# Relativistic particle in a box

• actionintegral
In summary, if the particle is moving relativistically, you would need a new prescription for studying things. However, otherwise, you should be able to solve the Dirac equation for a free electron subject to the constraint of the boundaries.
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?

actionintegral said:
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.

Ok I will try that.

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...

Well, the Schrodinger 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.

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.

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?

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?)

## 1. What is a relativistic particle in a box?

A relativistic particle in a box is a theoretical concept in quantum mechanics that describes the behavior of a particle moving at relativistic speeds (close to the speed of light) confined to a finite space, such as a one-dimensional box. This system is used to study the effects of special relativity on quantum particles.

## 2. What is the significance of studying relativistic particles in a box?

Studying relativistic particles in a box allows us to better understand the behavior of particles at high speeds and how special relativity affects their properties. It also has practical applications in fields such as particle physics and cosmology.

## 3. How does special relativity affect the energy levels of a particle in a box?

In a classical particle in a box system, the energy levels are evenly spaced. However, in a relativistic particle in a box system, the energy levels become closer together as the particle approaches the speed of light, reflecting the effects of special relativity on the particle's energy.

## 4. What are the differences between a non-relativistic and relativistic particle in a box?

A non-relativistic particle in a box is described by classical mechanics and follows the Schrödinger equation, while a relativistic particle in a box follows the Klein-Gordon equation. Additionally, the energy levels in a non-relativistic particle in a box are evenly spaced, while in a relativistic particle in a box they become closer together as the particle's speed increases.

## 5. How is the wave function of a relativistic particle in a box affected by special relativity?

The wave function of a relativistic particle in a box is described by the Klein-Gordon equation, which takes into account the effects of special relativity. The wave function becomes more complex and its shape changes as the particle's speed increases, reflecting the relativistic effects on the particle's behavior.

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