No, not every wave function can be an eigenfunction. In quantum mechanics, wave functions describe the behavior and properties of quantum systems. Eigenfunctions, on the other hand, are special types of wave functions that represent states of definite energy or other observable quantities. When an operator acts on an eigenfunction, the result is a scaled version of the original function.
In general, for a given operator, only certain wave functions satisfy the eigenvalue equation for that operator, where the eigenvalue represents the result of a measurement of the corresponding observable. These eigenfunctions form a complete set, meaning that any wave function can be expressed as a linear combination of the eigenfunctions.
For example, in the case of the Hamiltonian operator, which represents the total energy of a system, the eigenfunctions are the energy eigenstates. These eigenstates represent specific energy levels that a quantum system can possess. Other operators, such as the momentum or angular momentum operators, also have their corresponding eigenfunctions associated with their respective observables.
So, while any wave function can be expressed as a linear combination of eigenfunctions, not every wave function can be an eigenfunction itself. Eigenfunctions are special solutions that satisfy specific conditions and correspond to measurable properties of the quantum system.