I know the Schrodinger Equation is a key part of quantum mechanics. I am trying to understand it’s applications. Let’s say a team of engineers wants to build a laser from scratch, assuming they have the raw materials and tools at hand. At what point in the process would they need to use the Schrodinger Equation in order to complete the task.
When engineers are given a task to design and build a system, it usually involves a set of specifications. In the case of a laser, the specifications would naturally include the wavelength of the required laser radiation, among other things. If we assume that the raw material does not already include the material for the active medium associated with that wavelength, then the engineers would have to do some calculations to determine what material they need that would produce light with that wavelength. This calculation would require some rudimentary quantum mechanics, because the wavelength is associated with the energy separation between two energy levels in the material.
There are also other specifications, such as the amount of power that the laser must produce. Perhaps, it would even include the power efficiency of the system. To design a system that satisfies these requirements, the engineers would have to use the laser rate equations, which comes from Einstein's analysis of stimulated emission. These equations also involve basics from quantum mechanics.
So, although the Schroedinger equation may not be used directly in the design process, other aspects of quantum mechanics play a role in the design process.