What is the physical reasoning behind the mathematical derivation of double-slit interference pattern and single-slit diffraction pattern?

Question

Why is it that there are two different equations for the minima of a single slit diffraction pattern and a double slit interference pattern?

In the double slit experiment, the two slits act as two point sources of coherent light, thus maxima occur when the two rays have a path difference that is dsinθ=n λ and minima occur when dsinθ=(n λ)/2 where d is the separation of the two slits and n is an integer. In other words, if their path difference is an integer multiple of wavelength then there will be constructive interference and if there is half an integer multiple of wavelength then there will be destructive interference.

However, a different equation is used for finding the minima of a single slit experiment; (b/2)sinθ=(n λ)/2, where b is the slit width of the single slit. The reasoning behind this derivation for the minima of single slit diffraction is that according to Hyugen's principle, there are an infinite number of light source in a single slit. Thereby, by taking a light source at the top of the slit and a light source at the middle of the slit, their path difference will hence be (b/2)sinθ. Additionally, two coherent sources of light interfere destructively if their path difference is half an integer multiple of wavelength, and hence (b/2)sinθ=(n λ)/2 for minima. From that equation, it can then be simplified to bsinθ=n λ.

Comparing the maxima equation for the double slit and minima equation for the single slit, we get that they are the same. Additionally, the minima equation for double slit and single slit are opposite of each other, despite both are built on a similar concept.

Another question is that, in the single slit experiment, why must the light source at the top and middle of the slit be used to derive where the minima occurs? Why can't it be the light source at the top and bottom of the slit? By taking the first point and the last point, their path difference will be bsinθ, and for destructive interference to occur, bsinθ=(n+1/2)λ, which would hence correspond with the equation for minima in a double slit experiment.

Last question is that, why isn't there an equation for maxima in a single slit?

Answer 1

For single/double slit experiments interference/Huygen's principle is employed which is a classical/historical explanation for observations. The math used provided an explanation for the observations and there have been discussions on the conflicts. Classical is still taught but a more advanced explanation is available that better explains slit diffraction:

In the modern understanding, whether it's photon or electron, each one has a wave function to describe it's propagation. As a single event a wave/particle is traveling towards the slits and chooses one based on probability, the probability is based on the varying EM field of the wave/particle interacting with the varying EM field of the slit(s) (lots of electrons moving in the slit material), the varying causes the randomness of which slit is chosen. The wave propagation function also causes the banding pattern seen, it's a solution of the wave propagation equations with the EM properties and dimensions of the slit(s). Maxwell's laws for EM are used.

The classical model could not explain diffraction when photons/electrons were used fired one at a time, no inference between photons was possible yet the diffraction pattern still emerged. The single photon is said to be "interfering with itself" going through the slit(s) but using field theory (the photon is a wave in the EM field) it would be better explained as an interaction with the slit(s).

The term "interference" can not explain what happens in a double slit experiment and leads to a lot of debate, it's outdated. The classical theory math was great as a first step but as you note it has inconsistencies. Huygen's principle explained water waves interacting with objects, although I'm not sure how accurately.