Scientific publications
The wave function of relativistic electron moving in a uniform electric field
Fritz Sauter, Milton Plesset and Vernon Myers solved the Dirac equation for the case of a uniform electric field. Their solutions have the following disadvanteges.
- One can easily prove that non-stationary are all solutions of the Dirac equation for the motion of a charged particle in a uniform electrostatic field of infinite extent.
- The uniform electric field is used in electrostatic accelerators where accelerated particles behave almost like the free ones. They easily pass through accelerating tube and are easily focused. That is why one could expect a bispinor representing the Dirac particle moving in that field should resemble the free bispinor.
Therefore, I present another solution to that equation.
Can quantum particle move along classical trajectory?
While I still were a student I had noticed that if the free electron could move along classical trajectory then thanks to the fact that such an object could exist in a given moment of time t only at one position r(t) would disappear the problem of normalization of free wave function because there would be no sense to perform the following integrals
∫|ψ(r(t))|²d³r and ∫r|ψ(r(t))|²d³r.
The fact that |ψ(r(t))|² is equal to 1 at every point of space would mean that the probability of finding a particle at the point where it is currently located is equal to 1, and at any other 0, not because |ψ(r(t))|² is equal to 0 , but due to the fact that particle possessing trajectory is not allowed to be there. Confirmation of this fact would obviously change wave function interpretation. But how to prove that?
On the Search for Stationary States in Quantum Mechanics
Among the works on relativistic quantum mechanics, we found a few, upon reading which, one can have an impression that physicists have become so used to the existence of stationary states that they are convinced that there are stationary states in any electromagnetic field. Unfortunately, this is not true.
Perhaps this can be explained by the fact that a significant part of experimental research in quantum physics concerns atoms whose states are solutions of quantum equations for the Coulomb field, about which we know from experience that it has stationary states.
However, a mathematical proof of this fact in the case of the Dirac equation for the hydrogen atom is not trivial, and some of its mathematical details that are especially complex are omitted in the majority of quantum mechanics textbooks.
Moreover, some of these textbooks superficially present the method of finding stationary states, which may make physicists feel that this task for the Dirac equation is straightforward. But this is not true.