I'm sure you can find classical equations that model some aspects of p-n junctions but you're ultimately going to see that p-n junction physics is literally quantum physics of tunneling electrons in atomic solids.
Oh, I see what you're saying now. Your teacher showed you the equations explaining a classical (pre-semiconductor) diode, then showed those equations predict some aspects of semiconductor diodes.
Yeah, that doesn't mean that diodes don't work in a fundamentally quantum way. There are a number of details about diodes (for example, the emitted frequency of light in an LED) that are very specifically due to quantum energy transitions of electrons in outer shells. It doesn't get any more quantum physics than that.
This was the equations for a doped semiconducting device, not a vacuum tube. All the equations we had previously been using to describe buffered solutions, etc. also happened to work perfectly well for semiconductors.