Exchange rate of virtual particles?

[arivero]

Suppose two particles bound by electromagnetic interaction, kind of hidrogen atom. How do you calculate the number of virtual photons exchanged by unit of time? Has this frequency of exchange some meaning in usual textbooks?

 

[arivero]

I think I can simplify my question. The original motivation comes from other thread, https://www.physicsforums.com/showthread.php?s=&threadid=13679, where a many-particle interaction is needed. But perhaps here I can reduce the question to a particle bound in a central potencial.

Giving such particle, orbiting in a given eigenstate of the hamiltonian, do we have an idea of how many quanta per unit time does this binded particle exchange?

 

[arivero]

Even more concrete

The question becomes clearer in the framework of Feynman perturbation theory. Let the particle be in the eigenstate $$\psi_n$$ of energy $$E_n$$. Let K be the free propagator, then we can develop perturbatively the propagator of the previus post, say $$K_{int}$$, using the path integral formalism.

$$e^{-iE_n t}=\int \psi^*_n(x_b) K_{int}(x_b,x_a,t) \psi_n(x_a) dx_a dx_b=$$
$$=\int \psi_n^* K \psi_n + \int \psi_n^* K V(x_1) K \psi_n + \int \psi_n^* K V(x_2) K V(x_1) K \psi_n +...$$
Each term in the sum is the contribution of 0, 1, 2, 3,... quanta exchanged between the particle and the potential. I should expect for a given time interval $$t$$, the main contribution to exp(-iEt) to come from only one diagram. Is it so? That should be then the expected number of quanta for a time interval t. Still, I can not find this estimate in the textbooks :-(

 

[jeff]

Originally posted by arivero
Suppose two particles bound by electromagnetic interaction..How do you calculate the number of virtual photons exchanged by unit of time?

In quantum theory transition rates are necessarily probabilistic. Modulo irrelevant details, we can say for example that the exchange of a single virtual particle by tree level processes like

  \              /
   \            /
    \.../
    /          \
   /            \
  /              \

will occur with a transition rate of probability P per unit time. The combined rate for all other processes would then be 1-P per unit time. Of course experimentalists usually report not the rate, but the rate per flux known as the cross-section.

Added comment: We can measure initial and final states, but not individual mediating virtual processes.

 

[TeV]

IMO,question about number of virtual photons exchanged per unit of time doesn't have sense for me if understood literarly.
Number of real photons emitted from the system as the consequence of electromagnetic interaction has more sense to ask about.
In formulation of QED theory virtual photons are imagined as carriers of EM force without any frequency of propagation specified* and equal sumation momentum could be realized through the contribution of various numbers of momentum mediators having various local field borrowed energies.Hence "rate per flux" turns to be fundamental startpoint.
_______
*Velocity of virtual photon turns to be infinite following the theory consequences.

 

[arivero]

Consider this loop

       ...
     .         .
------------------------
          \          \
           \          \
            \          \
             x          x

ie a self energy loop, via a virtual massive particle M, and two external potential legs so that the loop fits between. The massive particle M gives a scale of time. The question is for which parameters of the potential will the above loop become relevant. Intutitively it depends of another time scale, the "average time" between potential legs

 

[TeV]

I see what you ask.
The loop relevance has to do with the potential (ie. treshold in potential for given QM process you diagramed) but potential parameter of number of virtual quanta exchanged per unit of time would be figure tightly bounded to uncertainity becouse of the dynamics of two particles system.

Regards

 

[arivero]

Now I think we can be more precise. If the perturbative series for the propagator K(x,y,T) is asymptotic, as usually happens, we can call n to the term where the maximum precision is get; ie beyond n the series goes worse. Then the "time between interactions" can be defined to be T/n.

Then one could research if this quantity T/n is stable against variations of T for a fixed eigenstate propagation $$\int \int \psi^*_n(x) K(x,y,T) \psi_n (y)$$. I could expect it to be related to the De Broglie frequency for energy $$E_n$$

 

[jeff]

Look, what you're doing is nonsense. Consider an exchange rate Γ. Naively, Γ^-1 should represent the time it takes for an exchange between two particles satisfying given boundary conditions, say of their positions. The problem is that for this to be correct requires that these positions, and hence momenta, be assumed fixed over this time interval in violation of the uncertainty principle.

 

[arivero]

No jeff, consider that the exchanged particles are virtual, this is, there are off-shell. Thus its energy its not connected to its momentum.

If you still do not believe, check the previous message. For a given eigenstate of the energy and a given T, the value of n seems well defined to me, in terms of the precision of asymtotic series. Remember that even in classical mechanics general perturbation theory induces asymptotic series.

 

[jeff]

Originally posted by arivero
No jeff, consider that the exchanged particles are virtual, this is, there are off-shell. Thus its energy its not connected to its momentum.

I wasn't referring to the momenta of the virtual particles, but of the particles between which they're being exchanged.

 

[arivero]

Surely the original question, two particles, is more complex than the reduced one, particle exchanging with an external potential. Still I do not see why I should require determinacy on position and momentum. You are telling me that I require classical mechanics, I am showing you that I can do it with quantum mechanics equations.

Even classically, the exchange rate does not imply the exchange speed; assumming that only one particle can be "in process" at a given instant, the most you can say is that exchange rate gives a bound for exchange speed, or reciprocally. No determinacy involved between both concepts. One is the rate of emission of quanta, another is the speed of traveling of quanta.

In the quantum world it goes even better, for instance the quanta can be issued with a wavelength (thus position uncertainty) of the order of the distance between both particles.

Originally posted by jeff
Look, what you're doing is nonsense. Consider an exchange rate Γ. Naively, Γ^-1 should represent the time it takes for an exchange between two particles satisfying given boundary conditions, say of their positions. The problem is that for this to be correct requires that these positions, and hence momenta, be assumed fixed over this time interval in violation of the uncertainty principle.

 

[TeV]

There is also problem of strenght of local scalar potential and how is related to uncertainity ,quantum uncertainity not just math,which isn't linear depedence even for 2-dimensional non-relativistic quantum oscilator.
I.e. type of problem and relevance to koherent state* vs. "average number of virtual quanta exchanged" gotten by formalism of perturbation theory (virtual particles are not called called virtual for nothing.They are only part of mathematical convention helpful in describing behaviour of fermions in QM fields:coordinates (p1,x1)->(p2,x2)).
If one makes observation of system reducable to 2-dimensional quantum oscillator ,and uses kanonic transformation ,from inital state /0> onward than selfstates of hamiltonian H could be in principle obtained and we call them koherent states.
Koherent state is superposition of all the states /n> with definable number of quanta and probability of Poisson.
Thus,even if we neglect problems about sense of definition of
quanta exchanged (issues of locallity and source origin are always issues) and stick only to math problem there is a problem to find aproximate methods and solutions.Example:After introducing Green's functions or propagators for fermions ,and bosons, and specificaly the spectral functions and the conection between their complex poles and particle energies the problem becomes how to calculate the Green's function for the system.Apropriate approximation methods are needed:To develop a perturbation expansions for Green's function,method of Feynman diagram,define the particle self energy,derive Dyson equation or to use another approach:equation of motion method that usually requires a "decoupling" approximation,where higher order Green's function is replaced with (anti)symmetrized combination of lower order functions.The second method is suitable for solving the bilinear hamiltonian,which enables one to formulate QM description of irreversible phenomena,e.g. decay of a particle or state. Even though T/n figure introducing in particular problem case may have certain sense , I highly doubt it is necessary to reach conclusion on behaviour of the same.

 

[jeff]

Before I address the specifics of your post, let me just challenge you to find anywhere in the physics literature a calculation of the sort you're thinking about here. I can assure you that you won't since what your supposing is simply wrong.

Originally posted by arivero
Surely the original question, two particles, is more complex than the reduced one, particle exchanging with an external potential.

My remarks apply in an obvious way to this case as well. We would need to assume the position and momentum of this single particle remain fixed over the some finite period of time in violation of the uncertainty principle. The calculation of numbers of exchanges per unit time requires we simultaneously know the values of conjugate observables. Now, if one used a particle detector (like a geiger counter or something) that could be treated classically, we'd be in business. But the "particle detector" here, a quantum particle, is no good because of the constraints put on it by the uncertainty principle.

Originally posted by arivero
Even classically, the exchange rate does not imply the exchange speed

Where in my post was anything said or implied about exchange speed?

Originally posted by arivero
...I do not see why I should require determinacy on position and momentum...I am showing you that I can do it with quantum mechanics equations...quanta can be issued with...position uncertainty...of the order of the distance between both particles.

Under what conditions does quantum theory allow us to say the particles are at given positions at a certain distance from each other and how long we may assume they remain there? Focus on the meaning of what quantum theory allows us to measure: Can we calculate from probabilities per unit time numbers of exchanges per unit time? Is there some other observable that no one but you knows about from which such information can be obtained? What do we mean when we say that quantum theory is probabilistic?

 

[arivero]

Originally posted by jeff
Other evidence of basic defects in your physical understanding of quantum theory were on display in the Eigenfunction & Uncertainty thread in which you seemed completely unaware that since the particle state was an eigenstate of the hamiltonian, it's momentum was uniquely determined by the corresponding energy eigenvalue E_n = p²/2m up to sign. This should be cause for concern in you since this basic fact is something that undergraduates learn in the first few of hours of any introductory course in quantum mechanics. [/B]

This is almost personal insult. I refer the audiency to the mentioned thread,
https://www.physicsforums.com/showthread.php?s=&threadid=15428
so there anyone can check by himself the statements done. If you do not agree with the calculations in these thread, ie that for the given wavefunction we have <p> equal 0 and <E> greater than zero, I do not consider worthwhile to keep on this discussion. If on the contrary you agree, I will ask you to retire your previous statement.

 

[jeff]

Originally posted by arivero
This is almost personal insult.

How members react to my characteristic bluntness about physics depends on how closely tied the latter is to their egos. It just seems like your physics hasn't caught up to your math. But of course, I apologize for making you feel personally attacked. I was addressing only the physics, and for me that's a different thing than attacking one's character.

Originally posted by arivero
If you do not agree with the calculations in these thread, ie that for the given wavefunction we have <p> equal 0 and <E> greater than zero, I do not consider worthwhile to keep on this discussion.

Well, the good thing about this issue is that it's not a matter of opinion: At least one of us is wrong. I'm going to post the reason I feel the way I do in the other thread.

 

[arivero]

Originally posted by jeff
How members react to my characteristic bluntness about physics depends on how closely tied the latter is to their egos. It just seems like your physics hasn't caught up to your math. But of course, I apologize for making you feel personally attacked. I was addressing only the physics, and for me that's a different thing than attacking one's character.

Actually my ego was substituted by mathematical physics, time ago. So I expect you to apologize about the physics, no about my name (what is a name, after all?).

Turin has already made some interesting remarks on the other thread for you to notice.

 

[jeff]

Originally posted by arivero
Actually my ego was substituted by mathematical physics, time ago. So I expect you to apologize about the physics, no about my name (what is a name, after all?).

Turin has already made some interesting remarks on the other thread for you to notice.

You're quite right arrivero, I'm extremely embarassed and of course I offer you my deepest apology.

 

[arivero]

Thanks very much. To be honest, I'd remark that the question in the other thread is a very touchy one and it is easy to go wrong. Besides the issue of the integration interval or the support of the wavefunction, a lot of operators there are hermitian but not selfadjoint, and a lot of times it happens that the selfadjoint extension is not unique.

Going back with this, I will try to work out an example for one particle in one-dimensional harmonic oscillator; it should take me a couple of days and then we can see if it has some sense after all, or perhaps not.

 

[arivero]

I am back but no news :-(

I have calculated $$<\Psi|K(T)|\Psi>$$ and $$\int_t<\Psi|K(t)V K(T-t)|\Psi>$$ for the gausian wavepacket, with K the free propagator and V the potential of the harmonic oscilator whose vacuum is this wavepacket... but I am sorry to tell the analytical expresions are not very intuitive.

I supposse I need to study with more detail which expansions are asymptotical series and for which order do they approach more exactly to the right solution. A lot of study ahead, thus. We will see.

 

[TeV]

Nothing surprising.In most of the cases ,left alone convergence/divergence issues,the solution can't be obtained in a closed math form but often in infinite serie of nonelementar integrals.
Well chosen math methods for particular cases are must.I suspect,and my interesting speculation is,that very same method used could heavily fail in one case while doing quite well for another one.


As regards 1-dimensional driven quantum oscillator.This is probably the only case in theory which is nicely solvable.In strong enough fields results approach the the result gotten by classical Maxwell EM theory.Otherwise (weak fields),coherent states and their correspondent Poisson distributions are considered.

OK.,this is for one particle.How about two?Consider for example "intuitively" simple system of positronium:positron and electron orbiting each other in stable configuration (no radiation).
Feynman diagram is pretty elementar here.Consider potential,energy,and radius of the system known*.How would you define and calculate exchange rate of virtual quanta between potential and electron (or positron) here?Btw,I still don't understand why do you need that figure at all?

regards,
TeV

 

[arivero]

Originally posted by TeV

Btw,I still don't understand why do you need that figure at all?

Suppose the particle in the potential is also suffering a self-interaction loop via a massive particle. Usually you can absorb this self-interaction into the renormalised mass and charge, but if the diagram for potential interactions begins to compete with the self-interacting, life becomes more complex. The question is, when does this competition begins?

My guess if that if you can control the potential in a separate way, then you can use this effect to measure the mass of the self-interaction mediator.

 

[TeV]

You are trying to link these formal virtual particles exchanges to the tresholds of coupling ofVector bosons to the nucleus??
Well,that is a nasty complication I must say.I'm rather pesimistic.
A simple 2 particle EM system is a hard nut for the treatment in the same way as you can see.
Good luck in your efforts.

 

[arivero]

Well of course I got into this due to my previous quest with the nucleus. But the mechanism does not depend of having the particle trapped in a nuclear potential or in any other one which could be under experimental control. It could be an alternate, new, way to measure the mass of any boson contributing to a self interaction.

 

[TeV]

Originally posted by arivero
Well of course I got into this due to my previous quest with the nucleus. But the mechanism does not depend of having the particle trapped in a nuclear potential or in any other one which could be under experimental control.

The mechanism does not depend OK ,but nuclear potential is more complex math wise and possible situations more rich in physical phenomena than just a Columb potential QED.

 

[jeff]

Originally posted by arivero
Suppose the particle in the potential is also suffering a self-interaction loop via a massive particle.

It sounds like you just want to learn how to calculate corrections due to the presence of background external fields. Try this paper http://arxiv.org/abs/hep-th/0312304.

 

[arivero]

Er... it is a 129 pages PhD thesis! Still, I'll try.

 

[arivero]

Jeff, thanks. It does not help very much because it is about electromagnetism after all. Which could be of some help is a explicit calculation of the "mesonic radiative" corrections for a particle bound in a potential.

IE it seems I am looking for an expresion for the vertex correction

------...---------
        \       / 
         \     /
          \   /
           \ /
            X

where ... is massive with a mass lot greater than -------

The quest is if for some scale of the interaction X, this term would dominate over the self-interaction terms, ie over

     /\
    /  \
---...-----   ---
              \ /
               X