wave packets

[Norman Albers]

I have finally solved the exact result of the Fourier transform of the wave packet I study in my paper on photon localization (cache listed below). I represent a Gaussian form in three dimensions of falloff over "many frequency cycles" by starting with vector potential in<y,z>: [math]A_y= A_0 cos (k_0 X) e^{-a^2(X^2+y^2+z^2)} [/math] where X=x-ct, with the assumption that [math]k_0>>a[/math]. This means the falloff of envelope is of many cycles. The dimensionless constant a/k is presumed to be the fine structure constant. The difficult part of the Fourier transform is the integration over x, since it contained in both terms (X is the co-moving center). I speak now only of the form of the [math]k_x[/math] terms in the expression of the transform, since the other integrations are simple. Leaving out the coefficients and a page of development, the essence is: [math] \int_{-\infty} ^\infty dX cos[(k_x-k_0)X] e^{-a^2 X^2} =\frac{ \sqrt \pi}{2a} e^{-(k_x-k_0)^2/4a^2} [/math]. This can be seen as a generalized delta function, since in the limit of small a it becomes a true delta function. This can be clearly seen if you go back to the original expression which has in the integrand: [math] cos{(k_0 X)} cos{(k_x X)} [/math]. I would appreciate comments from people who are familiar with wave packet construction in quantum theory. I was taught a similar idea starting from the idea of a 'momentum spread'. We can see here, the width of the frequency spike is where [math] \Delta k /k = \alpha [/math]; here the argument of the frequency comb is rapidly decreasing the term.

Edited June 28, 2008 by Norman Albers
further...

[KALSTER]

presumed to be the fine structure constant

Does this imply the involvement of some kind of medium?

 

Are you describing photons as solitons, maybe topological solitons?

 

Could this be extrapolated to describe particles?

 

If my questions don't make sense or aren't even pertinent, it is because I am a layman that doesn't know what he is talking about. "No, no and stay out of this thread, idiot" would then be an acceptable response.

[Norman Albers]

We have for a hundred and fifty years or so worked with the idea of the vacuum having polarizability: [math]\epsilon_0[/math] is the 'electric permittivity of the vacuum'. Ideas about an aether medium failed because they were not Lorentz-transformable. This means that physics is the same to different observers in relative motion, so this 'medium' is more subtle than any measurable mass or energy. The term I like to use is dipole availability, and if there are wave packets such as I analyze, it implies the vacuum is responding like a superconductor. Where there is a changing magnetic field (dA/dt) there is a displacement current and resultant charge densities, and these I describe as a sheath. Our classical theory of light allows no buildups of such charge density in the 'vacuum' and so it fails to describe anything but spreading wavefronts or infinite plane waves. If you diagram electric and magnetic fields in plane waves from Maxwell's equations, consider the displacement current identifiable as dE/dt. Where does it go? . . . . . For a possibly useful metaphor, what can a fish know of water? If it is not a leaping fish or a bottom feeder, it does not know or even exist in the absence of water. Surely fish know all relative changes in water: temperature, salinity, etc. I see the vacuum medium like this; it is the warp on which light and mass are woven. To your question about massive particles, Kalster, Google on 'wave packet' and you'll see plenty of such discussion. I'm not yet conversant there, however, which is why I seek exchange with other folks.

Edited July 3, 2008 by Norman Albers

[KALSTER]

Well, thanks a lot for that description. I have my own little hypothesis (devoid of math) that involves the medium behaving like a superfluid, with the constituent "particles" being sub-Planck lengthed to get rid of the base reference frame problem. I apparently also have wave packets in it mostly as representing photons (I have been calling them wave bundles though, more complex ones being particles). Yours looks much more detailed and thorough though. Good luck to you.

Edited July 3, 2008 by KALSTER

[Norman Albers]

I have a colleague, on this forum called 'solidspin', who just this week gave me a very positive response when I asked if we could possibly sense an alternating charge density sheath by getting interaction with an attosecond laser in the field of a lower frequency photon. He thinks this is reasonable since these sorts of things are being done in his area of physical chemistry. He said they are looking at molecules in their transition states, which is pretty exciting, no pun. As far as your thinking and mine, the concepts of superconductors and also of Lorentz invariance are important. My colleague also made a comment I intend to pursue with him, that there is physics of interest at the Planck length...

Edited July 4, 2008 by Norman Albers

[KALSTER]

I have been having a lot of fun with my theory trying to judge its qualitative compatibility with observation and current theory (lacking the maths required for quantitative explorations) as I expand on the emergences of the premise. I am glad to see someone doing serious research on a somewhat similar model. I will watch your progress with interest.

[Norman Albers]

We have brilliantly jumped through hoops wringing quantum electrodynamics from homogeneous field theory. When you read a text on QED, you are taken through the Fourier transform of the system of Maxwell's equations. In k-space, many things are simplified. Differentiations become multiplications, divergence becomes a vector product with the k being considered as independent variable. We find that a linear combination gives "normal modes" which are essential reductions. Then comes the weirdest page in the book, where we are taken over into quantum mechanic formalism and constructions which have the same forms as the electrodynamic expressions. I am not being antagonistic here. I am saying there are things unfinished. I work with a right-hand-side source current, where we usually put zero, in the normal mode equations.

Edited July 4, 2008 by Norman Albers

[Jacques]

Hi Norman

I found that little article that is related to your work.

How Long Is a Photon?

Hope it help...

[Norman Albers]

Jacques this is a very beautiful article. I shall digest what it offers and talk later. For the last two days I have been trying to understand what I might or might not be able to analyze if the falloff of my envelope is comparable to the characteristic wavelength. BINGO! NORM

Edited July 8, 2008 by Norman Albers

[scalbers]

Interesting support to the idea of a photon pulse in the above article. I wonder if the experimental creation of single photons bears any relationship?

 

http://www.physorg.com/news127049722.html

[Norman Albers]

My colleague solidspin described pulses of 'a few hundred attoseconds' duration, so those described here last 65 attosecs. Were this the time for one cycle it would be 20 micron waves. My impression is that these are higher frequency sources on free electron lasers at accelerators. Is this correct here? In the paper from Jacques my model is depicted in Fig. 2.

[scalbers]

I think the article I posted indicates 65 femtoseconds, and if that's the case the wave train would be 1000 times longer. It's unclear to me if they think this is the actual size of the photon, and why they say it is shorter than other single photon pulses. How could photons of presumably similar wavelengths have different "physical" lengths? Or is this the wave-particle duality again rearing its head?

 

Here are more details:

 

Single-photon ultrashort light pulse

 

1 May 2008

 

P.J. Mosley (Oxford University) and his colleagures obtained a record-short light pulse consisting of a single photon. Photons in pure quantum states were obtained by parametric conversion, i.e. by splitting photons in a nonlinear birefringent crystal into pairs of photons at doubled wavelength. The high state purity (more than ≈ 95%) in pairs of photons was achieved by a special choice of dispersion properties of the crystal, angle of incidence of the beam and light wavelength, so that the group velocity of the initial photon was equal to the group velocity of the photons produced by splitting the initial photon. It was therefore possible to eliminate quantum correlations between photons of the pair. The wave corresponding to the photons obtained was 65 fs long which is shorter by a factor of 15 than record-short photons generated previously. In fact even shorter pulses were reported earlier (see Physics Uspekhi 48 254 (2005)) but only in wave packets consisting of many photons. Source: Phys. Rev. Lett. 100 133601 (2008)

 

http://www.ufn.ru/en/news/

Edited July 12, 2008 by scalbers

[Norman Albers]

Yes, pardon me. This is different, being single photon production. The first questioner below asked also, what's the momentum, or frequency? In my modeling, I can allow the envelope decay parameter, 'a', to approach the propagation vector 'k'. As long as a/k<1, I can still generate an expansion series in the integration of dU/dt. It looks like the sheath does not lose its character. There is still an inner and outer, oppositely charged, double helix. One might imagine it as two opposite e-p dipole pairs manifesting from the vacuum on either side of center.

Edited July 12, 2008 by Norman Albers

[scalbers]

You might also pardon my confusion as this article mentions both a 20 micron pulse and 65 femtoseconds.

 

http://www.universetoday.com/2008/04/11/shortest-single-photon-pulse-generated-implications-for-quantum-communications/

 

Scientists at Oxford University have developed a method to generate the shortest ever single-photon pulse by removing the interference of quantum entanglement. So how big are these tiny record-breakers? They are 20 microns long (or 0.00002 metres), with a period of 65 femtoseconds (65 millionths of a billionth of a second). This experiment smashes the previous record for the shortest single-photon pulse; the Oxford photon is 50 times shorter. While this sounds pretty cool, what is all the fuss about? How can these tiny electromagnetic wave-particles be of any use? In two words: quantum computing. And in an additional three words: quantum satellite communications…

 

Quantum entanglement is a tough situation to put into words. In a nutshell: If a photon is absorbed by a type of material, two photons may be re-emitted. These two photons are of a lower energy than the original photon, but they are emitted from the same source and therefore entangled. This entangled pair is inextricably linked; regardless of the distance they are separated. Should the quantum state of one be changed, the other will experience that change. In theory, no matter how far away these photons are separated, the quantum change of one will communicated to the other instantly. Einstein called this quantum phenomenon "spooky action at a distance" and didn't believe it possible, but experiment has proven otherwise.

 

The Oxford University experiment

 

So, in a recent publication, the Oxford group are trying to remove the entangled state of photons, this experiment isn't about using this "spooky action", it is to get rid of it. This is to remove the interference caused when one of the photon pair is detected. Once one of the twins is detected, the quantum state of the other is altered, contaminating the signal. If this effect can be removed, very short-period "pure" photons can be generated, heralding a new phase of quantum computing. If scientists have very definite, identical single photons at their disposal, highly accurate information can be carried with no interference from the quirky nature of quantum physics.

 

"Our technique minimises the effects of this entanglement, enabling us to prepare single photons that are extremely consistent and, to our knowledge, have the shortest duration of any photon ever generated. Not only is this a fascinating insight into fundamental physics but the precise timing and consistent attributes of these photons also makes them perfect for building photonic quantum logic gates and conducting experiments requiring large numbers of single photons." - Peter Mosley, Co-Investigator, Oxford University.

[Norman Albers]

Arrghh, my decimals were also off! Now that we all know a femtosecond is E-15 sec., and an attosecond is E-18 sec., the pulse described multiplies to speed of light. What, however, is the "period"? It is the length of the pulse, and is the pulse one wavelength?

[scalbers]

That also is my main question, what is the relationship of the pulse length to the photon length? This might be different from the wavelength (which in turn is the inverse frequency - perhaps in the ranges for visible light). And for fun we can contemplate the yoctosecond...

 

http://en.wikipedia.org/wiki/Orders_of_magnitude_(time)

[Norman Albers]

We're talking septillionths here. [sorry, you out-Greeked me.] I hear the free electron lasers give pulses of a 'few hundred' attosecs. This corresponds to a light-length of 0.06 microns. Now I need to find out the frequency here, also.

Edited July 12, 2008 by Norman Albers
multiple post merged

[scalbers]

On another note that speaks to the length of a photon, you might consider this variation I haven't considered before. How does a single photon know the thickness of the glass is that it reflects off of?

 

http://quantumweird.wordpress.com/category/photon-clock/

 

One of the weird aspects of photons involves reflection from glass of varying thickness. Send a laser pointer beam perpendicular to a pane of glass and about 4% of it will reflect back, on average, but, by carefully selecting glass of various thicknesses, the reflections vary from 0% to 16%. Glass a foot thick can be slightly adjusted in thickness to not reflect at all! All the light goes into the glass - perfect transmission. QED easily shows how this works for light beams. Rays from the back of the glass interfere with the rays coming in the front so as to cancel the reflection if the wavelength is a multiple of ½ wavelength.

 

However, the cancellation at ½ wavelength also works for individual photons for thick glass, and there seems to be no answer other than “quantum weirdness”. How does an individual photon know how thick the glass is the instant it hits the front surface when the back surface is thousands of wavelengths away? The reflected photon would be six feet away before a copy could make a round trip through a foot thick piece of glass. (Two feet round trip at 1/3 speed of light in air)

 

............

 

This site seems to be pretty good with lots of thought experiments. Here is one on radio length photons:

 

http://quantumweird.wordpress.com/2007/07/23/thought-experiment-photons-at-radio-frequencies/

Edited July 12, 2008 by scalbers

[Norman Albers]

Do you know the reflection happens without delay?

[scalbers]

I'm assuming the delay is much less than what one would guess from the thickness of the glass and that this is why it is considered "weird". It's hard to find any clear statement on this except that Feynmann has lectured on this topic of partial reflection in thick glass. Still this might apparently open the prospect of faster than light communication if you can shave the glass a bit to alter the reflection probability on one side - then how fast could we see the probability be altered on the other side?

 

This sort of reminds me of the question of how a single photon can know the shape of an entire large telescope mirror to know that it has to reach its focal point diffraction spot within a certain probability. At least in this case all the path lengths of travel are equal an there wouldn't be any differential delay.

Edited July 12, 2008 by scalbers

[Norman Albers]

Does the photon have to 'sense' the entire mirror? Don't we build mirrors of many square meters to catch one here, one there?

[scalbers]

I think each photon (or at least the "pre-collapsed" wave function if there is one) would have the know the shape of the whole mirror. This provides the ability for diffraction theory to do its thing and predict that a larger mirror will result in a more tightly focused and specifically sized star image, even when individual photons are coming in at a slow pace.

[Norman Albers]

So each photon has less diffraction, and it's not just a matter of signal to noise from more photons at distinct reflection points? I thought that in double slit experiments the two slits had to be within a few wavelengths to get strong interference fringes.

[scalbers]

I think as corroborating information I can mention that the diffraction pattern comprising the star image at the focal point consists of the Airy disk (center spot) surrounded by a series of rings (of an interference nature). The size of this disk in radians is close to the number of wavelengths in the mirror diameter.

 

As for distinct reflection points this might be impractical since the shape of the mirror at small scales might have imperfections that could produce local slopes that throw off the photon reflection from the precise angle. Yet even with such a locally rough mirror you can (I think) still get most of the light to be focused in that small diffraction spot (or at least fall within that well defined pattern of ring sizes), as dictated once again by the overall mirror size/shape.

 

Perhaps more to the point visible light interference fringes can be seen with interferometers with components separated by a number of meters, and of course with radio telescopes separated by thousands of miles.

 

It's true though that I'm unable to be confident that all of these instruments have been tested with "single photon" flux rates, if one can clearly define this.

Edited July 12, 2008 by scalbers

[Norman Albers]

It is good if a little bird poop does not throw off the usefulness of the device. We dutifully integrate over all possible paths.

 

One photon has no fringes. The pattern is a buildup, yah?

Edited July 12, 2008 by Norman Albers