Help- "Harmonic Well"

That is a truly horrifying looking equation isn't it, but everything before the "H" is just a constant number to scale it properly. The Hn(x) means the nth Hermite Polynomial which is a special function with n-1 bends in it. (The square root part is also just scaling.) The last part is called a gaussian distribution and it just flattens the function as it gets further from 0.

This quantum system can be thought of as a quantum analogue to a pendulum or a spring.

In all the simulations on this site, the particle we are looking at is only allowed to move in one dimension which is represented by the Red axis. Left/Right

The reason it is called the "harmonic" potential is because its the basis for systems with resonant frequencies. If you drive a spring at the correct frequency, you get more energy out and pendulums have a specific frequency they swing at, hence their famous use in clocks. The further away from the middle, or "equilibrium", the particle wants to go, the more energy it needs to put in to go further.

The specific equation is E=Kx²÷2 :x=position; E=energy

The number K is called the spring constant, because it corresponds to the strength of a spring's restoring force. If you increase this number you should notice the particle becoming more confined to the middle. It is important not to confuse this with the other k often seen in quantum mechanics which is the wavenumber and corresponds to momentum. I will use upper case K for spring constant and lower case k for wavenumber.

See Also: Simple Harmonic Motion, Hooke's Law