# What is the relativistic particle in a box?

I know people try to solve Dirac equation in a box. Some claim it cannot be done. Some claim that they had found the solution, I have seen three and they are all different and bizarre. But my main issue is what would make the particle behave differently (in the same box). How useful is it, aside from the physicist curiosity.

To improve the question I am adding some clarification.

In this paper Alhaidari solves the problem, but he states in the paper that “In fact, the subtleties are so exasperating to the extent that Coulter and Adler ruled out this problem altogether from relativistic physics: “This rules out any consideration of an infinite square well in the relativistic theory”[4].

Ref [4] B. L. Coulter and C. G. Adler, Am. J. Phys. 39, 305 (1971)

But also there are attempts in this paper.

And I quote from this paper (page 2).

“A particular solution may be obtained by considering the Dirac equation with a Lorentz scalar potential [7]; here the rest mass can be thought of as an x-dependent mass. This permits us to solve the inﬁnite square well problem as if it is were a particle with a changing mass that becomes inﬁnite out of the box, so avoiding the Klein paradox [8].”

So my question is why all these discrepancies.

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I'm definitely not an expert in this subject, but one obvious difficulty with any 'relativistic particle-in-a-box' problem is that in QFT the number of particles is not fixed. The smaller the box, the higher the ground-state energy and the more probability that quantum fluctuations can create another Dirac fermion. Once the size of the box becomes of the order of the Compton wavelength of the particle then the single-particle approximation completely breaks down. David Tong gives a lovely exposition of this point in his first lecture here – Mark Mitchison Nov 14 '12 at 11:53
Heuristically potential gradients imply pair creation. In this case you have an infinite gradient. If you have a Dirac field in a potential, pairs would spontaneously form out of nowhere. This violate the applicability of the wave equation which is a single particle equation. – Prathyush Nov 15 '12 at 0:41
@Prathyush , you right, but there are quite few physicists like I have listed that think otherwise. They seem to say That there is a solution as long as you don't go below certain distance or potential, not sure really. – QSA Nov 15 '12 at 1:00
The edits help, but what discrepancies are you talking about? I think the question could still use some clarification. – David Zaslavsky Nov 15 '12 at 4:38
@DavidZaslavsky, Well, as you can see. Adler claims that there is no solution, Alhaidari finds a particular solution and Vidal Alonso finds a solution where mass is proportional to the width of the box, and there are others. So it is not clear to me what is going on. I usually try to ask questions related to fundamental issues that seem to be either glossed over or not well explained in the textbooks and literature. – QSA Nov 15 '12 at 15:08

I have checked the three articles linked by you and I do not find any discussion of this. For instance, if $\psi(x)$ is a solution to the Dirac equation then $|\psi(x)|^2$ is not the probability density of finding the particle at $x$ because $x$ in Dirac theory is not observable [2]. Moreover, their treatment is far from being completely relativistic. They are working in a pseudo-relativistic approach as in the Coulomb-Dirac approach.
[2] This is the reason why $x$ is downgraded from operator status to parameter in QFT.