Does an electron have an internal structure?

[malawi_glenn]

we are discussing bound states, then you can't argue with a continum.

The proof is about 4pages, depending on where you want to start. There should be millons of websites presenting that proof. It uses ladder operators.

 

[JustinLevy]

we are discussing bound states, then you can't argue with a continum.

It was presented as if the commutator alone allowed proof of quantization. I felt uncomfortable with this, because it seemed like there were clear counter-examples.

With several assumptions added on, maybe it becomes possible.
Is there a way to prove that bound states of finite number of particles cannot have a phase space region allowing continuously valued states?


As for arguments against preons having non "multiple of half integer" values, I've only seen the spin-statistics theorem proof.

Anyway, thanks for pointing me in the right direction. This sounds really interesting as I have never heard of this quantization argument before.
I'll definitely read up on it tonight.

 

[malawi_glenn]

eh bound states with phase-space region?

Also in continum, consider your incoming particle against a target with impact paramter which you say that you can continuous vary. But then I just say that you decompose your incoming partile plane wave into spherical harmonics.. recall quantum mechanical scattering.

 

[JustinLevy]

eh bound states with phase-space region?

Sorry, I probably wasn't explaining myself very well.
Basically: Is there a proof that in a bound system, there does not exist a range of energy (or similarly for another observables) in which there exist a continuum of states?

Also in continum, consider your incoming particle against a target with impact paramter which you say that you can continuous vary. But then I just say that you decompose your incoming partile plane wave into spherical harmonics.. recall quantum mechanical scattering.

I don't consider such decomposition convincing. That sounds similar to arguing that since an irrational number can be represented as an infinite sum of rational numbers that it too must be a rational number... which of course it is not.


Regardless, I need to read that proof in Sakurai before I can discuss it intelligently with you. I'm sure (or at least hoping) that reading it will clear up any confusion I have on this.

 

[per.sundqvist]

"You are assuming in your argument two things:
- if the electon has inner structure, the components are identical
- all spin configurations will have the same energy"

Yes you are right, It was a very simplistic way to describe the "spin-problem". I know that S=3/2 is an excited state for protons, and also if you have 3 electrons in an external parabolic potential you normally have S=1/2 as the groundstate (but S=3/2 for strong confinement).

In fact this may be an argument for the possibility that we only can see the S=1/2 state for electrons, since excited states with higher spins have so much higher energy! (if it is composite). The difference in exchange, between different spinstates has to be worked out in the vicinity of a leptonic "glue" attractive force between the electron's components (which is heavily speculative).

 

[granpa]

http://en.wikipedia.org/wiki/Delta_baryon#List

so the spin 3/2 delta baryons and their ultimate decay products disregarding neutrinos and photons are:

Δ⁺⁺ > proton + positron
Δ⁺ > proton
Δ⁰ > proton + electron
Δ^- > proton + electron + electron

the first and last having no equivalent spin 1/2 particles.

 

[malawi_glenn]

Sorry, I probably wasn't explaining myself very well.
Basically: Is there a proof that in a bound system, there does not exist a range of energy (or similarly for another observables) in which there exist a continuum of states?


I don't consider such decomposition convincing. That sounds similar to arguing that since an irrational number can be represented as an infinite sum of rational numbers that it too must be a rational number... which of course it is not.


Regardless, I need to read that proof in Sakurai before I can discuss it intelligently with you. I'm sure (or at least hoping) that reading it will clear up any confusion I have on this.

Sorry but it is simple addition of spin in "spin space" which is a direct product space. Make an ansatz that the electron is made up of 2 spin 1/4 particles. The direct product spin space you can construct will have 1/2 or 0 as magnitute of spin. You CAN't form a continum of states in this direct product spin space.

Regarding continoum states in energy, look for instance at nuclei, all states that are above E = 0 but still below the potential edge are called continoum states, but those are unstable since the wavefunction can tunnel out of that region. But that is for energy, not spin.

 

[per.sundqvist]

Sorry but it is simple addition of spin in "spin space" which is a direct product space. Make an ansatz that the electron is made up of 2 spin 1/4 particles. The direct product spin space you can construct will have 1/2 or 0 as magnitute of spin. You CAN't form a continum of states in this direct product spin space.

Yes true, but consider you divide the electron into N (even number) particles, such that each spin is s=1/2N. Then Stot=0,2/(2N),...,1/2 (magnitude) And then you let N->infinity. In this limit the spinstates are like a continuum. For odd number M you would get 1/2M as the groundstate, but as M-> infinity this goes to zero. The ultimate z-projections would be (+/-) 1/2 for the continuum electron, but all values in between are allowed. But this is not right.

Its like a QW with large L, where the energies are quantized, but it looks like a continuum if L->infinity (you could use semi-classical theory with DOS etc. to describe it, if dE<<kB*T).

As for the quarks in the protons, it is the strong confinement energy that makes up the "total mass" of mp=1.7e-27. The rest-mass of the d and u quarks md and mu is about only 2% of mp. The size is 1e-15m roughly. Now consider an electron of size 1e-18m, would then not the restmass of the sub-particles of the electron be ridiculously small, i.e., mq<<<<me? The ratio e/m, which determines the Pauli spin would then not be easy to control... I don't even understand it for the quarks with qd=-1/3e and md=0.02mp -> S>>1/2 (S_electron). Have to check this from the Dirac equation, could be a relativistic factor here? ...

 

[per.sundqvist]

To the original question, I now think I must answer no! I have checked the Dirac equation a little and I made a simplified model which shows the absurdity of getting the small electron mass as a result of such extreme confinement, radius a<10^-18m.

Assuming one of the sub-particle having rest mass m0 and bound in a box-potential with V=0 for r<=a and infinity else one can derive the following from the Dirac equation:

-\frac{\hbar^2}{2m_0}\nabla^2\Psi=\lambda\Psi

\lambda=\frac{\hbar^2}{2m_0}\left(\frac{\pi}{a}\right)^2

E\equiv M_ec^2=m_0c^2\sqrt{1+\frac{2\lambda}{m_0c^2}}

Assuming simple addition of energies of N sub-electrons one can solve for m0 in terms of the total (known) electron mass Me, to get:

m_0=\sqrt{\left(\frac{M_e}{N}\right)^2-\left(\frac{\pi\hbar}{ac}\right)^2}

The last part in the sqrt could become large if radius a is small. Using the radius of a proton a=1e-15 one gets this term (which is a mass) to be=1.1e-27kg, which is close to the mass of the proton (surprisingly good for such a crude model!). The problem with the electron is that the value of a must be a<1e-18 (or even mush less, due to experimental observation). Even with a small number of N (number of sub particles) the sqrt becomes imaginary if

a_{el}\leq \frac{\pi\hbar}{M_e c}=1.2\cdot10^{-12}m

This value is much bigger than 1e-18 so the rest mass of the constituents of the electron must be imaginary! This seems to me to be physically absurd, why I conclude that this little "proof" (very rough) strongly indicates that the electron cannot be built up of sub-particle and that it therefore has no internal structure. Does it make sense?

 

[enotstrebor]

"You are assuming in your argument two things:
- if the electon has inner structure, the components are identical
- all spin configurations will have the same energy"

You are assuming much more.
-Your assuming the structure is a particle structure rather than some other structure (e.g. string like/type).
-Your assuming that the experimental point like radius (<10^18 m) is the size rather than the resultant effective size (e.g. gyroscopic objects react about the center point dispite the extent)

The mathematics (dirac spinor) and experiment actually suggest that there are two spin structures (the electron is mistakenly picture as having only a spin up and spin down state, but in actuallity the mathematics indicates four spin orientations). These two spin structures are spining at 90 degrees to each other (physically 90 degree orthagonal vs spinor mathematical 180 degree orthagonal).

Note that with two 90 degree spin sturctures (magnetic quadrupole) things like the stern-gerlach experiment now makes physical sense. The first magnetic field only orients one of the two orthagonal magnetic spins (up or down, 50/50 probability) with a while the second spin plane in unoriented. A second stern gerlach magnet field at 90 degrees to the first thus also result in a 50/50 probability of spin up or down because it is orientated to this second unoriented spin plane.

This thought experiment is not proof of structure, but...?

 

[ZapperZ]

You are assuming much more.
-Your assuming the structure is a particle structure rather than some other structure (e.g. string like/type).
-Your assuming that the experimental point like radius (<10^18 m) is the size rather than the resultant effective size (e.g. gyroscopic objects react about the center point dispite the extent)

The mathematics (dirac spinor) and experiment actually suggest that there are two spin structures (the electron is mistakenly picture as having only a spin up and spin down state, but in actuallity the mathematics indicates four spin orientations). These two spin structures are spining at 90 degrees to each other (physically 90 degree orthagonal vs spinor mathematical 180 degree orthagonal).

Note that with two 90 degree spin sturctures (magnetic quadrupole) things like the stern-gerlach experiment now makes physical sense. The first magnetic field only orients one of the two orthagonal magnetic spins (up or down, 50/50 probability) with a while the second spin plane in unoriented. A second stern gerlach magnet field at 90 degrees to the first thus also result in a 50/50 probability of spin up or down because it is orientated to this second unoriented spin plane.

This thought experiment is not proof of structure, but...?

Can you please provide citations where these have been published? And how would you reconcile this with all our observations of spin-triple and spin-singlet pairings, not to mention, results from every single electron paramagnetic resonance experiments?

Zz.

 

[malawi_glenn]

And what is the string composed of? How big is the string?

 

[JustinLevy]

Sorry but it is simple addition of spin in "spin space" which is a direct product space. Make an ansatz that the electron is made up of 2 spin 1/4 particles. The direct product spin space you can construct will have 1/2 or 0 as magnitute of spin. You CAN't form a continum of states in this direct product spin space.

I don't really agree with this logic since angular momentum need not be spin. In most multi-particle states, S_z is not even a good quantum number. What we want is J_z of this bound state.

I read the argument you mentioned in Sakurai that gives angular momentum quantization just from the commutator relations. It leaves much lacking for my tastes, as it essentially only shows that "measurement J_z" < Sqrt[ "measurement J^2" ] and that there is an operator which takes you from a state with Jz = n hbar to a state with Jz = (n+1) hbar (unless that would violate the previous inequality). That is very interesting (and I've seen similar algebra worked out), but I don't consider it a proof that given the commutator relations J^2 is quantized nor do I consider it a proof that there does not exist an operator which can increase Jz by a value less than integral hbar.

But this is getting off topic. I understand that angular momentum is quantized in many cases (just from the experience of working out problems), and I believe this is indeed the case under fairly general circumstances. If there is a proof showing this for all circumstances, I'd like to see it. Please answer in a new thread though. I don't want to pull this thread any further offtopic.

 

[JustinLevy]

The mathematics (dirac spinor) and experiment actually suggest that there are two spin structures (the electron is mistakenly picture as having only a spin up and spin down state, but in actuallity the mathematics indicates four spin orientations). These two spin structures are spining at 90 degrees to each other (physically 90 degree orthagonal vs spinor mathematical 180 degree orthagonal).

What!?
Are you thinking of right handed vs. left handed electrons?
If so, it does not mean that the electron has "two spin structures are spining at 90 degrees to each other".

Regardless, this sounds more like a personal theory of yours instead of something that can be traced to Dirac, for this is highly unconventional. I agree with ZapperZ, please cite a source that lead you to believe this.

To the original question, I now think I must answer no! I have checked the Dirac equation a little and I made a simplified model which shows the absurdity of getting the small electron mass as a result of such extreme confinement, radius a<10^-18m.

...

This seems to me to be physically absurd, why I conclude that this little "proof" (very rough) strongly indicates that the electron cannot be built up of sub-particle and that it therefore has no internal structure. Does it make sense?

I would have to disagree here. This argument does not allow you to rule out preon models (otherwise, indeed, all preon models would have been ruled out long ago).

The problem is that you assumed the particles are non-interacting in your calculation. If the effective potential is very negative within the bound state, the total energy and kinetic energy due to "confinement" can still work out fine.

The ratio e/m, which determines the Pauli spin would then not be easy to control... I don't even understand it for the quarks with qd=-1/3e and md=0.02mp -> S>>1/2 (S_electron). Have to check this from the Dirac equation, could be a relativistic factor here? ...

You've mentioned something similar before as well. Where are you getting this idea that spin is determined by e/m? If it were, since spin comes in quantized units, and charge comes in quantized units, so too would m. This is obviously not the case.

 

[granpa]

the angular momentum due to spin is S=√s(s+1) or S=s√(1+1/s)
s=1/2 or 3/2 or 5/2...once you know the angular momentum of the particle then you multiply by ge/2m to get the magnetic moment. g is the fudge factor. its 2 for electrons and 5.9 for protons.

orbital angular momentum L=√l(l+1) or L=l√(1+1/l)
l=0 or 1 or 2...

J=L+S

 

[JustinLevy]

Can the fact that g agrees with QED calculations which assume no internal structure for the electron, be used to reduce the length scale for any possible "internal structure" even further?

True, it is not direct probing, but maybe it still provides good contraints?
(What is the main motivation behind preon models anyway... it seems like they would have to be highly contrived. What are they striving for which would make such methods "worth it"?)

 

[malawi_glenn]

JustinLevy: of course you are not thinking it as a proof, you are not a master physicists as Sakurai ;-)

Preon models of the electron, tell me which classy institute of physics are dealing with such?

If the electron has substructure, it can not have substructure of the kind "point particles" described by todays physics. It must be something else, like strings.

 

[Orbb]

Out of line of your discussion, but concerning the topic:

Imaging spin as the internal angular momentum of a particle, isn't the spin of the electron an indication for its finite size? I know that in experiments as well as in current theory, the electron appears as a point particle without any spatial extent. However, the upper bound is some 10^-18m, so before reching the Planck scale, there are many orders of magnitude yet to be uncoverd (if ever possible). If one regards spin as an emergent phenomenon like angular momentom (from circular motion), an internal structure having some orientation in space would be a necessity to give rise to spin, which of course has such orientation.

 

[JustinLevy]

Out of line of your discussion, but concerning the topic:

Imaging spin as the internal angular momentum of a particle, isn't the spin of the electron an indication for its finite size?

No, it doesn't give any indication of that. If you start with the generators of rotation, there is a term which corresponds to spin and to (essentially) r x p. It is not necessary for spin to be described by a orbital angular momentum on a smaller scale.

JustinLevy: of course you are not thinking it as a proof, you are not a master physicists as Sakurai ;-)

Oh come on now, how does that help me learn anything?

If you are saying his algebra shows more than essentially
1] "measurement J_z" < Sqrt[ "measurement J^2" ]
2] there is an operator which takes you from a state with Jz = n hbar to a state with Jz = (n+1) hbar (unless that would violate the previous inequality)

and that this somehow allows one to prove
3] J^2 is quantized
and
4] there does not exist an operator which can increase Jz by a value less than integral hbar

then please do help me learn the missing pieces.
As I said before, I understand that angular momentum is quantized in many cases (just from the experience of working out problems), and I believe this is indeed the case under fairly general circumstances. However I am not sure if I can take this as a requirement in general. If there is a proof showing this for all circumstances, I'd like to see it.

Preon models of the electron, tell me which classy institute of physics are dealing with such?

A quick search in INSPEC shows that preon models do get published in decent journals. I don't really follow any preon modelling since there currently isn't any phenomenology from experiment to guide it (so it just seems like shots in the dark to me). Most physicists probably don't spend time on preon models for the same reason.

Oh, I guess I have read one recent attempt. One of Lee Smolin's post-docs tried to build up the standard model particles in spin-network theories of quantum gravity by using geometric braiding for "preons".

Anyway, I'm not trying to advocate any preon models. My point is merely that you cannot declare them ruled out. You can only use experiment to show there is no substructure down to 10^-18 m. I would have thought this would be easily agreed upon, but yet people keep using over simplified statements which seem to claim otherwise, such as even your last statement:
"If the electron has substructure, it can not have substructure of the kind "point particles" described by todays physics. It must be something else, like strings."

I don't think current experimental data allows you to make such a sweeping general "disproof" of anything non string-theory. Can we at least agree on that?


EDIT: Huh? Why was granpa banned?

 

[malawi_glenn]

But spin is not derived by assuming any internal structure. Point particles and spin works totally fine mathematically - math is language of physics - not intuitive reasoning. Maybe it is our concept of size which we have from our daily (classical mechanics) life which is unaccurate to describe elementary particles?

 

[malawi_glenn]

JustinLevy: pick up a textbook on quantum angular momenta. I think Edmonds is the best which I have tried. Quantum angular momentum is very well understood mathematically.

No spectrum has been obtained for the electron eiether, or anyone else of the leptons.

Of course I cannot disprove substructure of electron from experiment, but with todays limit, and the sucess of the Standard Model and rules of quantum angular momenta and the absence of excitation spectras also of the muon and the tau - makes a decomposition of the electron into smaller point-lime particles extremely hard to believe. There is not even a single sign of the electron or any other lepton to have such substructure. But you cannot, as you say, 100% rule things out.

So it was not a "proof" I gave you in "If the electron has substructure, it can not have substructure of the kind "point particles" described by todays physics. It must be something else, like strings." - it was my summary of what I believe is the current status of elementary particle physics.

Also imagine what these new substructure particles of the electron would cause? - well a new force, new particles etc. In general, this is not what physicists wants, but physicsists wants to describe and discover how nature is, so it is always a dilemma...

granpa has a temporary banning I was told

 

[per.sundqvist]

What!?
Are you thinking of right handed vs. left handed electrons?
If so, it does not mean that the electron has "two spin structures are spining at 90 degrees to each other".

Regardless, this sounds more like a personal theory of yours instead of something that can be traced to Dirac, for this is highly unconventional. I agree with ZapperZ, please cite a source that lead you to believe this.


I would have to disagree here. This argument does not allow you to rule out preon models (otherwise, indeed, all preon models would have been ruled out long ago).

The problem is that you assumed the particles are non-interacting in your calculation. If the effective potential is very negative within the bound state, the total energy and kinetic energy due to "confinement" can still work out fine.


You've mentioned something similar before as well. Where are you getting this idea that spin is determined by e/m? If it were, since spin comes in quantized units, and charge comes in quantized units, so too would m. This is obviously not the case.

I was not ignoring interaction, just assumed that the effective potential for one sup-particle (like the one you get from a Hartree model) was like a potential well. It is in fact (E-V)/2m0c^2 that makes the relativistic correction, so if you shift V down in case of some negative attractive potential, it should not matter to much.

We have of course no idea about what force that could bind together the leptonic sub-particles, so an effective hard-wall potential is fair enough. Anyway it can be shown that if you add a strong confining potential (like parabolic) to a Coulomb attraction potential, the eigenstate becomes more and more dependent only of the parabolic potential. Veff=-A/r+k*r^2, where ka^2>>A/a, where a is a typical distance from scaling theory. I had something like this in mind when I thought of the effective constant potential.

 

[per.sundqvist]

You are assuming much more.
-Your assuming the structure is a particle structure rather than some other structure (e.g. string like/type).
-Your assuming that the experimental point like radius (<10^18 m) is the size rather than the resultant effective size (e.g. gyroscopic objects react about the center point dispite the extent)

The mathematics (dirac spinor) and experiment actually suggest that there are two spin structures (the electron is mistakenly picture as having only a spin up and spin down state, but in actuallity the mathematics indicates four spin orientations). These two spin structures are spining at 90 degrees to each other (physically 90 degree orthagonal vs spinor mathematical 180 degree orthagonal).

Note that with two 90 degree spin sturctures (magnetic quadrupole) things like the stern-gerlach experiment now makes physical sense. The first magnetic field only orients one of the two orthagonal magnetic spins (up or down, 50/50 probability) with a while the second spin plane in unoriented. A second stern gerlach magnet field at 90 degrees to the first thus also result in a 50/50 probability of spin up or down because it is orientated to this second unoriented spin plane.

This thought experiment is not proof of structure, but...?

What is this argument about two magnetic fields? The first and the second? To my knowledge B=B1+B2 holds well, why the new vector B-field is only reoriented in another direction and this will NOT cause any mystic effect in the Stern-Gerlach experiment!

 

[per.sundqvist]

The mathematics (dirac spinor) and experiment actually suggest that there are two spin structures (the electron is mistakenly picture as having only a spin up and spin down state, but in actuallity the mathematics indicates four spin orientations). These two spin structures are spining at 90 degrees to each other (physically 90 degree orthagonal vs spinor mathematical 180 degree orthagonal).

Hmm, do you talk about the two lower spinors here? In the time-independent solution all physics is obtained exactly by the two upper (space-like, Phi) and the lower two (momentum-like, Chi) spinors are always given in terms of the upper ones

<br /> \chi &amp;=&amp;<br /> \frac{\kappa(\vec{r})}{2m_0c}\left(-i\hbar\nabla-e\vec{A}\right)\cdot\tilde{\vec{\sigma}}\phi,<br />
<br /> \kappa(\vec{r}) &amp;=&amp; \frac{1}{1+\frac{E-V(\vec{r})}{2m_0c^2}},<br />

Parhaps I missunderstood the B-field as well. Maby you was talking about an arrangement like in the first pages of Sakurai, with the discussion of spin polarization? In that case the four spinor (the lower two "chi") has nothing to do with it.

Also, a satisfactory many-particle Dirac equation has not yet been presented, to really understand the meaning of "spinor" in that case (some presented models gives unphysical and singular solutions). Have tried to formulate this by my self a time ago but got stuck because of to many degrees of freedom. The thing I use now is to set up the spinor with 2^n elements like in a binary table (n=number of electrons). (up-down, up-up,... etc for two electrons).

 

[enotstrebor]

Can you please provide citations where these have been published? And how would you reconcile this with all our observations of spin-triple and spin-singlet pairings, not to mention, results from every single electron paramagnetic resonance experiments?

Zz.

I thought that the ``Dirac electron'' (spin 1/2 particle) was mathematically a 720 degree particle requiring four spin flips (a spin flip is orthogonal but mathematically is 180 degree) was common knowledge (SEE http://en.wikipedia.org/wiki/Spin-%C2%BD" subtopic ``SYMMETRY'')

The (four) spinor also has positive and negative helicities. (See R. Penrose, {\it ``The Road to Reality'' and associated zigzag picture of electron},

As this mathematical aspect gives the correct results, one might suggest that it reflects the true physics but to be physical physics the orthogonality would be two (one positive and one negative helicity) spin planes at 90 degrees.

As indicated this physical view of the mathematics makes the Stern-Gerlach experimental results physically sensible.

And how would you reconcile this with all our observations of spin-triple and spin-singlet pairings,...?
Zz.

The zero spin state has both spin axes alighned the other triplet has one aligned and one anti-aligned. One can not have a stable state where both are anti-aligned.

..., not to mention, results from every single electron paramagnetic resonance experiments?
Zz.

Flipping between the two orthogonal spin states is what the mathematics says. This has been given the mathimatical picture of spin up and down which are mathmatically 180 degree orthagonal. The only thing that changes is that the physical picture is physically orthagonal and the spin flip is from one of the two spin planes to the other orthogonal spin planes.

(Note that according to the mathematics a 360 degree rotation is a change in phase not a 180 degree flip. The electron flips 180 degrees from a spin up (right helicity arrow up) to a spin down (left helicity arrow up) then flips a second 180 degrees to change phase, i.e spin up (right helicity arrow down), then flips a third 180 degrees (same phase), i.e spin down (left helicity arrow down), then flips a fourth time to return to the same state. i.e. spin up (right helicity arrow up)

And how would you reconcile this
Zz.

This dual spin picture of the electron is in keeping with all experimental evidence that I am aware of and in keeping with the mathematics.

It simply a new picture and it makes all experiments sensible.