What are the differences and similarities between eigen function and wave function?

Answer 1

As you may know, the eigenvalues which one customarily computes in connection with the infinite square-well are energyeigenvalues: E0, E1, E2, ...

Corresponding to each of these energy eigenvalues is a wavefunction φ0(x),

φ1(x), φ2(x), ...

These particular wavefunctions, φn(x), are said to be the energy eigenfunctions associated with the infinite square-well. Thus, for example, a particle in the state φn(x) will have energy En.

But the state of a particle in the well doesn't have to be just a particular one of these φn(x). The state could be any normalized complex-valued function ψ(x) whose value is zero for x outside of the well. Such a ψ(x) is said to be a wavefunction for a particle in an infinite square-well.

Thus, every energy eigenfunction is a wavefunction, but not every wavefunction is an energy eigenfunction.

Nevertheless, it turns out that any such wavefunction ψ(x) can be written as a superposition of the eigenfunctions φn(x). That is, we can write

ψ(x) = ∑nanφn(x) ,

for some complex coefficients an, where

∑n|an|2 = 1 .

Answer 2

Eigenfunction refers to a function that, when acted upon by a mathematical operator, produces a constant multiple of the original function. Wave function, on the other hand, is a mathematical function used in quantum mechanics to describe the behavior of particles. Both eigenfunctions and wave functions are used to describe the state of a quantum system.