What is the energy due to Spin?

[jonjacson]

Hi folks,

A simple question, in classical mechanics for example you have:

-Because of the speed --> Kinetic energy =1/2 m v^2

-Because of position in a gravitational field ---> Potential energy

In quantum mechanics, let's talk about an electron on an hydrogen atom. From the point of view of the energy on an excited state, the electron will have kinetic energy because of the movement, potential energy because of the proton, but WHat is the mathematical expression for the energy due to the intrinsic spin?

THanks

 

[blue_leaf77]

WHat is the mathematical expression for the energy due to the intrinsic spin?

The effect of spin on the energy levels is manifested in the interaction between electron and a magnetic field. That electron has an intrinsic angular momentum leads to the property that it also has intrinsic magnetic dipole moment. This intrinsic magnetic moment may interact with external magnetic field. When there is only one electron around the nucleus (H-like atom), there is only spin-orbit coupling. This is actually an interaction between electron's magnetic moment and magnetic field created due to the relative motion between the electron and nucleus. As the number of electrons increases, the magnetic moment of one electron can interact with magnetic field generated by the spin of the other electron. This kind of interaction is called spin-spin interaction. Another example where spin of electron influences energy levels can be observed in Zeeman effect where atoms are subjected under a constant magnetic field.

 

[jonjacson]

The effect of spin on the energy levels is manifested in the interaction between electron and a magnetic field. That electron has an intrinsic angular momentum leads to the property that it also has intrinsic magnetic dipole moment. This intrinsic magnetic moment may interact with external magnetic field. When there is only one electron around the nucleus (H-like atom), there is only spin-orbit coupling. This is actually an interaction between electron's magnetic moment and magnetic field created due to the relative motion between the electron and nucleus. As the number of electrons increases, the magnetic moment of one electron can interact with magnetic field generated by the spin of the other electron. This kind of interaction is called spin-spin interaction. Another example where spin of electron influences energy levels can be observed in Zeeman effect where atoms are subjected under a constant magnetic field.

Very interesting, and another question. If the electron is alone in the Universe, What is its energy due to its intrinsic angular momentum?

IN clasical mecanics if we had a rigid solid rotating on its symmetry axis we would say it has an energy = 1/2 I w^2 .

Where I is the moment of inertia.

What would be the equivalent for the electron due to its intrinsic angular momentum? (And nothing else, no interaction, no magnetic field, no other particles)

 

[blue_leaf77]

If the electron is alone in the Universe, What is its energy due to its intrinsic angular momentum?

If this electron roams the space at a non-relativistic velocity, its spin will contribute nothing to its energy. As for when it moves with relativistic-speed, if I am not mistaken, the solution of the energy operator equation (called Dirac equation) suggests that the inclusion of spin leads to the existence of negative energy solution. I cannot comment further on this part though since I have been long since studying this subject.

What would be the equivalent for the electron due to its intrinsic angular momentum?

Electron spin does not have classical analogy, moreover it's not correct to view it as a consequence of electron undergoing a rotation around some axis.

 

[jonjacson]

If this electron roams the space at a non-relativistic velocity, its spin will contribute nothing to its energy. As for when it moves with relativistic-speed, if I am not mistaken, the solution of the energy operator equation (called Dirac equation) suggests that the inclusion of spin leads to the existence of negative energy solution. I cannot comment further on this part though since I have been long since studying this subject.

Electron spin does not have classical analogy, moreover it's not correct to view it as a consequence of electron undergoing a rotation around some axis.

-How is that possible? How is possible that a particle has angular momentum different from zero, but energy equal to 0? I don't get the point in non relativistic velocity.

-That is what I was looking for! I will search information about that.

-Yes I know, it was just an example to show what I meant.

Thanks for your replies!

 

[Jilang]

It depends where you put the zero!

 

[jonjacson]

It depends where you put the zero!

That looks a trick to avoid the question.

 

[blue_leaf77]

How is possible that a particle has angular momentum different from zero, but energy equal to 0?

What do you mean by zero energy, I thought I didn't say anything about that?

 

[jonjacson]

Well you said there is no contribution of the spin to the energy. Why not?

I mean Spin is an angular momentum, and momentum entails energy, Isn't that correct?

 

[blue_leaf77]

Well you said there is no contribution of the spin to the energy. Why not?

I mean Spin is an angular momentum, and momentum entails energy, Isn't that correct?

Spin has no contribution to the energy doesn't mean the energy has to be zero. Free particles still have kinetic energy.
In every case, spin is not the only angular momentum relevant in the system. Remember there is also orbital angular momentum. In the case of free particle, it can be shown that the wavefunction is expandable in spherical harmonics of the form $Y_{l0}(\theta)$.

 

[jonjacson]

Spin has no contribution to the energy doesn't mean the energy has to be zero. Free particles still have kinetic energy.
In every case, spin is not the only angular momentum relevant in the system. Remember there is also orbital angular momentum. In the case of free particle, it can be shown that the wavefunction is expandable in spherical harmonics of the form $Y_{l0}(\theta)$.

Well, maybe I didn't explain it well.

I only want to know the contribution of the Spin to the energy, neglegting everything else, so if you say that the contribution of the Spin to the energy is 0 I understand the energy due to the Spin is 0.

 

[blue_leaf77]

if you say that the contribution of the Spin to the energy is 0 I understand the energy due to the Spin is 0.

Yes, that's what I meant.

 

[jonjacson]

Yes, that's what I meant.

Is it possible to deduce this result from first principles? Or is there a demonstration in any book?

 

[vanhees71]

Sure, look for "Pauli equation" in the non-relativistic and "Dirac equation" in the relativistic context.

 

[jonjacson]

Well I will have a look at Pauli equation, thanks for your answers.

I know the answer for this question is going to be a big NO, haha but I have to ask.

Imagine that somehow you could be able to change the spin of a particle from 1 to 0, Could you extract some energy from that process?

 

[blue_leaf77]

Imagine that somehow you could be able to change the spin of a particle from 1 to 0, Could you extract some energy from that process?

If not impossible, that's very unlikely to happen because a particle is assigned with one spin. At least up to what we have observed, there is no way a particle can have two or more distinct spins. If two particles have different spin then they must be of different kind.