Fri 28 May, 2010 11:50 am
For a quantum particle bounded on the left ( x=-a) and right ( x=a). The wave function that described it has discrete energies
1. E(n) = n^2 pi ^2 h^2 / 8 m a^2, n=1, 2, 3 , ....
The lowest energy level is when E(n), but one can surely imagine the case when n=0, E(0)=0.
The wave function R for the particle when n=0 is 0, or R=0. So, the probability | R|^2 = 0.
It seems we can formulate the above into:
2. If E=0, then the probability is 0.
Why is it a mystery? The condition in 2, "E=0" is never true in this world. It seems in all possible worlds, E is not zero given what we mean by E in the QM sense. In all possible world that don ` t obey QM, we don` t care. In all possible world that do obey QM, 2 is always false. All this point to 2 being a necessary false. Don` t one feel that 2 ought to be contingent?
The ZPE is only 0 by definition, sometimes adopted for convenience in calculation. The actual energy of the particle at zero is when n=1 in your example series. I would further make the argument that all scientific knowledge is contingent, that QM is scientific knowledge, and therefore (2) is contingent in all possible worlds.
However, a nitpick: You seem to be conflating probability with energy levels. I'm not understanding your statement "If E=0, then the probability is 0".