In my last post, I talked about how the solution of the two-electron well becomes simple in the two extreme cases: the very large box, and the very small box. And how these cases correspond to having a fixed size of box while varying the strength of the electric repulsion. The funny thing is, the very small box corresponds to the case of weak repulsion, and the very big box is the case of strong repulsion. Why is this?
The answer turns out to be almost obvious: The repulsive energy is a 1/r force, but the kinetic energy goes as 1/r-squared. You know about the 1/r from basic electrostatics; you see it in the fomula for the potential of a spherical charge. It comes from the 1/r-squared law of forces. Since energy is the integral of force along a path, it always comes to something in 1/r for spherical geometry. The kinetic energy, however, is a purely quantum-mechanical thing. For the particle confined in a box, the kinetic energy is the square of the momentum operator. The momentum operator is differention (well, it's the del operator in three dimension). If you make the box twice as small, the derivatives are twice as big, so when applied twice, you get a factor of 4. In other words, it's an inverse square relationship on the dimension of the containment. So for a very small box the kinetic term dominates the potential term.
The interesting thing is that you can see this in a physical case: it's called the isoelectronic series of helium, and it goes all the way back to my second post about why the helium atom doesn't have a miniature replica in the hydrogen atom. It has to do with the scaling effect being different for the potential and kinetic energy terms, and I did some really cool calculations to show how this works for certain atomic energy levels: hence, "the isoelectronic series of helium".
You may know that when you solve the equation for the hydrogen atom, you automatically get by analogy the solution for a whole series of other atoms in their ionized states, when they have only one electron. For example, singly ionized helium is exactly the same as hydrogen except the wave functions are compressed by a factor of two and all the energy levels are four times greater; ditto for doubly-ionized lithium, except it's three times on size and nine times on energy. Etcetera. It's called the isoelectronic series of hydrogen.
The series for helium is a little different. You don't get the same geometric scaling effect, and the reason is because you're basically putting two eletrons in a box, and it makes a difference how big the box is. You can see it pretty clearly if you look up (as I did) the energy levels for the atomic states belonging to this sequence.
In my next posting I'm going to explain some more about this.