Is Dirac's equation valid for point charges?

what_are_electrons

I've read that Dirac's equation for explaining spin does not hold if electrons are defined as true point charges. Is that correct?

dextercioby

WHATTTTTTTTTTTTTTTTTTTT??????????????Hell,no.What said/wrote that was a big LIAR.

Daniel.

Tom Mattson

Dirac's equation is written for point charges, and there's nothing ill-defined about it.

What's the source of this statement?

selfAdjoint

Maybe he's worrying about self-action at zero distance?

marlon

Perhaps he is wondering about the fact that in QFT, the position of some particle is not exact and therefore you cannot point out exactly where the particle is. It is situated inside a little cloud, if you will, expressing the region where you have a certain propability to find such particle...Perhaps he is just referring to orbitals ???

marlon

dextercioby

Marlon,don't speculate & perform a bad tasting mixing of QM & QFT...
Let us allow the OP to come up with the source of missinformation...

Daniel.

marlon

Err, how can you mix QM with QFT, dexter ???

marlon

dextercioby

U did that...Not me...If you're claiming that "orbitals" and QFT live happily ever after,then you're wrong...

Daniel.

marlon

So are you saying that the concept of orbitals does not exist in QFT ???

I hope not...

You do know that all the principles of QM are incorporated into QFT. I did not invent this (unfortunately), many other more "ingenious" minds did...

marlon

ps : and are you saying that uncertainty does not exist in QFT ???

dextercioby

UNCERTAINTY EXISTS IN QFT,ORBITALS NOT.That's (the second part,really) what i meant by mixing things...Concepts...

Daniel.

marlon

Ok, then if orbitals do not exist in QFT, this implies that the spherical harmonics also don't exist in QFT. Are you really serious now ???

marlon

Let's not fall into a discussion on personal interpretations...stick to the physics...we might as well be discussing your favorite actor

Hans de Vries

The Dirac's equation hasn't anything to do with this.


Observable spin effects like the spin angular momentum and the magnetic
momentum can not come from a spinning point charge. A spinning charge
with the classical electron radius (2.817940285 10^-15 meter) is also much to
small to support the measured values.

To get for instance the measured magnetic moment at this radius means
that the charge needs to "rotate" with a speed much higher than the speed
of light.

If we use classical equations to get an indication of the required size then
it turns out that one gets more realistic values ([itex]\leq [/itex] c) if you assume a "radius"
of circa 137 times the classical electron radius: ~ 3.8616 10^-13 meter, where
1/137.03599911 is the finestructure constant.


You get this radius if you apply Heisenbergs Uncertainty Principle:
Take the electron's rest energy of 0.5109892 MeV as the uncertainty in
energy. This corresponds with an uncertainty in momentum of 0.5109892
MeV/c.

Now if you use the equation: [itex]\Delta x \Delta p \ = \ \hbar [/itex] then you'll get the radius above
of ~ 3.8616 10^-13 meter as the uncertainty in position. The corresponding
rotation frequency becomes [itex]\approx [/itex] the 1.235 10²⁰ Hz: The rest frequency of
the electron.

In a similar way one can obtain the Bohr Radius and the ground state
frequency of the hydrogen atom from the orbital angular momentum and
the 13.6 eV ionisation energy of the ground state.


So to conclude:

Spin and orbital angular momentum as well as the spin and orbital
magnetic momentum are in "the wave" and not in "the particle"



Regards, Hans

dextercioby

It's just that.Personal interpretations... To me,"orbital(s)" is a term not pertaining to QM,but to chemistry and their idea of "doing" QM.It's very deluding.It derives from "orbit" and hence the old theory of Bohr.No respectable physicst teaching QM would not use this word...I think chemistry teachers have created a "passion" for it...

If u saw a QM course mentioning the term "orbital",then the author was not a physicist...Was a moron...



Daniel.

Tom Mattson

Since the original poster hasn't explained the remark, it's not really possible to say. Dirac's equation does indeed describe the mechanics of spin-1/2 point particles. If we take the OP at face value, it says that there is some problem with that description. What you have explained is why classical spinning cannot be the same as quantum mechanical spin.

marlon

I am sorry but what you say is wrong. Orbitals are just the squared spherical harmonics. They are nothing else then a probability...But why orbitals, well, because these spherical harmonics are the eigenfunctions of the equations for L² and L_z...They only depend on the two angular degrees of freedom. This combined with the name of L² and L_z, you have the explanation for the word ORBITAL... Nothing is orbiting here

marlon

Hans de Vries

Classical spinning isn't quantum mechanical spin OK. But the gap is not
that big. Dirac included Pauli spinors in his equation to account for the
spin 1/2 behavior. It was the (relativistic) classically derived Thomas
Factor of 2 which convinced Pauli of the existence of the electron spin.
Tomonaga showed that this approach predicts other angular interactions
and energy levels in the atom as well.

The conclusion remains the same: spin angular and magnetic moment
can not stem from an arbitrary small sized particle. It must come from
the quantum field.

Regards, Hans

Tom Mattson

Yes it is that big. It's huge, in fact. Take a spin-1/2 particle and rotate it through 2π radians. You'll find that you do not arrive at the initial state, but rather the negative of the initial state. Something is very much at odds with classical angular momentum here.

Dirac included Dirac spinors (4 components) in his equation. They aren't the same as Pauli spinors, which have 2 components. And I don't think it's accurate to say that he included them to account for spin-1/2 particles. The mapping of Dirac's theory to spin-1/2 particles emerges naturally from the linearization process on the relativistic Hamiltonian, if we take the gamma matrices to be of the lowest possible dimension.

How does this address the fitness of the Dirac equation with point charges?

In QFT the particles are excitations of the quantum field, and they carry angular momentum (including spin). So it's not surprising that angular momentum must come from the quantum field. But we weren't talking about where angular momentum comes from, we were talking about an alleged problem with the Dirac equation.