Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Mr Andrew on 25/10/2007 02:06:32
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How does planck's law require the absorbtion of light to be quantized? Planck's constant is simply a proportionality constant. A constant multiplied by a continuously variable quantity gives a continuously variable quantity. Where does quantization come in? Can someone please explain this one to me?
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The quantisation comes in the integer solutions to the wave equation.
If you didn't only allow integer solutions, the electron, because it is under acceleration in its orbit around the atom, would be constantly radiating energy and it would spiral into the nucleus. This, clearly, doesn't happen, so this aspect of the quantum theory is justified.
I was well chuffed when I was told this at Uni, all those years ago.
As you say, the Planck constant is just the constant of proportionality which relates the energy of a photon to the frequency.
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But I'm talking about where Planck got this idea in the first place. His law for black body radiation is where he first used his constant and in that equation, the intensity of light absorbed is a function of the frequencies absorbed. It is proportional by h (to the cube of nu or something like that...I don't remember). But frequencies can be absorbed at any wavelength (especially for black bodies) so putting a continuous function into an equation, no matter how many constants are in it, gives only another continuous function. So I still don't know how you can have "integer solutions" to an equation if you are only putting in continuous variables as input. You have to have discrete input to have discrete output.
-Note- I understand how quantization evolves from the Schroedinger Equation, but how did Planck come up with it the first time, without wave quantum mechanics?
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How does planck's law require the absorbtion of light to be quantized? Planck's constant is simply a proportionality constant. A constant multiplied by a continuously variable quantity gives a continuously variable quantity. Where does quantization come in? Can someone please explain this one to me?
If you talk about Planck's law I assume you talk about the black body's emission:
I(ν,T) = (2hν3/c2) / [e(hν/kT) - 1]
I(ν,T) = intensity of light emitted at temp. T between the frequencies ν and ν + dν
k = Boltzmann's constant.
"Absorption of light must be quantized" means:
given an oscillator at frequency ν, it cannot, as assumed in classical physics, have any arbitrary value of energy continuously between 0 and +oo but only an integer value of times a fundamental one:
E = E0, 2E0, 3E0, ...
with: E0 = hν.
About your second post, the quantization I'm referring up, comes in the computations of the average energy of a single oscillator; Planck's law express this average energy, that is, the result of the computation, so you can't actually see where is this quantization (indeed you question was quite right!)
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Mr Andrew. Let me give a simplified view of Planck's thinking when he discovered the constant that is named after him. Think of a lump of material being heated until it is red hot or white hot. Physicists were trying to understand what was going on. They knew about electromagnetic radiation caused by the accelerating charges associated with the atoms bumping together. Stefan and Boltzmann had shown that the total energy radiated by a hot body should increase as the fourth power of the temperature. Rayleigh and Wien had made models using classical electromagnetic theory that showed that as the atoms bumped into each other electromagnetic radiation would be radiated and this radiation would be greater at higher frequencies but there was absolutely nothing to stop the radiated energy increasing without limit as the frequency got higher. However the predictions by the model were good at low frequencies but in measurements the radiation the peaked and tailed off to nothing at high frequencies so something was missing in the model. Planck realised that this something was that the higher frequency radiations must be limited to specific bursts of energy and so tried models in which the energy was "quantized" ie could only be released in bursts of a specific energy and not at a continuum of energy. Eventually the model that worked was that the energy quanta was defined by a constant (Planck's constant ) times the frequency. This was very difficult to do mathematically and was quite a feat well before the age of computers. All this happened before conventional quantum theory relating to atomic spectra was defined although the fact that atoms tended to radiate characteristic frequencies had been known and used for many years.
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That's well put - SS.
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To a certain extent the electromagnetic problem solved by Planck by proposing and proving that electromagnetic radiation was quantised is similar to the problem of the potential for gravitational collapse to a singularity. This too will probably be solved by a proper quantum theory of gravity and space-time. There are several very good hypotheses but the conditions under which they could be tested are so extreme that the proof of the correct theory is very difficult over and above the fact that most if not all quantum gravitiational interactions take place behind the shield of an event horizon inside a black hole.
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I guess I'm not understanding what is 'quantized' in the assumption of Planck's. The intensity of the radiation released at a given T and v? Or is it the energy of the radiation emitted at those frequencies? To what equation did Planck add a variable that can only take on integer values? The one for energy, intensity...I can't find it. I read his paper on quantization and he only meantions a constant, h, which by no means gives a discrete functional output. It gives a single number for each given T and v. Where is the mathematical manifestation of quantization in Planck's Law?
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I guess I'm not understanding what is 'quantized' in the assumption of Planck's. The intensity of the radiation released at a given T and v? Or is it the energy of the radiation emitted at those frequencies? To what equation did Planck add a variable that can only take on integer values? The one for energy, intensity...I can't find it. I read his paper on quantization and he only meantions a constant, h, which by no means gives a discrete functional output. It gives a single number for each given T and v. Where is the mathematical manifestation of quantization in Planck's Law?
I assume you have missed my post?
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Let me explain clearly in words.
In the classical view electromagnetic radiation is continuous and if you have a transmitter that works at a particular frequency you can emit continuous radiation at any power level however small by effectively reducing the current in the transmitting antenna. In the quantum case this is not true. There is a lower limit to the power a transmitter can send out in one burst this is called a quantum. The value of the energy in this quantum is a function of the frequency of the radiation, This is where you get the formula.
E (energy) = h (Planck's constant) x Greek letter nu (Frequency)
For radio frequencies this lower energy limit is so tiny you only get involved with it in extremely sensitive detectors like nuclear magnetic resonance. For higher frequencies like light this energy is quite a significant amount of energy and quanta can easily be generated and detected individually. For extremely high frequencies like gamma and cosmic rays this energy can be very large indeed and quite destructive.
To go back to my original description Atoms are tiny things much smaller than the wavelength of light so in theory the electromagnetic radiation from the field disturbances due to their interactions should be very short at all normal temperatures and in the classical theory most of the radiated energy from interacting atoms should be way up in the ultra violet region however the interactions are relatively gentle and do not produce enough energy to radiate a whole quantum at these frequencies so no quanta are radiated the energy has to radiate at lower frequencies where the quanta have lower energies that match the energy of the interactions.
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There are two aspects to this 'quantisation' thing.
Firstly, em radiation can be regarded as consisting of photons or 'quanta'.
The next thing is the integral solutions to the wave equation.
The integral solutions bit is only relevant , practically,for a single atom, isolated from others - as in a low pressure gas.
We are dealing with Fermions, which cannot have the same set of quantum numbers. When there is more than one atom in your system, the two atoms have an energy 'situation' between them, which introduces more quantum numbers into the sums. It, effectively splits the line spectra up into pairs of lines or groups of lines . For a large enough number of atoms (as in a lump of something), the lines become a continuum and, although you could, in principle, think in terms of millions and millions of quantum numbers for each atom - so that they could each have a different set, it is easier to talk in terms of energy bands and statistics, instead.
You have to go further than just talk in terms of the simple Hydrogen atom wave equation if you want a model which which links to Planck's law. Your 'nu' can have a continuum of values, corresponding to all the 'oscillators' in your radiating body.
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Yes sophie but the really important thing to remember is that all this came much later as interactions within atoms gradually came to be understood. When people are presented with all the peculiarities of quantum theory "up front" they tend to think that the scientists have gone mad and missed something important and thats why we get so many quantum theory and relativity "denyers" around. It has not been pointed out how these facts emerged perfectly naturally from basic studies in classical practical physics and are well understood and not just something out of the mind of a theoretician.
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Fair enough but the original question seemed to have a problem reconciling a continuously variable quantity vs a 'quantised' quantity with discrete values. The original work involved a statistically large number of oscillators with a continuum of possible values. That's what you get with condensed matter, effectively, even when you include quantum theory, isn't it? The integer solutions from the wave equation of a gas atom thing is a red herring and that seems to be what was Mr A's original problem.
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OK, I read up on this a little more and I now see that what was assumed to be quantized by Planck was the energy or the harmonic oscillators in the walls of the black body or atoms. What does this translate into physically, this oscillation of atoms? Was that where Einstein came in and said that it was the electrons in the atom that vibrated and whose energy was quantized thus leading to the photoelectric effect and light quanta and all of that?
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First think of a radio transmitter you put an oscillating voltage into a wire or current into a coil and some of the energy is radiated as radio waves at the oscillation frequency. That is, unshielded, changing electrical currents result in electromagnetic radiation in this case radio waves. Thats a nice simple physical idea that can easily be demonstrated and tested. Now think of dynamic atoms and molecules inside a solid liquid or gas moving around and bumping into each other. Atoms consist of bundles of electromagnetic charge that are largely neutralised when still and at a distance but when they move about and bump into each other as a result of being warm this neutralisation is slightly unbalanced and the atoms behave on the small scale like there are unbalanced and there are moving charges and varying currents flowing. These currents will try to radiate electromagnetic radiation. You would expect this radiation to be around the frequencies associated with the impacts. You can calculate this by looking at the speeds that they are moving the mean free paths and the mean free times. This is quite easy with gases (look up the kinetic theory of gases for a full explanation). Solids and liquids are more closely packed and complex and difficult to analyse from first principles. (remember that the temperature of an object is just a measure of the speed that its atoms are moving about. ie the amount of energy of motion each atom has to itself)These impacts are very brief and would tend to represent light or ultra violet frequencies but at normalroom temperatures these frequencies are not radiated because the quanta are too large so under these conditions the collisions are perfectly elastic and no energy can be radiated so any radiation has to come out at lower frequencies associated with larger scale and slower varying field patterns in the infra red region where the quanta are small enough.
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Let me get this straight...atoms are moving around and when they collide with each other they deform their electron clouds slightly and become miniature dipoles. These dipoles are moving around and thus creating a current (AC since they are constantly bouncing off of each other and changing direction) which emits EM radiation.
If this is true than the wavelength of the light emitted depends on the distance between the atoms...which increases with temperature. This would mean that longer wavelength light would be produced at higher temperatures--exactly the opposite of reality. And I thought Planck talked about harmonic oscillators. These atoms are not oscillating, they are bouncing off of one another and then travelling at a constant speed until they bounce off of another atom (like bouncy balls in a box being shaken). Did I misinterpret this?
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Why should you think that the wavelength is related to the distance between the atoms? it is not. The frequency of oscillation is the critical factor that determines the radiation frequency. (it is quite possible to have a low frequency radio transmitter and a short aerial its just that short aerials do not transmit very efficiently).
The bouncy balls analogy is standard.
At room temperature in solids it is not really the individual collisions that are radiating because that's not allowed because the energies involved are to low to generate a quantum at that frequency. It is a larger chunk of the whole material. You must also realise that the radiation is not at a precise frequency like a spectrum line but over a range of frequencies like a red hot poker which emits a wide range of red and infra red radiation.
It is quite easy to visualise electrons jumping between levels and emitting quanta at precise frequencies to create a spectrum line but I have never seen a good visual analogy of thermal radiation from a solid liquid or gas. The discussion of "harmonic oscillators" in the theory is the sort of theoretical bodge that a theoretical physicist uses to make the problem tractable as he is trying to create a model to analyse, but it is likely to lead non expert theoretical physicists in the wrong direction.
The best analogy that I can think of at the moment is this. Think of all the violent action and jostling in the material as being a bit like a large crowd of people moving and interacting at random, but there can be slower collective motions in the crowd a bit like a swaying football crowd. It is these motions that emit the infra red radiation quanta. These motions are not at any particular frequency but they have slight constraints over a range of frequencies. In the radiation case the upper constraint is the fact that the energy is quantised and only if you get a collective motion large enough can you get radiation the lower one is that the motions are random and as you get bigger and bigger averages the deviation becomes smaller with respect to the mean.
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This idea of oscillators is just a model which helps to explain and predict. Hardly anything (in fact nothing at all) is exactly like the Scientific model that is used to describe it. Models are just constructs to help understanding - unless you really know your stuff, you can't extrapolate the model and expect it to work.
In many cases, it helps to think macroscopic, rather than microscopic. The fashion is always to try and explain using quantum theory when, in fact, it is very hard when using 'schoolboy' theory.
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So, the energy of the domains of atoms that are oscillating is quantized? The harmonic oscillators were not individual atoms but groups of them. That makes more sense. Using the bouncy ball analogy, several bouncy balls bouncing on parallel paths to each other generate the EM radiation? But, they can only generate EM radiation if they are bouncing with more than a certain threshold energy for each frequency, am I correct? That is, if the energy of the domain that is oscillating is equal to 5h then light will be emitted at E = 5h = hν => ν = 5 Hz? If that's true, then I think I finally understand!
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The harmonic oscillators were not individual atoms but groups of them.
It might be better to say that the total number of quantum numbers which would describe the energy state of the whole system (consisting of 'ten to the power of a lot' atoms) would be very high. You can regard it as a continuum. Take a photon of any frequency and it will find a possible resonance, corresponding to one possible energy transition of the system. That means that it can be absorbed or radiated. This is just an alternative way of saying that there are a lot of oscillators in the object. Two alternative models describe the same phenomenon - one is more 'classical' than the other, that's all.
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So, the energy of the domains of atoms that are oscillating is quantized? The harmonic oscillators were not individual atoms but groups of them. That makes more sense. Using the bouncy ball analogy, several bouncy balls bouncing on parallel paths to each other generate the EM radiation? But, they can only generate EM radiation if they are bouncing with more than a certain threshold energy for each frequency, am I correct? That is, if the energy of the domain that is oscillating is equal to 5h then light will be emitted at E = 5h = hν => ν = 5 Hz? If that's true, then I think I finally understand!
If you consider a *single* electron in a metal, this can oscillate to almost any frequency, so it would be associated to an infinite number of oscillators itself.
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I feel that for the broad spectrum case the concept of an "oscillator" is not a good one because this implies a long term periodic cyclic process. It is quite possible for electromagnetic quanta to be very brief just like a single pulse. A better visualisation of the process might be a bit like viscosity or friction where there is an energy loss into the electromagnetic domain. Inside the material there are also gains from the electromagnetic domain when the fields set up by the quanta affect the motion of the atoms and the whole process is in equilibrium if the temperature is uniform. At the edge of the material some of this radiation escapes and gives the sense of the temperature of the object.
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I feel that for the broad spectrum case the concept of an "oscillator" is not a good one because this implies a long term periodic cyclic process. It is quite possible for electromagnetic quanta to be very brief just like a single pulse.
Fourier transform; where's the problem?
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Have you ever had to do a Fourier Transform with an old-fashioned 10 key calculator? Or with an even simpler slide rule? You will hate them for the rest of your life. It is one of the ways sonic responses from seismic surveys are filtered and "cleaned up." The signals are from point sources and are a measure of single events below ground, gathered to form a "wave".
I had to do it in geophysics class in college before the hand calculator came out. Pain in the buttocks.
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An oscillator can have any 'Q' factor you like. The effective Q of an electron in a metal (if that's the model you choose to use) is very low - so it can absorb / emit any frequency of photon you like.
It is risky to insist on one particular model to describe a given phenomenon - it may be the 'hard way' of understanding what is going on.
Jim Bob - tell us about the war Daddy! Not long ago, a 'computer' (or computor) was a person - just like you were; cannon fodder for the guy in the end office.
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I guess I'm not understanding what is 'quantized' in the assumption of Planck's. The intensity of the radiation released at a given T and v? Or is it the energy of the radiation emitted at those frequencies?
Planck’s hypotheses was that the energy of a wave of a given frequency cannot be arbitrary but assumes only discrete values. Within the framework of classical physics it was impossible to explain how an electromagnetic wave can have only discrete values of energy, this is one of the reasons that Maxwell’s theory of Electromagnetic radiation is out of favour. Planck’s work was impeccable, some of the later hypotheses of QM are not.
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I agree sophiecentaur that you should chose your models to match the conditions. Lightarrow Fourier analysis is not relevant in this particular case.
This discussion brings me back to a question that I am still looking for a definitive answer to.
Does an individual quantum posses bandwidth? and is this a measurable property of a quantum? Groups of quanta from specific sources and quantum detection devices can both posses bandwidth ie a quick transition produces a broad band quantum and a high q detector is a narrow band detector and vice versa but does this apply to a single quantum if so bandwidth as well as energy should be given as a property of a quantum.
McQueen you are wrong. All physical theories must be considered as approximations. Maxwells electromagnetic theory is perfectly adequate for many normal electromagnetic experimentation as is Newtonian gravity the additional theories of quantum mechanics and relativity are only needed for extreme cases where the accuracy of the theories breaks down.
The breakdown of electromagnetics into quantum mechanics started with experimentation observation and modelling as is being discussed here. the great genius of Einstein was that the breakdown of Newtonian gravity to relativity was almost entirely theoretical and would not have been required by practical physics for several more years.
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Planck’s hypotheses was that the energy of a wave of a given frequency cannot be arbitrary but assumes only discrete values.
But I think you would agree - and it is the basis of calclus - that , given enough discrete values and if they are close enough together, you can regard it as a continuum. The discreteness of which Planck is talking is not just limited to the obvious big-steps of the energy levels in a hydrogen atom.
Yes, of course, classical explanations are not sufficient for the full story -we have moved on since late victorian times.
Soulsurfer- your question about quanta and bandwidth. I think you are really asking about photons and bandwidth. Because a photon has a finite effective lifetime and pulse length, it must have a bandwidth - it s bandwidth relates to the reciprocal of the length of the impulse of energy it carries. It also relates to the effective Q factor of the oscillation or the time actually taken for the energy transition to occur.
So the 'envelope' of the photon could depend on the process that gave rise to it. I imagine that the high coherence of the light from a laser could imply a narrower bandwidth.
However, there is probably much more to it than we are getting to in this discussion. The duality thing must come into it - if we are trying to discuss bandwidth then we are, by implication, talking about a wave - which is not the particle aspect of electromagnetism. Perhaps it's the wrong question to ask?
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Lightarrow Fourier analysis is not relevant in this particular case.
Why? You know that it's possible to write every waveform, as short as you like, as an infinite sum of sinusoidal plane waves.
This discussion brings me back to a question that I am still looking for a definitive answer to.
Does an individual quantum posses bandwidth?
I presume yes, because I know for certain that photons don't have a precise energy, also depending on the specific kind of transition, as sophiecentaur says.
and is this a measurable property of a quantum?
I don't know the answer; I think it could be done simply, e.g., sending individual photons, all emitted from the same source in the same conditions, through a diffraction grating an collect the resultant pattern in a screen, exactly in the case of "common" light; however the photons cannot be exactly all equal because of an intrinsic statistical inequality.
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However, there is probably much more to it than we are getting to in this discussion. The duality thing must come into it - if we are trying to discuss bandwidth then we are, by implication, talking about a wave - which is not the particle aspect of electromagnetism. Perhaps it's the wrong question to ask?
Maybe; however you have the interference pattern even sending individual photons; how would you interpret this fact?
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Because a photon has a finite effective lifetime and pulse length, it must have a bandwidth - it s bandwidth relates to the reciprocal of the length of the impulse of energy it carries.
I am sorry I can’t understand this, a photon is supposed to have an infinite lifetime. That is if it is not absorbed it will maintain its energy, indefinitely.
McQueen you are wrong. All physical theories must be considered as approximations. Maxwells electromagnetic theory is perfectly adequate for many normal electromagnetic experimentation as is Newtonian gravity the additional theories of quantum mechanics and relativity are only needed for extreme cases where the accuracy of the theories breaks down.
That is just the point a model is appropriate providing it is an honest one. Take the treatment of EM by Quantum Mechanics for instance, forget about quantization, what about the second quantization and the third and the fourth and after each of these, normalization takes place where infinities are reduced to zero. This is still OK except for the claim that the theory has an accuracy of 10^^12 !! Under these circumstances where could quantum mechanics possibly be needed?
Again, Maxwell’s theory even though it might work, does not satisfy the fact that EM is quantized, therefore it is not an acceptable theory. Why was P.Drude’s explanation of electrical conduction dropped ? It was dropped because it didn’t correspond to new facts that came to light. Note that Drude’s theory did work quite well to a certain extent. But it was still wrong in the end.
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Because a photon has a finite effective lifetime and pulse length, it must have a bandwidth - it s bandwidth relates to the reciprocal of the length of the impulse of energy it carries.
I am sorry I can’t understand this, a photon is supposed to have an infinite lifetime. That is if it is not absorbed it will maintain its energy, indefinitely.
He meant to say that the interval of time during which the photon is generated is finite, so the "train of EM waves" (if a photon is a "train of EM waves) has a finite lenght, and this means it cannot have a unique frequency (it can be seen through Fourier transform).
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Why? You know that it's possible to write every waveform, as short as you like, as an infinite sum of sinusoidal plane waves.
You should be more accurate if you want to get it right.
Fourier ANALYSIS takes a repeating waveform and replaces it as an infinite series of harmonically related sinusoids.
The Fourier TRANSFORM can be carried out on any function. The FT of a sinewave (in the time domain)which has been modulated by a pulse (i.e. a burst of sinewave) has a peak (in the frequency domain) at the 'carrier frequency' and, potentially, a continuum of sidebands (sinusoids of infinite duration) which stretch out on either side. The shorter the pulse in time , the more the spread of the sidebands in frequency.
you have the interference pattern even sending individual photons; how would you interpret this fact?
The two slit interference pattern would give a density pattern, corresponding to the probability of a particle / photon hitting a particular bit of the screen. That is the same pattern as the wave interference approach would give. (As the probability density function of the electron around the atom relates to the wave function.)*
The detailed pattern, seen on the screen (i.e. the actual spacing of the half power points, for instance) would depend on the range of frequencies (bandwidth) of the light / population of photons. It would be interesting if someone had compared the detailed interference patterns from different sources. Ah, yes, of course there would be spectral spreading due to Pauli exclusion - as with high pressure sodium lamps.
No two photons would be of the same frequency - even in a low pressure gas .- Also, because of Heisenberg uncertainty, you could not be sure of the precise frequency of any photon.
As for the lifetime of a photon McQueen, - it can be as long as you like - but, once you have measured it, it is ended. It starts life 'sometime', when an atom (or system of atoms) changes energy level and it ends when it has interacted with another system. (Yes, Lightarrow -I have just read your post)
*You get the same thing with electrons interfering as they pass through a thin carbon sheet.
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Why? You know that it's possible to write every waveform, as short as you like, as an infinite sum of sinusoidal plane waves.
You should be more accurate if you want to get it right.
Fourier ANALYSIS takes a repeating waveform and replaces it as an infinite series of harmonically related sinusoids.
The Fourier TRANSFORM can be carried out on any function. The FT of a sinewave (in the time domain)which has been modulated by a pulse (i.e. a burst of sinewave) has a peak (in the frequency domain) at the 'carrier frequency' and, potentially, a continuum of sidebands (sinusoids of infinite duration) which stretch out on either side. The shorter the pulse in time , the more the spread of the sidebands in frequency.
Exactly, and any arbitrarily short function of time f(t) (in the sense that it's ≠ 0 in an arbitrarily short time interval) can be written as an infinite sum of pure sinusoids, that is an integral, where the function inside the integral is the FT of f(t):
f(t) = (1/Sqrt(2π))∫F(ω)eiωtdω
F(ω) = (1/Sqrt(2π))∫f(t)e-iωtdt
and the integrals are from -∞ to +∞.
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Sure thing boss.
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Sure thing boss.
[:)]