# Eigenfunctions v.s. eigenstates

Is there a difference between "eigenfunction" and "eigenstate"? They seem to be used interchangeably in texts, which is confusing. My guess is that an "eigenfunction" has an explicit representation/argument, e.g. in terms of position, $x$, whereas an "eigenstate" is abstract.

Thanks in advance for any clarification! :)

But then again, some texts violate this rule, so I'm prolly wrong...

• Yes, they are practically synonyms. Because any wave function is a valid representative of a state. – Vladimir Kalitvianski Sep 1 '11 at 14:50

## 1 Answer

Maybe in principle, not really in practice. Most people talking casually will use the interchangeably.

If you want a rule that makes sense though, "states" are what we call the objects that live in a hilbert space (we also call them vectors, thus eigenvectors!) so the generic, "coordinate-independent" ket should be the eigenstate.

It's projection onto a definite coordinate basis (say space or momentum) then becomes a mathematical function of that coordinate, so we should call that guy the eigenfunction (which then becomes a basis dependent notion)