## Eigenfunction vs wave function

What is the difference between eigenfunction and wave function?

I'm always get confused when i am asked to write wave function and eigenfunction..
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 Recognitions: Homework Help An eigenfunction is a type of wavefunction that has an eigenvalue when operated on. It is said to be "an eigenfunction of the operator".

 Quote by Simon Bridge An eigenfunction is a type of wavefunction that has an eigenvalue when operated on. It is said to be "an eigenfunction of the operator".
is this right?

ψ(x,t) = ψ1(x) + ψ2(x)
wavefunction = eigenfunction1 + eigenfunction2

Recognitions:
Here's the full description of the situation: You have a quantum system which has only two possible states, 1 and 2. That means there are two quantum states, ψ1 and ψ2 describing the system in either state 1 or 2 (1 and 2 can be spin up or spin down, for example). In general, then, any arbitrary wavefunction can be written as a linear superposition of these two states, $\Psi(x,t) = \alpha \Psi_1(x,t) + \beta \Psi_2(x,t)$. This is possible because the set of eigenstates (ψ1,ψ2) are complete and form a basis.
 Recognitions: Homework Help Since the LHS is a function of time as well as position while the LHS is position only, not really. However ... lets say we have a set of wavefunctions $\{ \psi_n \}$ which has been selected so that $$\mathbf{H}\psi_n = E_n\psi_n$$ .... then each $\psi_n$ is said to be an eigenfunction of the Hamiltonian with eigenvalue $E_n$. A system prepared in a superpostion state may have wavefunction $$\psi = \frac{1}{\sqrt{2}}\left ( \psi_1 + \psi_2\right )$$ (assuming each $\psi_n$ are already normalized.) In this case $\psi$ is not an eigenfunction of the Hamiltonian. In general, the set of eigenfunctions of an operator can be used as a basis set. Any wavefunction can. Therefore, be represented in terms of a superposition of eigenfunctions ... including eigenfunctions of another operator. (Just in case someone infers that superpositions of eigenfunctions cannot be eigenfunctions.) It is also possible for a wavefunction to, simultaniously, be an eigenfunction of more than one operator. Notice how careful I was in the way I phrased things above? In QM it is very important to be careful about what exactly is being said about a system ... when you are starting out it is as well to get really pedantic about this.