of it in terms of a semi-infinite Krönig-Penny model.)

What obscured this well-known phenomenon was the formulation of the problem in terms of elementary quantum mechanics books, where barriers are designed ad-hoc and electrons may be injected at energies far from the ground state. Such scenarios are possible in particle accelerators, but not in most situations where we deal with tunneling through insulators, such as tunnel junctions or STMs.

An interesting consequence of this analysis is that, if the incident electron has the energy above $E_g/2$, it will be able to pass through the material almost without scattering, which is probably a manifestation of the Klein paradox (with $E_g$ being the solid state equivalent of $mc^2$. But again, this scenario is rare in real devices.

I would like to thank @annav for suggesting me to think in terms of the band structure (in her comment to my other post).

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