What does it mean to find eigenvalues and eigen functions of an infinite well?
When you solve the Shrodinger equation for an infinite well potential and obtain the eigenvalues and eigenfunctions you are finding the allowed states of a particle subjected to this potential. The only energies that a particle like this could ever be measured to have are the energy eigenvalues. The eigenfunctions represent the possible wavefunctions of the particle when it has a definite energy. Remember that the interpretation of the wavefunction is that its magnitude squared at a given point gives the probability density for the particle to be located at that point
It is possible for the particle to have wavefunctions other than the eigenfunctions. The particle can have a wavefunction that is any linear combination of the eigenfunctions. In the case of the infinite well potential the eigenfunctions are the sin and cos functions, so the possible wavefunctions are very general: any function with a Fourrier series. But, if the particle does not have an eigenfunction as a wavefunction then we can not be sure what energy the particle will be measured to have. The probability that a given eigenvalue of the energy will be measured corresponds to the coefficient of the eigenfunction in the linear combination.
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