# Is "Particle in a box" actually a misnomer?

That's an interesting thought but no, we are talking about a particle in a box when we talk about a particle in a box and we can (and do) separately talk about a particle on a ring (or a torus). The differences between considering an infinite potential boundary (the case of boxes) and a periodic boundary (the case of rings/torii) are absolutely physical and the two are different physical systems.

In particular, the boundary condition for an infinite potential boundary is that the wave-function should vanish at the boundary, i.e., $$\psi(0)=\psi(L)=0$$. On the other hand, the boundary condition for the case of a periodic domain such as a ring is that the wave function should be periodic, i.e., $$\psi(x)=\psi(x+L)$$. You can verify that the former boundary condition implies that eigenfunctions be of the type $$C(e^{ip_nx}-e^{-ip_nx})$$ along with the condition that $$p_n=n\pi/L$$. Whereas the second case is simply a free particle on a periodic boundary, so the eigenfunctions would simply be $$C \exp(ip_nx)$$ but to satisfy the periodicity, $$p_n$$ would have to be of the form $$2n\pi/L$$.

So, there is a measurable physical difference between imposing infinite potentials and imposing periodic boundary conditions.

I'd like to address the OP's comment to one of the answers:

... I can see mathematics says they are different physical systems. The odd thing is, my intuition keeps saying they should be the same. I just can't grasp why, it just doesn't sit right with me.

Consider what happens with a classical particle in both cases.

In the particle in a box problem the particle goes to the right, hits the barrier and reflects, goes to the left, hits another barrier and reflects, and repeats this motion. If the particle is composite, these hits may break it down or at least deform and heat up.

In the case of a particle on a ring, the particle goes on and on without any reflections, like on a merry-go-round. It never hits anything, and the only force it could feel is the centrifugal force (if we view the "ring" literally), which is constant in magnitude.

What might be confusing you is the way you warped your piece of paper: you left the potential "walls" in place, just putting them together instead of erasing, so effectively your picture is now illustrating a different problem: particle on a ring with an impenetrable wall at one point of the ring. This problem (if we ignore centrifugal effects) is indeed equivalent to the original particle-in-a-box problem.

The ring scenario you call a "hypertorus" is an important one. For example, it's used to model the behavior of an electron in an effectively infinite crystal lattice.

But the two problems are not equivalent.

For one thing, $$\lambda=2l$$ is a solution to one of these situations but not the other.