This is a special form of the general, time dependent Schrödinger
equation
Many of you are familiar with electron wave-functions from chemistry.
There one describes, for example, s and p
states of various shapes, which can be occupied by the electron, binding
it to the nucleus. The shapes of these wave-functions in large part
determines the chemical bonding of the atoms. The energies of these
wave-functions determines the frequencies of the light they can absorb
and emit. Unlike waves on a string, each solution does not separately
resonate with light of a particular frequency.
Instead, light is emitted and absorbed when the electron
changes from one state to another, and the frequency of the
light is determined by the difference in energies between the two
solutions:
We'll solve the problem by "shooting": we pretend we know the derivative of at X=0, solve the equation, and then fiddle around until it has the right value at the other boundary. For our problem, we'll fiddle around with the energy E: raising and lowering the energy will change how the electron wave function acts at the right-hand boundary.
The default potential is a square well, with infinite walls. Try varying the energy to make the wave-function zero at the right-hand side! You'll find there are several solutions, with different numbers of nodes, just like there are several different shapes of wave-functions for electrons in your chemistry class.
Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).