Similarity of particle in a box and free particle

Question

It can be shown that particle in a box and free particle have the same energy at certain wavenumbers (at an integer multiple of $pi/L$ , where $L$ is the length of the box)
I am aware that the general wavefunctions of the two particles spoken of are different, but I can't get over the fact that both particles have a $k^2$ dependence on energy.
If we can do the correlation, it seems as if the free particle becomes a particle in a box at these wavenumbers. Can I visualize the free particle becoming a standing wave in such situations? But a length is not defined for the free particle so that wouldn't make sense for such a particle.
Is there any similarity between them?

J. Murray · Accepted Answer

The particle in a box is simply a free particle which lives in a compact interval $[0,L]$ rather than $mathbb R$, and whose energy eigenstates are chosen to vanish at the endpoints.  In other words, the free-particle energy eigenfunctions$^dagger$ satisfy $-frac{hbar^2}{2m}psi''(x) = Epsi(x)$, while the particle-in-a-box energy eigenstates satisfy $-frac{hbar^2}{2m}psi''(x) = Epsi(x)$ and the boundary conditions $psi(0)=psi(L)=0$.
Any function on $[0,L]$ can be extended to a function on $mathbb R$ by copying and pasting it along the whole real line.  From the above argument, it follows that (the extensions of) the particle-in-a-box energy eigenstates are also free particle eigenstates which additionally obey the prescribed conditions at $x=0$ and $x=L$.

$^dagger$The free particle energy "eigenstates" are non-normalizable, but this is somewhat beside the point of this disccusion.

Similarity of particle in a box and free particle

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