In quantum mechanics, if a wave function is an eigenfunction of an operator (such as momentum, energy, etc.) then the wave function will yield a definite eigenvalue of that operator when measured. For example, if p^ψ (x) = pψ(x), then the wave function ψ(x) is a momentum-eigenfunction that represents a particle whose momentum will be measured to be a definite eigenvalue p. (Note that the Schr¨odinger equation is simply the equation that yields energy eigenfunction/eigenvalues: H^ψ(x) = Eψ(x), where E is the energy eigenvalue of a particle whose wave function is ψ(x)). Wave functions that are not eigenfunctions of an operator represent particles that will not yield definite eigenvalues of that operator, and can only be determined probabilistically.(a) Show whether or not the wave function ψ(x) = Ae^(ipx/h-bar) of a free particle is a momentum-eigenfunction with a definite momentum.(b) Is this same wave function an energy eigenfunction? If so, what is its energy? Explain.(c) Show whether or not the wave function un(x) = √(2/L) sin (nπx/L) of a free particle is a momentum eigenfunction with a definite momentum.(d) Is this same wave function an energy eigenfunction? If so, what is its energy?(e) A superposition state is a wave function that is a linear combination of energy eigenfunctions, e. g. the wave function given by a linear sum of wave functions of the form in(d): ψ(x) = 1/(√2) u1(x) +1/ (√2) u2(x). Show whether or not ψ(x) itself is an energy eigenfunction with a definite energy.