In quantum mechanics, bound states can be put together by choosing a potential energy and solving the a wave equation. The first non toy example that students learn is the hydrogen atom (or any other single electron atom) studied using Schroedginer’s wave equation. Our interest in this blog is more general bound state problems, but there is a lot we can learn by examining the hydrogen problem and its solutions.
The primary motivation for studying the hydrogen atom was to find an explanation for the light that was given off or absorbed when the atom switched from one energy state to another. For this reason, it was natural to look for the bound states as classified by their energies. That is, we will be looking for wave functions that correspond to sharp energies. If we wanted a more general solution, we can always combine different energy solutions by linear superposition.
The equation we wish to solve is where “H” is the quantum operator for energy (which is a sum of an operator for kinetic energy and one for potential energy), is the energy, is the wave function, and n =1, 2, 3, … is an index that distinguishes different energy solutions. It turns out the energy will be proportional to . Our solutions do not depend on time so we will leave off the t. And instead of writing (x,y,z), we will just write (x), or even leave it off completely like this: .