Using periodic boundary conditions to analyze momentum eigenstates on the circle $S^1$, we have the momentum operator $P=-i\hbar\frac{d}{d\phi}$ which satisfies $P|n>=\hbar n|n>$ (for clarity, |n> is meant to represent ket(n)).
I don't understand what the second half of that statement means. The |n> on either side of the equation makes this seem like a Schrödinger equation to me, but I'm not sure where the $\hbar n$ comes into play—perhaps as an eigenvalue?
I also don't understand the significance of this condition being satisfied once we have shown that $P=-i\hbar\frac{d}{d\phi}$ is the momentum operator on $S^1$.
If anyone is able to clarify this, I would appreciate the help. Thanks in advance.