You have to realize that wave equations and interference phenomena had been studied and understood by the nineteenth century.
Plane waves 
are the simplest mathematical solution of wave equations,
where k, is the wave’s wave number or more specifically the angular wave number and equals 2π/λ, where λ is the wavelength of the wave. k, has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a given distance at a particular point in time.
A list of explanations of the other terms exists in the link.
A plane wave hitting two slits will produce interference patterns in the context ( equations and boundary conditions) that they are a solution of. It is not necessary to derive the solutions over and over again since for most interference patterns the plane wave assumption is a good approximation.
The Schrodinger equation is a wave equation , and plane waves are solutions of the equation. Again in the double slit experiments the plane wave solutions are used to model the impinging particles on the two slits, quantum mechanically . The difference between classical plane waves and quantum mechanical plane waves lies in the postulates of the theoretical model used for studying the patterns.
Classical equations predict variations of energy density in space at a given time, it is the amplitude that changes by construction of the theoretical models. In quantum mechanics the wave pattern predicts a probability distribution in space at time t. Thus light waves and photons impinging on the two slits will produce the same interference pattern, because the probability distributions for photons are from a solution of a quantized Maxwell's equation , but as has been shown with single photons at a time, when viewed quantum mechanically the two slits give a probability distribution in space for photons , which eventually builds up the classical ( energy deposition ) interference by each of the zillion photons leaving the appropriate energy at the appropriate (x,y) of the screen (the red histograms in the video). Physics is continuous between quantum mechanics and classical.
The mathematics for the probability distributions for electrons impinging on a double slit
are again plane wave solutions of the Schrodinger equation, the only complicated concept being that they represent probabilities of finding the particle in space at time t, not energies or mass and this is evident in the slow build up of the interference pattern.