Quantum Mechanics for Scientists and Engineers

We have seen how to solve some simple quantum mechanical problems exactly and, in principle, we know how to solve for any quantum mechanical problem that is the solution of Schrödinger's equation. Some extensions of Schrödinger's equation are important for many problems, especially those including the consequences of electron spin. Other equations also arise in quantum mechanics, beyond the simple Schrödinger equation description, such as appropriate equations to describe photons and relativistically correct approaches. We postpone discussion of any such more advanced equations.

For all such equations, however, there are relatively few problems that are simple enough to be solved exactly. This is not a problem peculiar to quantum mechanics; there are relatively few classical mechanics problems that can be solved exactly either. Problems that involve multiple bodies or that involve the interaction between systems are often quite difficult to solve.

One could regard such difficulties as being purely mathematical, say that we have done our job of setting up the necessary principles to understand quantum mechanics, and move on, consigning the remaining tasks to applied mathematicians or possibly to some brute-force computer technique. Indeed, the standard set of techniques that can be applied, for example, to the solution of differential equations can be (and are) applied to the solution of quantum mechanical differential equation problems. The problem with such an approach is that if we blindly apply the mathematical techniques, we may lose much insight as to how such more complicated systems work.