quantum mechanics - Why is $n=0$ allowed for a particle on a ring?

Is there a simple intuitive explanation for why a particle on a ring has no zero point energy? That is, if we write the energy as:

$$ E_n = \frac{n^2\hbar^2}{2mr^2} $$

then the integer $n$ is allowed to take the value zero. If we consider the apparently similar system of a particle in a 1D infinite potential well, where the energy is given by:

$$ E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} $$

then the integer $n$ is not allowed to be zero. Why the difference?