Posted on 10/25/2004 1:46:25 AM PDT by accipter
CONSIDER a verbal description of the effect of gravity: drop a ball, and it will fall.
That is a true enough fact, but fuzzy in the way that frustrates scientists. How fast does the ball fall? Does it fall at constant rate, or accelerate? Would a heavier ball fall faster? More words, more sentences could provide details, swelling into an unwieldy yet still incomplete paragraph.
The wonder of mathematics is that it captures precisely in a few symbols what can only be described clumsily with many words. Those symbols, strung together in meaningful order, make equations - which in turn constitute the world's most concise and reliable body of knowledge. And so it is that physics offers a very simple equation for calculating the speed of a falling ball.
Readers of Physics World magazine recently were asked an interesting question: Which equations are the greatest?
Dr. Robert P. Crease, a professor of philosophy at the State University of New York at Stony Brook and a historian at Brookhaven National Laboratory, posed the question in his Critical Point column and received 120 responses, nominating 50 different equations. Some were nominated for the sheer beauty of their simplicity, some for the breadth of knowledge they capture, others for historical importance. In general, Dr. Crease said, a great equation "reshapes perception of the universe."
(Excerpt) Read more at nytimes.com ...
In your experimental arrangement, you have electrons always moving in the direction of propagation. If that were the case in Compton's arrangement, all of the electron's would be gone and a huge charge left on the block. Additionally, some of those electrons in Compton's block have significant v's in the other direction. No blue shifted frequencies were noted. So I don't think replacing alpha with Compton's factor and introducing h in eq 19 does anything but to show that in your particular arrangement, the same result is found. Your analysis doesn't apply to Compton's arrangement.
Re: the disscussion section, par. 2, Compton scattering, the momentum transfer to free electron's is always observed and is understood as a momentum transfer. A continuous plane wave would be expected to simply oscillate the electron. If the envelope of a plane wave is not sinuisodal, other frequencies are present, so depending on the shape, any momentum could be given.
I'd have to look at it more to comment further.
Math lies in the minds of sentient beings. Without them, it doesn't exist. One reason it was created, was to know and understand the world in a logical way.
Excellent taunt!
I'm not sure that statement makes sense. It seems to presuppose the existence of the rules of logic, independent of the minds of sentient beings. Why should the principles of mathematics be different from the rules of logic?
Or, looking at it from the other direction, if we accept your statement as correct, how can we claim that the rules of logic are any different from those of mathematics -- and therefore non-existent except within the minds of sentient beings?
There are some fairly obvious difficulties with the latter interpretation, such as: does a statement like "A is A" really make sense if it is nothing more than a construct of sentience? In his Pensees Descartes pretty much bogged down on this point -- he couldn't really get past "I think, therefore I am."
It seems more likely that the former interpretation is better: that rules of mathematics, like those of logic, are in some sense independent of sentient minds. But that, too, has problems -- what are "rules of logic?" They don't seems to be "physical" in any normal sense of the word -- so does it make sense to apply "A is A" to them?
I prefer my geometry when it is limited to three dimensions, and am most happy when it is limited to two.
I would certainly hope that your egghead mind is being put to constructive use in New Mexico. You wouldn't happen to be at Los Alamos slaving away on bunker busting nukes, would you?
By the way, stochastically speaking, what's your opinion on maximum entropy inference?
One of the laws of nature is that a = a. What I have written is a representation of the fact that physical objects always act the same and according to their nature and whatever properties a physical object has, it will always have. Identity is inherent in Nature. It is not within the minds of sentient beings that, either hold mistakes, or falsehoods, for whatever reason they have. Note, nonsentient physical objects don't reason, justify, nor require justification, they always act according to a = a.
From there the rules of logic and mathematics are representations and method that allow rational beings to comprehend and know Nature and to relate to other beings. W/o the beings their are no representations, nor methods, only the unwritten laws of physics contained within the nonrational, nonsentient objects that always behave as a = a. ie. an electron will always be and act like an electron for as long as it is an electron. If it's annihilated, it will act according to how electrons behave when faced with the appropriate annihilators and physics. The converse applies when they are created.
Only beings refuse to act according to a = a. They are either mistaken, or deliberately dishonest according to their own reasons(laws), that they made up.
"I think, therefore I am."
He was honest. He perceived and knew himself and created this logical statement . By the same representations and methods, he could conclude that other things exist also.
They are written down in various places. Yes, a=a applies to them also. It means they exist and are what they are, regardless if anyone in particular does not know what they are.
They are written down in various places. Yes, a=a applies to them also. It means they exist and are what they are, regardless if anyone in particular does not know what they are.
Express the statement "Without you, I am nothing" mathematically.
ME - SHE = 0
ME = SHE
M = SH (division by E)
Now multiply both sides by IT...
Electrons in the Compton experiment are considered effectively free, not bound. The shift goes from blue to red in my paper, just as in the QM interpretation.
In your experimental arrangement, you have electrons always moving in the direction of propagation. If that were the case in Compton's arrangement, all of the electron's would be gone and a huge charge left on the block.
Nope. X-Rays are short pulses of intense sinusoidal em waves. In a femtosecond the electrons can't move very far.
Additionally, some of those electrons in Compton's block have significant v's in the other direction. No blue shifted frequencies were noted. So I don't think replacing alpha with Compton's factor and introducing h in eq 19 does anything but to show that in your particular arrangement, the same result is found. Your analysis doesn't apply to Compton's arrangement.
Compton's experimental result shows scattered frequencies in all directions, too. The particular case he analyzed in the equations is the one I do, too -- that of an electron initially at rest. Look at his spectral plates. There's a peak at the shifted frequency at a given angle but its a curve, not a spike.
Re: the disscussion section, par. 2, Compton scattering, the momentum transfer to free electron's is always observed and is understood as a momentum transfer. A continuous plane wave would be expected to simply oscillate the electron. If the envelope of a plane wave is not sinusoidal, other frequencies are present, so depending on the shape, any momentum could be given.
No, an em wave that's limited in time (eg. an em wave with a gaussian envelope) **cannot** transfer net momentum to an electron. Once the wave has passed by, the electron returns to its original momentum. If it was originally at rest, it returns to rest once the wave has passed by.
In addition, in a continuous, circularly polarized plane em wave, the electron is accelerated in the direction of propagation by v x B until it becomes synchronized with the wave, then downstream acceleration ceases and the electron has reached its "terminal velocity" (for lack of a better phrase). It follows a helical path with the axis in the direction of propagation. That terminal velocity is beta*c in my derivation. In reaching this terminal velocity, the wave will have transferred a photon's worth of energy and momentum.
The theory works. The math is complete within its domain. Take a look at the coherent state in QM. Compare it to the results of my paper (the electron traces out a gaussian helical path). This isn't a coincidence.
Not slaving away on bunker busters.
Maximum entropy inference isn't too bad. It's really good in "natural" or "physic" surroundings where the "system" tries to become a Boltzmann distribution. One can generate counter-examples though. Some of the advocates treat it with what I consider misdirected ardor though. It's just another tool.
"I have not thunk, so I can not am."
Depends on what the meaning of "am" be.
Sounds like an explanation for where the "missing" dollar went in the "problem" that was batted around earlier on the thread....
;-)
A classic. Like the one about the commuter who began to fear he was on the wrong train. Then he recognized Einstein sitting across the aisle. "Excuse me, professor," he asked, "does New York stop at this train?"
I have Compton's results in front of me at phi= 0,45,90 and 135o. Incident wvlen = 0.7ang. The scattered rad is all redshifted to longer wavelengths, there is no higher frequencies. At 135o the peak lies at ~0.745ang.
"X-Rays are short pulses of intense sinusoidal em waves. In a femtosecond the electrons can't move very far."
Compton used a monochomatic beam applied for a significant time. You are trying to use wave mechanics alone, yet are appealing to a particle concept. ie. the photon, the particle that arises out of the EM field.
"Compton's experimental result shows scattered frequencies in all directions, too. The particular case he analyzed in the equations is the one I do, too -- that of an electron initially at rest. Look at his spectral plates. There's a peak at the shifted frequency at a given angle but its a curve, not a spike.
Here's a page with some data. It's the same type I have. The photon always loses momentum to the electron. The process is momentum transfer. The same experiment could be done with an x-ray beam and a slow neutron source. That's why all these experiments show a redshift, never blue.
"No, an em wave that's limited in time (eg. an em wave with a gaussian envelope) **cannot** transfer net momentum to an electron. "
The shape of the amplitude envelope gives info only on the frequencies and phases present. The em field gives rise to particles called photons. They are particles of zero spin that obey Bose-Eistien statistics. If the field is large enough, there will be a fixed value for the density of these particles. If the field is below a certain value, the number of particles is uncertian. Regardless of the envelope shape, as long as there is a field, there will be some photons. Any photon impacting a particle will transfer momentum to that particle and be redshifted. Incident waveshape doesn't matter. Conservation of momentum applies and that is how Compton originallty analyzed and explained the process and demonstrated w/o doubt the existence of photons. One part of the history is the discovery of the particle properties of light. Like the photoelectric effect, there is no classical EM wave solution.
LOL!
Yes, my mistake. I was thinking about the emitted radiation from the electron. In front, it is blue shifted but of course the incident radiation was redshifted, net result is (1-cos theta) or no forward shift.
Compton used a monochomatic beam applied for a significant time. You are trying to use wave mechanics alone, yet are appealing to a particle concept. ie. the photon, the particle that arises out of the EM field.
A classical em waveform can be any shape: sinusoidal, triangular, square, etc. It can have any amplitude envelope. The computer you typed your message on uses nominally square em waveforms. Note: a square wave is *not* the sum of a bunch of sine waves. It can analyzed as such mathematically but it isn't physically.
Here's a page with some data. It's the same type I have. The photon always loses momentum to the electron.
Only in the case of an electron initially at rest. What happens when an electron is moving towards a photon before interaction? What is the energy/frequency of the emitted photon as a function of the emission angle (angle relative to the electron's motion)? Hint: search for "inverse Compton Effect".
Like the photoelectric effect, there is no classical EM wave solution.
My paper (I'm a crackpot) and the others from respectable physicists show otherwise with regards to the Compton Effect. You can analyze most of the effect via classical em and relativity as I have done. In time, all of the Compton effect will be explained in this way.
Going out to watch the eclipse for a while.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.