Approximate methods I: variational method and WKB

It is often (almost always!) the case that we cannot solve real problems analytically. Only a very few potentials have analytic solutions, by which I mean one can write down the energy levels and wave functions in closed form, as for the harmonic oscillator and Coulomb potential. In fact those are really the only useful ones (along with square wells)... In the last century, a number of approximate methods have been developed to obtain information about systems which can’t be solved exactly.

These days, this might not seem very relevant. Computers can solve differential equations very efficiently. But:

• It is always useful to have a check on numerical methods
• Even supercomputers can’t solve the equations for many interacting particle exactly in a reasonable time (where “many” may be as low as four, depending on the complexity of the interaction) — ask a nuclear physicist or quantum chemist.
• Quantum field theories are systems with infinitely many degrees of freedom. All approaches to QFT must be approximate.
• If the system we are interested in is close to a soluble one, we might obtain more insight from approximate methods than from numerical ones. This is the realm of perturbation theory. The most accurate prediction ever made, for the anomalous magnetic moment of the electron, which is good to one part in ${10}^{12}$, is a 4th order perturbative calculation.