Differential Equation Problem, Help Please

[jinydu]

I've been trying to prove a certain theorem in quantum mechanics, but I've gotten stuck at an intermediate point:

http://www.mersenneforum.org/showthread.php?t=5187

Basically, the question boils down to this:

Suppose we have a complex-valued function defined with spherical coordinates. Assume that the Laplacian of the function always has the same complex argument as the original function itself. Show that the function cannot approach 0 whenever the radial coordinate goes to infinity.

I've already posted this on two forums and haven't been able to find a satisfactory proof (or refutation).

[Keiji]

*moves to Q&A...*

[PWrong]

I'm not sure I understand the question. Surely the function has to approach 0. :?

From your other forum:

This problem comes from my trying to prove a theorem in quantum mechanics: That there are no normalizable solutions to the time-independent Schrodinger equation where the total energy is less than the potential energy at every point in space.

How can there be no normalisable solutions? What would happen to an electron if it's wavefunction suddenly didn't know what to do?

In any case, I think you should take Ewmayer's advice, and use separation of variables first.

[jinydu]

I'm not saying that there are no normalizable solutions at all. I'm just trying to set a lower bound on the energy of any normalizable wavefunction.

As a special case, this would imply that a system made up solely of two electrons could never have negative energy, since the potential energy is always positive at any point in (6-dimensional) "space".

[PWrong]

I think I understand the problem now. It seems obvious, but I'm not sure how to prove it. Anyway, I don't know about systems with more than one electron. I think we touched on it in physics, but it seemed like mostly guesswork. I haven't seen any multi-electron version of Schrodinger's equation.

[jinydu]

Actually, I was thinking about a system with only two electrons and nothing else. This is not the same as the multi-electron systems covered in quantum mechanics class; those normally contain protons as well.

I still haven't found a way to prove the theorem. I know that it is trivial in Newtonian mechanics, in which

Total energy = Potential energy + Kinetic energy

Since Kinetic energy = 1/2 mv^2 > 0 (mass must always be positive and the square of velocity is nonnegative), we must have:

Total energy >= Potential energy

But it seems that the corresponding statement in quantum mechanics is a lot harder to prove.