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?