# Bound states in relativistic quantum mechanics

1. Feb 28, 2009

### AxiomOfChoice

Suppose a particle is subject to a spherically symmetric potential $V(r)$ such that $V(r) = -V_0$, $V_0 > 0$, for $0\leq r \leq a$ and $V(r) = 0$ elsewhere. If we were considering a non-relativistic particle, we would have bound states for $-V_0 < E < 0$ (which I understand); however, since the particle is relativistic, apparently (or, at least, according to my professor), we have bound states for $|E| < mc^2$. Could someone please explain why this is? I'm pretty sure it has something to do with the fact that the relativistic energy-momentum dispersion relation is $E(p) = \sqrt{p^2c^2 + m^2c^4}$, but I can't wrap my head around it.Thanks.