Relativistic correction to quantum mechanical system

[Dilema]

Relativistic correction to quantum mechanical system: quantum well with a particle moving in high velocity relative to lab frame (the well moves together with the particle in it).

Without relativistic correction there will be a probability for the particle to be outside the well since it's tunneled through the well's wall. This implies the particles have a probability to exceed light speed which contradicts SR.
I expect the correction to provide with a wave function that prevents this probability as if the well would have infinite wall height to nullify any probability for a particle to exceed light speed.
Please comment.
If someone knows any successful correction please let me have the link or ref.

Thank you from advance
Dilema

 

[Bill_K]

You're certainly right that standard Schrödinger quantum mechanics does not know about the speed of light and will happily exceed it. It would seem plausible that you could correct for this and come up with a theory that was similar, but made to be consistent with special relativity. Unfortunately you cannot.

The relativistic "correction" for the Schrödinger equation is the Klein-Gordon equation, and at once this introduces all the complications: the ones associated with antiparticles and pair production. And you find that you're no longer describing a single particle, rather a field with particle-like excitations. There is no half-way.

 

[Dilema]

Dear Bill K.

I realized that there are other relativistic corrections for Schrödinger equation such as Dirac equation. Please review my understanding.
1) There is no general relativistic correction for Schrödinger equation which is the mother of all equations. Although physicists are trying to find one.
2) Each system has to be constructed with its own equation similar to boundary conditions.
3) Some equations may revile different particle/wave properties that have not been recognized in different equation. It is better to have supportive evidence for these properties before using this equation.
4) It is recommended to test Klein-Gordone or Dirac equations before trying to construct your own personal equation since those equations has been already explored.