Energy of electron + positron separated by distance

[xortdsc]

Hi,

given the scenario of an electron and a positron (both assumed to be stationary) being separated by a given distance how could i compute the energy of that system ?
The total energy would certainly contain the rest-mass-energies of both particles, but I'm struggling to incorporate the electro-static energy (the usual electro static potential does not seem to be of help as it would suggest it requires infinite amounts of energy to separate the orignally superimposed electron/positron in the first place, or do i miss something here ?).

So mathematically:
E = 2m_ec² + f(separation)
But what would f(separation) look like ?

Can somebody help me here ?

 

[xortdsc]

Hmm, is it that much of a tricky question ?

 

[Bill_K]

Isn't it just E = 2mc² - e²/r?

When the particles are infinitely far apart the energy reduces to 2mc², since the rest energy includes the Coulomb self-energy.

 

[xortdsc]

hmm, but that seems odd, as it would mean that at a certain distance E=0 and coming closer together E<0 up to E=-inf at r=0
Wouldn't make a whole lot of sense, or would it ?

 

[Bill_K]

The total energy of the system can't fall below zero. At short distances you have to take into account the effects of QFT. And in particular, the Coulomb potential is modified by vacuum polarization.

Quoting this paper, for example,

An important consequence of the polarization of the vacuum is a modification of the electrostatic interaction between two electrically charged particles at spatial separations of the order of or smaller than an electron Compton wavelength.

They then give the first order modification of V(r).

 

[xortdsc]

Ah thanks. That seems interesting. Gotta try out that formula and see what it spits out.

 

[xortdsc]

Hi,

so I tried out the suggested corrected coulomb potential, but it doesn't seem proper and I cannot see what I'm doing wrong.

my definition goes like this (using cgs gauss units as in the paper, I'd presume, because of alpha=e^2/(hbar*c)):

e = -4.80320451*10^-10 statC
m = 9.10938291*10^-28 g
hbar = 1.054571726*10^-27 erg*s
c = 29979245800 cm/s
alpha = e^2/(hbar*c) = ~1/137
qftCorrection[r] = 2*alpha/(3*Pi) * Integral[Exp[-2*(m*c/hbar)*x*r] * (1 + 1/(2*x^2)) * (x^2 - 1)^(1/2) / x^2, {x, 1, Infinity}]
coulombEnergy[r] = -e^2/r*(1 + qftCorrection[r])
pairEnergy[r] = 2*m*c^2 + coulombEnergy[r]

The main problem is that the energy for the electron/positron-pair (pairEnergy[r]) drops below 0 (which it shouldn't, right?) for distances smaller than around 2*10^-13 cm as can be seen in this plot.

Is there possibly some error in the units I'm using or did I misinterpret the formula somehow ?

 

[The_Duck]

The minimum radius of positronium (its radius in its lowest energy state) is of order the Bohr radius, very roughly $10^{-10}$ meters. Thus it always has positive total energy, because this is much greater than the radius you calculated at which the negative binding energy would exceed the positive rest mass energy of the constituents.

However, it seems to me that if you increased the charge of the electron enough, then you could make it so that the total energy of the positronium ground state would be negative. I'm unclear what this means--does the vacuum turn into a condensate of e+e- pairs or something?

 

[xortdsc]

well, I'm not really concerned with positronium in ground state, but rather with electron/positron pair creation itself.

 

[The_Duck]

OK, your worry seems to be "at the moment the e+e- pair is created, the potential energy is infinitely negative, therefore the total energy is infinitely negative."

If you want a classical picture of e+e- pair creation that resolves this worry, the e+ and the e- each start with infinite positive kinetic energy, canceling their infinite negative potential energy (and leaving a small finite positive remainder equal to the energy put into the e+e- pair by whatever process created it). This isn't too far from the correct quantum mechanical picture.

 

[xortdsc]

yes exactly.
but then how to get a actual number for the energy for a system in which a electron/positron pair is created which depart from each other, slowing down due to electro-static forces between them and at a distance of "d" actually stop (the "turning point") and start accelerate towards each other again ? That precisely what I want to calculate. But how exactly ?

 

[The_Duck]

Classically, the energy of such a system is V(Rmax), where Rmax is the maximum separation of the electron+positron.

Quantum mechanically, particles don't have definite trajectories.

 

[xortdsc]

but the classical V will always be negative for positron/electron, right ? So it would only permit computing differences in energies and the difference to the zero separation is infinite.