Relativistic Energy Quantization for Particle in Box

[Mindscrape]

How would you go about finding the energy for "the particle in a box" when the particle is relativistic? Since the energy is no longer p^2/2m, then the general quantization won't apply.

I know that the two principles that still apply even when a particle is relativistic are:
$$\lambda = \frac{h}{p}$$
and
$$E = h f = \frac {hc}{\lambda}$$
such that
$$E = c \sqrt{\hbar^2 k^2 +m_0 c^2}$$

From here, I am not really sure what to do with the wave-vector. I suppose that the wavefunction still has to satisfy the general solution that

$$\psi(x) = Asin(kx) + Bcos(kx)$$ for 0 < x < L

The upper bound will change from length contraction, but does that change anything about how the interior of the wave must vanish at x=0 and x=L? If not, then the wavefunction must still satify the equation that
$$Asin(kL) = 0$$
where the solution is that
$$kL = n \pi$$

or maybe...
$$Asin(\frac{kL}{\gamma}) = 0$$
in which
$$\frac{kL}{\gamma} = n \pi$$

Then depending on what value k is, I can substitute it into the relativistic energy equation, and get the equation. But which value is the right one for k?

Am I anywhere on the right track? I know that, ultimately, I need to regain that
$$E_n = \frac{\hbar^2 k^2}{2m} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$.

 

[Jhero]

I would approach this problem by first reviewing the principles that still apply when dealing with relativistic particles, as mentioned in the forum post. This includes the de Broglie wavelength and the energy-frequency relation. From there, I would consider how these principles can be incorporated into the general quantization approach for the particle in a box.

One possible solution could involve using the relativistic energy-momentum relation, E^2 = (pc)^2 + (mc^2)^2, to substitute for the energy in the general quantization equation. This would result in a modified version of the quantization condition, where the wavevector k is no longer a constant but is now dependent on the energy and the mass of the particle. This would lead to a new set of solutions for the wavefunction and the corresponding energy levels.

Another approach could involve considering the effects of relativistic length contraction on the particle in a box system. This would result in a modification of the boundary conditions for the wavefunction, and therefore a new set of solutions for the energy levels. It may also be necessary to consider the effects of time dilation on the system.

Overall, the key is to carefully integrate the principles of relativity into the general quantization approach for the particle in a box, while also considering any modifications to the system due to relativistic effects. This may involve some trial and error, and potentially even numerical methods, but with careful analysis, it is possible to find the energy levels for a relativistic particle in a box.