Talk:Planck's law of black body radiation

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1 Derivation (Statistical Mechanics)
2 Spin of photons is 1
3 expression of U and R
4 A Little Suggestion
5 Planck's law is wrong (?)
6 Boxes-- new guideline?
7 Table
8 History
9 references
10 Alternative derivation of energy density (with little mathematics)

[edit] Derivation (Statistical Mechanics)

Hey PAR, you didn't define $f$ at the end of your derivation.

Hey, my friend, its our derivation, mostly yours. I fixed the f, it was just the factor of 2 for the photon spin. PAR 18:34, 19 May 2005 (UTC)

[edit] Spin of photons is 1

I remember that it shall be 1, NOT 2.--GyBlop 08:52, 24 February 2006 (UTC)

And in another cases,3/2...1/2....5/2 something etc. like those,are for Fermion.--GyBlop 11:27, 24 February 2006 (UTC)

[edit] expression of U and R

In Beiser's and Blatt's books,it is said of U(frenquency)=something,and seems to be NOT get a R(something)=stuff... making others confused.

Actually I believe that U(frenquency)=something proposed in a place/page only will be better than proposed both U and R in a one. At least,it will be better in someplaces of math proceses.--GyBlop 08:58, 24 February 2006 (UTC)

EXAMPLE:

for $u(\nu,T)=\frac{8\pi h\nu^3 }{c^3}~\frac{1}{e^{\frac{h\nu}{kT}}-1}$

and we can calculate an expression of certain R which contains of those above.

$R=\mathcal\, \frac{c}{4}u \,$

Thus

$R(\nu,T)=\frac{cu}{4}\frac{8\pi h\nu^3 }{c^3}~\frac{1}{e^{\frac{h\nu}{kT}}-1}=\frac{2\pi h\nu^3 }{c^2}~\frac{1}{e^{\frac{h\nu}{kT}}-1}$

Where this formula is not good for teaching when talking about itself and Energy of EM-w in a cavity at the same time. Or easily gets confused. --GyBlop 09:28, 24 February 2006 (UTC)

It doesn't have to be confusing if the article is very clear in defining symbols and showing the relationships among them. -- Metacomet 12:42, 24 February 2006 (UTC)

The wavelength is related to the frequency by

$\lambda = { c \over \nu }.$

The law is sometimes written in terms of the spectral energy density

$u(\nu,T) = { 4\pi \over c } I(\nu,T) = \frac{8\pi h\nu^3 }{c^3}~\frac{1}{e^{\frac{h\nu}{kT}}-1}$

which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz).

The spectral energy density can also be expressed as a function of wavelength:

$u(\lambda,T) = {8\pi h c\over \lambda^5}{1\over e^{h c/\lambda kT}-1}$

as shown in the derivation below.

For those editing by the original author.

I am not sure what you did was for Stefan's law $\mathcal\,R={\sigma}T^4=\frac{c}{4}\rho_T\,$ or not. If it was,then some formula like : $\mathcal\,R({\nu},T)\,$ should be going to represent Planck's $U=\frac{8{\pi}V\nu^2}{c^3}\frac{h\nu}{e^{h{\nu}/kT}-1}$ (Of-course it is after some process of math)

It is because that I is for a radiance of a volume,but R is for a radiance of a surface.--GyBlop 17:34, 26 February 2006 (UTC)

[edit] A Little Suggestion

My information was from books of Modern Physics of Blatt's and Beiser's. I suggest the author could refer to one of their books for rewritting some parts of the article.

At least,for myself,a little confussion on me while my first and second times getting in to see some descriptions of yours.--GyBlop 17:45, 26 February 2006 (UTC)

[edit] Planck's law is wrong (?)

There must be a mistake in Planck's law here. These are the two reasons:

Seconds cancle out => impossible for an equation determining a power
When drawn in Excel the peak doesn't correspond to Wien's displacement law

I didn't have the time to verify it all the way through but it seems as though it should be

$I(\nu) = 2 h\nu^5/c^3/(e^{h\nu/kT}-1)\,$

This way it corresponds to equations I have found that stated lamda instead of the frequency and recalculating. It also corresponds to Wien's law like this. As I said, no guarantee for what I say here and I won't be able (or willing to invest the time) to redo the derivation.

Seems like there is work needed on that derivation. I hope someone does it correctly because it will look really odd, if I am going to change the equation without touching the derivation.

Thomas

Thomas - don't change it. You forgot to transform the differential

d ν

. In other words, you don't want to say

$I(\nu)=I(\lambda)\qquad\mathbf{(WRONG)}$

you want to say:

$I(\nu)d\nu=I(\lambda)d\lambda\qquad\mathbf{(RIGHT)}$

Since

λν = c

, you have

d λ = - c d ν / ν 2

and

d ν = - c d λ / λ 2

. If you redo the math, and realize that the signs on the differentials don't matter, you will see that everything is ok. PAR 16:23, 31 July 2006 (UTC)

[edit] Boxes-- new guideline?

Whats with the boxes outlining some of the equations/ Is this a new WP guideline?--Light current 21:10, 8 August 2006 (UTC)

I hope not! :) When I revised the derivation, inserted more details etc., I left the boxes with the main results intact. Count Iblis 21:24, 8 August 2006 (UTC)

You mean you yourself did not add the boxes?--Light current 21:49, 8 August 2006 (UTC)

NO, I didn't do it. Perhaps user:PAR might have done it, not sure though... Count Iblis 22:26, 8 August 2006 (UTC)

NO, I didn't do it, and I wouldn't mind seeing them gone. PAR 22:30, 8 August 2006 (UTC)

OK that sounds like a consensus. Now who is going to do the dirty deed?--Light current 22:44, 8 August 2006 (UTC)

I'm too busy editing a real article that was provisionally accepted (the referee wanted some more explanations which means changing large chunks of the text) :( So, why not have a go if you have time and we'll take a look later... Count Iblis 22:50, 8 August 2006 (UTC)

OK done it!--Light current 23:01, 8 August 2006 (UTC)

GOOD PAR 23:40, 8 August 2006 (UTC)

[edit] Table

Do we really need a table to define the variables? i think it tends to dominate the page. 8-(--Light current 21:55, 8 August 2006 (UTC)

My opinion: Get rid of it, but make sure you explain everything in the text. The explanation of units/dimensions is a waste of editing space. That's almost primary school level information. And it's irrelevant, because the whole point of units is that you can choose your own favorite units for the relevant quantities. Count Iblis 22:37, 8 August 2006 (UTC)

Not quite primary school level, but they could just be listed normally 8-|--Light current 22:49, 8 August 2006 (UTC)

[edit] History

Any one think history should be near the top of the page?--Light current 23:47, 8 August 2006 (UTC)

Yes, I think that would be better. Count Iblis 00:34, 9 August 2006 (UTC)

1) How about adding a link to what Planck called the "first order" Stirling's approximation? I added two paragraphs under "Stirling's formula" yesterday. That Stirling's thing is visible not only in Planck's original paper which is now Ref.2 here, but it was also used by Debye, Bose and Louis de Broglie. It worked, so there must have been a reason for $n! = n n$ , which is what I tried to teach for some time. A possible reason follows qualitatively in point 2 with a proposed correction in 3 below.

2) If you change the number W in Planck's equation 3, you change the entropy of the whole system. Imagine that you keep the temperature constant and manage to change the entropy just by changing the number W (you change the experimental set up, an element is absorbed, etc). Then the system energy has to change by very little (Boltzmann constant is small). This is because of the definition of entropy: delta S equals delta Q over T. The change of energy due to the change of the number W does not seem to be related to a single element, but rather goes to the whole system. Consequently, the way in which Planck, Debye and Bose used Stirling's formula led to a situation where they added some extra variable(-s) (in addition to energy)to the system. That worked as the black-body equation matched the results. One can reasonably guess (speculate) that the three writers thus managed to include some interactions that were unknown at the time of their writing (eg. spin, the uncertainty principle, Pauli's exclusion principle, etc?) and possibly something that is not known so far. All that or anything of that could have been hidden in the "first order Stirling's" disguise (which at that time served the developing theory very well). As a result, the various (then unknown) quantities were eliminated from the black-body equation. Not completely, however, as they are still there, in the approximations, so that, in conclusion, the quanta in the black-body equation are not completely separated from each other.

3) Therefore I propose the following corrections to the history section. It must not be too long here. So how about:

a) A small change to the words "a theoretical derivation" made by Bose. Perhaps "an alternative" or "a new" instead of "a" would be better, because how to call Planck's work then - only "practical"?

b) A sentence between the two paragraphs could refer the reader to the Stirling's formula. For example:

Interestingly, Planck and other early writers on the black-body radiation used the "first order" Stirling's formula which provided a greater number of interactions among the elements of the system than the number calculated from the idea of completely separated system elements. The approximation was strongly nonlinear for a small number of elements. It could have been equivalent to a joint effect of some variables or phenomena discovered later in quantum mechanics, such as spin or the uncertainty principle .

Any comments please? (That was me but the signature and time is sadly missing. So let me add now -- C. Trifle 21:41, 5 October 2006

I decided to change the above "interactions among ... the ... completely separated elements" to "complexions", as Planck wrote, because otherwise, if the elements were completely separated, the number of interactions should be zero, which would surely make all that mathematics useless. C. Trifle, 5 October 2006

[edit] references

Are there different standards for citing sources on mathematical derivations? I know the steps should be self-evident, and the math is in the public domain, but the way it looks now it appears to have come off the top of the authors' heads. I don't want to be pedantic, and if nobody cares, I can remove the little templates. Good writing, btw. Ojcit 15:55, 3 October 2006 (UTC)

I gave a self-contained derivation suitable for wikipedia. You won't find this derivation anywhere in the literature, because textbooks usually assume that the student knows the basic stuff and they'll proceed from there. Giving the standard derivation that you can find in textbooks serves no purpose, because the people who can follow that are precisely the ones who already know it. What I did here was to imagine that the reader knows nothing except what he/she can find on wikipedia. He/she must, of course, have some basic mathematical skills.

So, perhaps a sentence needs to be included at the start of the derivation saying that we give a self contained derivation and that other derivations can be found in books such as.... Count Iblis 16:29, 3 October 2006 (UTC)

[edit] Alternative derivation of energy density (with little mathematics)

The energy due to equilibrium radiation in a little volume $d^3x\,$ has three contributing factors:

Energy of a light quantum, the so called photon:

$h\nu\,$

Possible states:

$2\frac{dp_{x}dx\,dp_{y}dy\,dp_{z}dz}{h^{3}}=2\frac{d(\frac{h\nu_{x}}{c})dx\,d(\frac{h\nu_{y}}{c})dy\,d(\frac{h\nu_{z}}{c})dz}{h^{3}}=2\frac{d^{3}\nu \,d^{3}x}{c^{3}}$

The factor 2 is due to the possible polarisations, the rest due to the fact that phase space $dpdx\,$ could be occupied only in multiples of $h\,$ . In thermodynamic equilibrium no particular direction will be favored. Therefore we can integrate over all frequency values lying on a sphere

$2\frac{4 \pi}{c^3} \nu^2 \mathrm{d}\nu \, d^3x$

Expected occupation number for a state:

In thermodynamic equilibrium a photon with energy $h\nu\,$ is emitted and absorbed by an electron jumping between two energy states separated by an energy gap of $h\nu \,$ . The higher of the energy states will be less populated by electrons than the lower - at least in a thermodynamic equilibrium where the upper energy level has a lower occupation by electrons given by the factor

$e^{-\frac{h\nu}{kT}}$

The expected occupation number $b\,$ of the photon's quantum state determines how probable it is that a photon kicks an electron from the lower energy state up to the higher. Without external influence an electron will drop back to the lower. However, photons could additionally stimulate the electron to drop down. Assuming that the occupation of a photon state has the same influence in kicking electrons up as kicking them down the energy gap the equilibrium condition is

${b=e}^{-\frac{h\nu}{kT}}(1+b)$

determining the occupation number of a photon state in equilibrium:

$b=\frac{1}{e^{\frac{h\nu}{kT}}-1}$

With these three factors we get the energy in a little volume $d^3x \,$ and the frequency interval $d\nu\,$ :

$(h\nu )\,( 2\frac{4 \pi}{c^3} \nu^2 \mathrm{d}\nu d^3x )\,( \frac{1}{e^{\frac{h\nu}{kT}}-1} ) = \frac{8 \pi h}{c^3} \frac{\nu^3 \mathrm{d}\nu}{e^{\frac{h\nu}{kT}}-1} \,d^3x$