QM: Why does the creation operator not have eigenstates?

Question

I know that coherent states are the eigenstates of the annihilation operator and i assumed that $a^{+} |n \rangle =\sqrt{n+1}|n+1\rangle$ was the creation operators eigenstate, why is this not an eigenstate?

Answer 1

Given an operator $A$, an eigenstate of such operator with eigenvalue $a$ is a vector $|a\rangle$ such that the application of the original operator onto the mentioned vector gives back the same vector multiplied by a factor $a$, namely $A|a\rangle = a|a\rangle$.

In your example $$ a^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle $$ you can see that the state on the right hand side is not the same state on the left hand side.