Potential energy of spin anti-alignment

[Sky Darmos]

TL;DR Summary

Calculating the potential energy of spin anti-alignment

Hello everybody, I consider two electrons that have enough kinetic energy to reach their respective classical electron radius. This would be:

2.0514016772310431402e-13 J

The corresponding speed is v = 287336682 m/s.

The electric field is

E = \frac{k_{e}}{R_e^2} = 1.8133774657059088443 × 10^{20}

The magnetic field is:

B = \frac{v × E}{c^{2}}

Multiplying that by the relativistic gamma factor, which in this case is 3.5056494831959322035, we have:

B=2.0323868283603503304e12 T

The magnetic moment of an electron is:

μ_{S} = - 9.2847647043 × 10^{−24} J⋅T^{-1}

The potential energy of spin anti-orthogonality is then:*

∆U = 2 μB Or: ∆U = γ_{e} B ℏ

That yields:

3.7740466978888805979e-11 J

Obviously too much. It exceeds the total mass energy of the electron, and even that of the muon. That means the resulting mass would be negative. It can't be, so there must be something terribly wrong with my calculation.

Kind regards.

*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magpot.html7

 

[Nugatory]

Obviously too much. It exceeds the total mass energy of the electron, and even that of the muon. That means the resulting mass would be negative. It can't be, so there must be something terribly wrong with my calculation.

Something is indeed wrong with your calculation: we need quantum mechanics to describe the behavior of electrons at this scale, so these calculations based on classical notions of distance and trajectory aren't applicable.
But even setting this aside, you are making a common mistake about potential energy: Potential energy is a property of the system as a whole, not of any one particle within the system. So when you find an enormous amount of potential energy associated with two nearby and anti-aligned electrons, that's just telling you how much energy has to be provided to force them into that arrangement. Two electron rest masses is the total energy of a system of two electrons infinitely (well, ok, not "infinitely" but far enough that all interactions between them are negligible) far apart. Bringing them closer will require adding energy to overcome the forces between them, which increases the potential energy and mass. Add an arbitrarily large amount of energy and the total mass/energy of the system will become arbitrarily large.

 

[Sky Darmos]

Potential energy of spin anti-alignment does contribute to the total energy of the system. This is evident from the Zeeman effect, the doublet splitting of spectral lines.

Ok, yes, I guessed that this calculation is not quantum enough. So, how would I go about calculating this in a proper quantum mechanical treatment?

I just found this: The energy difference between the two Zeeman states is given by ΔE = E(mS = +1/2) - E(mS = -1/2) = geβeB0/h (in Hz).

So I guess this would be a more correct approach.

 

[Hyperfine]

Potential energy of spin anti-alignment does contribute to the total energy of the system. This is evident from the Zeeman effect, the doublet splitting of spectral lines.

I find the terminology "spin anti-alignment" confusing in that it seems to imply two electrons.

The Zeeman effect lifts the degeneracy of spins states within an applied, external magnetic field in the weak field limit in which spin-orbit interactions exceed the magnetic field effects. Doublets result only for J=1/2 where J is the total angular momentum exclusive of nuclear spin.

In any case, a quantum mechanical treatment is necessary.

 

[Sky Darmos]

Yes, I imply two electrons, and they have to be free. I guess the Zeeman effect is about electrons in orbit only. I would need something similar, but for free electrons.

 

[Hyperfine]

Yes, I imply two electrons, and they have to be free. I guess the Zeeman effect is about electrons in orbit only. I would need something similar, but for free electrons.

This makes no sense whatsoever to me.

 

[Sky Darmos]

It is easier to smash two electrons with opposite spins together. It takes more energy to do that with same spin electrons. That is the exclusion principle.

 

[topsquark]

It is easier to smash two electrons with opposite spins together. It takes more energy to do that with same spin electrons. That is the exclusion principle.

But in that case, the electrons are not free... They "see" each other's electric potential.

-Dan

 

[Hyperfine]

It is easier to smash two electrons with opposite spins together. It takes more energy to do that with same spin electrons. That is the exclusion principle.

I suggest that you reconsider your formulation of the Exclusion Principle.

 

[Sky Darmos]

Well, in my calculation I assumed that if we double the electron charges which are forced into the same area of space, from 2 to 4, then it is like dividing their confinement space into half, and so the energy needed to bring them together would be double. Consequently, when it is only 3 charges, the energy needed is 1.5 times more than normal. I got excellent results with that method. Predicted the muon mass with 99.91% accuracy and the tauon mass with 100% accuracy. However, I could do that only by scaling up the little discrepancy for the muon up to the tauon, assuming that it is spin related, causing a tiny mass decrease for the predicted muon mass (about 0.18 of an electron mass) and a not quite as tiny a mass increase for the predicted tauon mass (about 50 electron masses).

Anyway, it is a theory, and I am not here to convince anyone. I just need to figure out how to actually calculate that spin effect, and not just take the discrepancy to be that effect.

 

[Hyperfine]

I am sorry, but I have no intention of becoming involved in a personal "theory" particularly when that "theory" has not be articulated.

 

[Sky Darmos]

I find the use of the word "private theory" very amusing. I guess you mean a theory that was developed by a single individual. So, you are suggesting that Penrose's CCC model, which falls into that category, is somehow more wrong than string theory? String theory is an utter pseudoscience with no predictions and zero credibility. By this standard, string theory would be better than Einstein's relativity, because string theory is a collective delusion, while Einstein's work is that of a single individual (if we want to believe that he didn't know about Poincare and Lorentz).

Any theory that is profound will be that of a single individual, because if everyone jumps on it, then it must be so obvious that it isn't revolutionary.

The fact that after 50 years of complete stagnation in physics (since 1973), there are still people using the word "private theory" as an insult is like people using the word "conspiracy theorist" as an insult after 3 years of a fake global pandemic based on a new kind of sniffles where people are forced to wear face diapers to "stop the spread".

If you mean "private" in the sense of secret, then no, my theory is very public. There are things like google that you can use to look things up.

This is a post about a computational problem. I expect only answers that contain equations. If it continues like this, then it tells me that this can't be a professional forum. I have seen no professional answers so far.

 

[weirdoguy]

I guess you mean a theory that was developed by a single individual.

No, it means "theory" developed by someone who is not a professional scientist and does not publish his work in peer-revieved journals. Usually those kind of people lack serious knowledge in physics and hence their theories are just a bunch of nonsense.

I have seen no professional answers so far.

And the fact that you have not recognized those is weird, since you clearly have been given professional answers (e.g. post #2).

 

[topsquark]

Well, in my calculation I assumed that if we double the electron charges which are forced into the same area of space, from 2 to 4, then it is like dividing their confinement space into half, and so the energy needed to bring them together would be double.

The only way the Pauli Exclusion Principle would forbid two electrons from sharing the same state is if they are part of the same system. Hence there has to be some kind of mutual EM (or weak nuclear) potential field. And indeed, there is: the EM potential field due to their charges.

So, as you said before, you want the electrons to be free: they aren't so your concept of two interacting "free" electrons is impossible.

In general, yes, if the electrons in a system have the same spin it will be impossible for them to pass close by each other based on the Exclusion Principle. Or, more precisely, the probability density for both electrons to have the same position will be vanishingly small.

Sorry, no equations needed.

-Dan

 

[DrClaude]

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