Does Pauli's principle prevent an electron from having infinte energy?

[Mr Virtual]

Suppose we have an electron. Electrostatic potential of the electron at any point at a distance r from it,
V = kq/r = (9*10^9 * 1.6*10^-19)/r

Now, as r decreases, V increases. We say that V is the amount of work done by the electron to bring a unit poitive charge from infinty to r. If r is close to zero, then V will be quite high.
I believe that r cannot be exactly zero as that would mean the electron and the test charge are at the same place at once. According to Pauli's principle, no two particles/fermions can occupy the same space at once. Since 'r' cannot be zero, V cannot be infinte. Does this mean that Pauli's principle protects electron from having infinite energy (I hope I am not mixing QM with classical physics too much)?
If I am wrong, please supply explanations as to why that is so.

warm regards
Mr V

 

[CompuChip]

I don't think it has to do with Pauli's principle. The Coulomb force between two charges q₁ and q₂ is indeed proportional to q₁q₂/r². So when placing them "infinitely" far apart, they don't feel each other anymore. At r = 0 we get divergence, but this will actually not happen, since charged objects are rarely point particles and r can never become smaller than the diameter of such a particle.

 

[Gib Z]

In a sense, Pauli's principal is already incorporated in the equation, as a fraction can not have 0 as its denominator. And yes it does, either the Principle or the r in the denominator stops V from being 'equal to infinity' however r can be arbitrarily small and hence V arbitrarily big. That is assuming point particles though.

 

[Mr Virtual]

I don't think it has to do with Pauli's principle. The Coulomb force between two charges q1 and q2 is indeed proportional to q1q2/r2. So when placing them "infinitely" far apart, they don't feel each other anymore. At r = 0 we get divergence, but this will actually not happen, since charged objects are rarely point particles and r can never become smaller than the diameter of such a particle.

Aren't you yourself stating the Pauli principle? Why r cannot be less than the diameter ( actually, I think it should be radius) of the particle, is due to the reason that the test charge( aparticle) cannot reach into the electron(another particle) to make r=0. This is because both the test charge and the electrons are fermions, and thus, one cannot just pass through (or enter the body of) the other. This is what Pauli states: no two particles can be at exactly the same place at the same time.

In a sense, Pauli's principal is already incorporated in the equation, as a fraction can not have 0 as its denominator. And yes it does, either the Principle or the r in the denominator stops V from being 'equal to infinity' .

Exactly!

however r can be arbitrarily small and hence V arbitrarily big. That is assuming point particles though.

But as far as I know, there is no such thing as point particle in reality, is there? But yes, theoretically, it is very much possible to assume such point particles.

Thanks for your replies

regards
Mr V

 

[olgranpappy]

Aren't you yourself stating the Pauli principle?

no, he was not.

 

[Mr Virtual]

Again, please explain why that is so.
Isn't what GibZ said above correct ?


Mr V

 

[olgranpappy]

Okay. A many-particle system of fermions (e.g. electrons) is described by a wavefunction that must be antisymmetric if I interchange any two of the fermions. If I think in terms of single-particle "orbitals" (or "quantum states") this means that no two fermions can occupy the same quantum state (because the total wave function would be trivally zero). That is the Pauli principle and it has nothing to do with the fact that the electrostatic potential of a point charge goes like 1/r.

 

[CompuChip]

Aren't you yourself stating the Pauli principle?

The formula I stated was the Coulomb formula, which I think is valid for any particles (bosons and fermions) that interact "Coloumbically".

Why r cannot be less than the diameter ( actually, I think it should be radius

If the origin of one particle is 2 radii (= 1 diameter) away from the origin of the
other particle, they exactly touch.

But as far as I know, there is no such thing as point particle in reality, is there?

Indeed, not that I know of. But theoretically it is often convenient to deal with point particles, as in many applications whether the radius is 10^{-10} m or 0 doesn't really matter.

 

[jostpuur]

Talking about potential energy between particles makes sense only in the classical theory, or as an approximation in a quantum theory in a special case where particles are in localised wave packets and sufficently far away from each others. So I would say that the OP is on a wrong track.

Continuing with the hand waving quantum mechanics, it is the uncertainty principle that prevents electrons from gaining infinite energies as result of small distances, although exclusion principle does have some effect on the wave functions too.