Upper bound for wavelength of a photon inside an infinite square well

[Kostik]

TL;DR Summary

I need to show that a photon inside an ISW cannot have arbitrarily low momentum p=ℏω/c. In other words, I need an upper bound on the possible wavelength.

Obviously a particle inside an ISW of width L cannot have arbitrarily precise momentum because ΔP ≥ ℏ/2ΔX ≥ ℏ/2L. Therefore you cannot have a particle with arbitrarily low momentum, since that would require ΔP be arbitrarily small.

I need to show that a photon inside an ISW cannot have arbitrarily low momentum p=ℏω/c. In other words, I need an upper bound on the possible wavelength. My instinct says that the maximum wavelength must be connected to the size of the well L, but the photon doesn't have a size. How can I prove an upper bound on λ?

 

[vanhees71]

Just solve for the four-potential in the radiation gauge, $A^0=0$, $\vec{\nabla}\cdot \vec{A}=0$ with the boundary conditions for an ideally conducting box. You find the solution, e.g., in

J. Garrison and R. Chiao, Quantum optics, Oxford University
Press, New York (2008),
https://doi.org/10.1093/acprof:oso/9780198508861.001.0001

 

[Kostik]

You find the solution, e.g., in
J. Garrison and R. Chiao, Quantum optics, Oxford University
Press, New York (2008),
https://doi.org/10.1093/acprof:oso/9780198508861.001.0001

Thanks - I assume you mean Eqn. (2.15), p. 34, where they show that a photon inside a conducting ISW has quantized wave numbers k = nπc/L, therefore, its maximum possible wavelength is λ = 2π/k(1) = 2L/c. This is exactly what I need.