Ab initio the momentum operators can be constructed using de Broglie Plane waves
In one dimension, using the plane wave solution of the Schrodinger equation,the wave function
Psi = exp. i (kx -wt) ,
if one takes the partial derivative w.r. to x of the wave function
delta/delta x (Psi) = ik. Psi
and using de-Broglie relation p = hbar . k we get
delta/delta x (Psi) = i p/hbar . Psi
The above relation suggests the operator equivalence of momentum:
p-operator = -ihbar. Delta/deltax
so the momentum value p is a scalar factor, the momentum of the particle and the value that is measured, is the eigenvalue of the momentum operator.
As the partial derivative is a linear operator the momentum operator is also linear, (one can think of momentum as generator of translational symmetry)
and because any wave function can be expressed as a superposition of other possible states
when this momentum operator acts on the entire superimposed wave, it furnishes the momentum eigenvalues for each plane wave component.
