Falling p.7

Some Details on QM Falling

As previously stated, energy eigenfunctions have position probabilities that do not depend on time (although the energy eigenfunctions themselves do have a [complex] time dependence). Thus to see motion (i.e. things that depend on time), we must not be in a energy eigenfunction; the only alternative is to be in several energy eigenfunctions, i.e., to be in a superposition of energy eigenfunctions. Thus the idea is to create superposition of energy eigenfunctions which initially has a narrow range of position possibilities, and watch that range of position possibilities move.

The game then is a version of Fourier analysis: producing a sum-of-eigenfunctions that adds up to a given function. Orthogonality is what makes this easy; in particular, that our energy eigenfunctions are mutually orthogonal.

Our normalized energy eigenfunctions are displaced Airy functions:

These eigenfunctions are orthogonal:

So if we try to express a given function f(z') in terms of a sum of these functions:

orthogonality allows us to solve for the b_n:

In this particular example, I choose f(z') to have the shape of the ground state, but with the whole function moved over. Of course, we can't sum all the terms of the infinite sum. I've limited the sum to the 15 eigenfunctions nearest the E'=122.2 eigenfunction. Thus the sum using these 15 b_n does not exactly equal the initial f(z'). The below plot displays (in black) f(z') and (in red) the sum using these 15 b_n.

The missing terms would contribute only 5% additional probability, so it's a fair approximation. To see how this probability lump moves with time, we need only use the time dependence of the energy eigenfunctions:

The previous page shows the results: the lump of probability moves just like a classical particle (with broadening of the lump caused by Heisenberg-required uncertainty in initial velocity).