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The main motto of this question is to find whether atomic shells are real or it's just the property of electrons.

Professor, you are well aware of the fact that electrons (I am talking about only the Bohr model of the hydrogen atom for simplicity) behave weirdly in an atom-like quantization of orbit, the discrete spectrum of elements etc. This can be explained through the dual nature of electrons(de-Broglie proposal to quantization).

So imagine, in an external magnetic field causing electrons to follow a circular closed trajectory whose radius is given by $$\frac{mv^2}{r}=q(v \times B)$$ (just as they do in atom except here no nucleus). My question is what will be the electron's behaviour in the provided situation?

One of the differences I am encountering is the orbital radius is quantised in the atomic model but here we can change the orbital radius by an arbitrarily amount by changing the magnitude of B arbitrarily.

So are there more differences? And what does it implies to the basic notion of atomic shells? (Assume the velocity of an electron to be negligible compare with the speed of light so that Relativistic effects and maths can be ignored!)

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    $\begingroup$ Your circular orbits in a magnetic field are quantised as well. This is known as Landau levels. $\endgroup$ Jun 22, 2021 at 5:34
  • $\begingroup$ @JohnRennie, is that quantization only apparent on extremely small length scales, or would a macroscopic beam of electrons in for example a cyclotron also occupy measurably discretized orbital radii as they circulated around in it? That is, are landau levels detectable in a cyclotron? I thought they weren't (see my answer below) but now I'm not sure. I will delete my answer if it's wrong; please advise, thanks- NN $\endgroup$ Jun 22, 2021 at 7:15
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    $\begingroup$ @nielsnielsen in principle Landau levels exist in a cyclotron but in practice you would never observe it as the level spacing would be far too small ever to observe. Your answer is fine. $\endgroup$ Jun 22, 2021 at 7:42
  • $\begingroup$ @JohnRennie, thanks for your insight, it is helpful- NN $\endgroup$ Jun 22, 2021 at 18:56

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Let's start with a proton attracting a single electron, as in a hydrogen atom. The electron is occupying the 1s ground state, which occupies a very microscopic volume of space- but has plenty of higher-energy orbitals available for it to get promoted into.

In general, the higher the energy associated with the orbital, the further away it is from the proton and the closer together the available energy levels become. In the limit of "far away" from the proton, the available energy levels are so close to one another that they blend into a continuum and now the electron behaves not like something that can only occupy distinct and discrete energy levels while confined within a potential well, but like something unconfined that is free to propagate through macroscopic space with any energy level it wants i.e., it is a free particle without energy quantization.

If we then guide that electron into a strong magnetic field, its path gets bent into a circle and the faster it is moving (that is, the more energy it has), the larger the radius of the circle. In this macroscopic case, the available energy levels of the bent electron are so close together that for example in the case of electrons boiled off a hot object, a broad spread of radii result and any quantization effects are too tiny to detect.

This means the answer to your question depends on whether you are dealing with macroscopic beams of many, many unconfined electrons in free flight through a magnetic field (no atomic shells exist; quantization effects are undetectable) or if you are talking about one electron confined within a submicroscopic region (quantization effects are detectable as atomic shells).

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