Philosophical Corollaries of Schrödinger's Equation

For instance, take the Schrodinger wave equation:

-((h/2π)²/2m)(∂²ψ(x)/∂x²)+V(x)ψ(x) = Eψ(x)

If we take the potential energy, V, to be 0 everywhere (i.e. the particle is not being acted upon by its environment), the equation becomes:

-((h/2π)²/2m)(∂²ψ(x)/∂x²) = Eψ(x)

Valid solutions for fÕ(x) include Asin(cx) and Bcos(cx). We'll use Asin(x). However, since both sin and cos functions "look" the same everywhere, we find we have no useful information about the particle. The particle has an equal likelihood of existing anywhere; since it's probability of existence is 1 and it has an equal probability of existing anywhere in infinite space, it essentially does not exist. (i.e. 1/infinity = 0)

This can be interpreted as meaning that all particles can be said to exist only as they relate to other particles. An atom's existence is contingent on its interactions with other atoms. Furthermore, since potential energy is almost always dependent on distance, an object's distance from everything else in the universe actually has an effect on the probability of that object's existence.

It was only in writing this that I realized a more profound philosophical significance to this. While there are numerous interactions WITHIN an atom (between quarks, protons, neutrons, electrons, etc.), the atom still cannot be said to exist if all the interactions occur within it. A component of the atom cannot act upon the whole atom, since it cannot act upon itself. Thus, the existence of a system is dependent on what lies outside the system.

This leads to a rather profound paradox when one takes the universe to be the system. If the universe is all that is, how can it exist without something else to define a potential for the universe?

The simplest solution to this paradox would be to claim that the Schrodinger wave equation is not true, or is simply a simplification, which I believe is its current status in physics.

While it is true that the solutions to the Schrodinger Equation given a constant potential are plane waves, one has to keep in mind that a particle need not be in an energy eigenstate. In fact, since infinite quantities (the uncertainty in position in this case) are widely assumed to be unphysical, the particle must be localized to some extent. The particle could very well be almost completely-localized, since the Fourier transform of a delta function is unity (see waves).

Perhaps a more realistic paradox is this: if only a single atom exists, and its energy is measured, collapsing its wavefunction into an energy eigenstate, we get a completely delocalized particle. This implies that infinite quantities--the uncertainty in position here--are physical.