In the single slit diffraction pattern there exists the following formula:
$$d \sin \theta = n \lambda$$
I was looking at and thinking about the quantisation condition of energy, $E = nh\nu$, and saw the formula above, and my brain sparked with the idea that the $n$ might be interpreted in relation to quantisation as such. I think this is quite a valid thing to think about because there are phenomena like electron diffraction and so on.
What do you think?
The number of a particular diffraction minimum $\color{blue}{n}$ in the single-slit diffraction formula $$ d \sin \theta_\color{blue}{n} = \color{blue}{n} \lambda$$ doesn't have a connection to the number of excitations $\color{green}{n}$ of an electromagnetic field mode (in $E_\color{green}{n} = \color{green}{n} h \nu$).
The only thing which is quantized in the diffraction pattern is the number of intensity minima, because $\sin \theta \in [-1, 1]$. Therefore, depending on the slit size $d$ there are cases with $\color{blue}{0}$ minima (if $d < \lambda$), cases with $\color{blue}{2}$ minima (if $\lambda < d < 2 \lambda$), and so on.
The (real part of the) dielectric constant and the number of wave lengths of path difference are unrelated although they happen to be both called $n$. This is a case of mistaken identity.
