Relativistic particle in a box

[actionintegral]

Hello,

I was wondering about the Shroedinger equation. I know about the "particle in a box" solution, but I was wondering what happens when the box is moving by at velocity v? And then as v approaches c? What does the equation look like?

 

[StatMechGuy]

Hello,

I was wondering about the Shroedinger equation. I know about the "particle in a box" solution, but I was wondering what happens when the box is moving by at velocity v? And then as v approaches c? What does the equation look like?

If it's moving relativistically, you would need a new prescription for studying things.

However, otherwise...

I would suggest that you try writing down the classical Lagrangian for such a system in a moving frame, and then come up with the Hamiltonian for such a system, and operatorize the Hamiltonian to get the new quantum Hamiltonian. It's an interesting exercise to see what happens to QM in a moving reference frame.

 

[actionintegral]

Ok I will try that.

 

[Severian]

You need to remember that the volume of the box changes with frame. So your particle density (which is $$|\psi|^2$$ in the SE but will need to be modified for a relativistic equation) should too...

 

[XVX]

Well, the Schrödinger Equation is not Lorentz invariant. While your question seems simple enough, I don't believe there is a consensus on how to correctly do it.

 

[vanesch]

From the moment that your particle starts to have kinetic energies comparable to its rest energy, a single-particle quantum theory will not do anymore. You have to switch to, at your choice: a multi-particle theory, or a field theory. Happily, both are equivalent.

 

[actionintegral]

This morning I did the free particle schroedinger equation using d'alembert's solution (x+vt, x-vt). I guess a particle in a box would be a standing wave solution with nodes at the edge of the box. No problem.

But in a moment of reckless whimsy, I set v=c and the equation still seemed to work. What's up with that?

 

[quetzalcoatl9]

what would prevent one from solving the Dirac equation for a free electron subject to the constraint of the boundaries? (i have not actually tried this myself, maybe it could be done numerically with something like GAMESS?)