Talk:Wien's displacement law

From Wikipedia, the free encyclopedia

	Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
???	This article has not yet received a rating on the assessment scale. [FAQ]
???	This article has not yet received an importance rating within physics.
Help with this template Please rate this article, and then leave comments to explain the ratings and/or to identify its strengths and weaknesses.

[edit] Alternative form correction

Since

I deleted the alternative form section (sorry, metacomet). It is possible to derive a frequency version of Wien's law, but I don't have it on hand at the moment. ...and I forgot to do an edit summary, so that's mostly the reason for this entry. ...wow, I am soo inept. --UltraHighVacuum 02:54, 23 February 2006 (UTC)

Don't apologize for fixing something that was incorrect. If it's wrong, it's wrong. I didn't write that section, I merely updated it so that it was possible to follow the logic. I didn't realize that the logic was flawed, or I would have removed it myself. -- Metacomet 10:18, 23 February 2006 (UTC)

On the other hand, it depends on what the definition of nu-max is. If it is the frequency of peak emission, then perhaps you are correct. But if the definition is the frequency that corresponds to the wavelength of peak emission, then the section that you removed was actually correct as written, although it would probably need a brief explanation of the difference in definitions. -- Metacomet 10:21, 23 February 2006 (UTC)

I have added a new, corrected section entitled "Frequency form" to the article. The source is Weisstein's World of Physics, and the information is consistent with the discussion below. -- Metacomet 11:10, 23 February 2006 (UTC)

In all my experience (and quantumm mechanics books)nu max is defined as the actual maximum frequency, so this was my motivation. It seems a definition of nu-max as being the corresponsing frequency for lambda max is confusing, and maybe also a bit arbitrary, since there is no reason for the definition not to be the other way around. But you're right, this is kind of ambiguous, so thanks for that section in the article. --UltraHighVacuum 18:04, 23 February 2006 (UTC)

Isn't that exactly what the new section says? -- Metacomet 23:07, 23 February 2006 (UTC)

Your frequency version is incorrect.

That is because the curves for frequency and wavelength are different. In fact, using a similar derivation it can be shown that

This gives

It turns out that

I will go ahead and make the changes, but it would still be nice for someone to check me on this. --129.93.63.14 23:08, 9 February 2006 (UTC)

Any point is putting the frequency version in?

--Audiovideo 13:09, 16 May 2005 (UTC)

It's said that "This equation cannot be solved in terms of elementary functions. It can be solved in terms of Lambert's Product Log function but an exact solution is not important in this derivation. One can easily find the numerical value of x"

But this derivation is horrible. Looking back at history einstein used the wien displacement law in his paper on radiation, but in no was was the wien displacement law a derivation from plancks radiation theory. This should be noted. I suck at writing or else i would.

I put a link to adiabatic invariant which reproduces Wien's original argument, and shows its connection to Einstein-Bohr-Sommerfeld quantization. Wien didn't win the nobel prize for nothing. Likebox 18:21, 11 September 2007 (UTC)

[edit] visual system and solar radiant power spectrum

The wavelength coincidence between the peak of the solar radiant power spectrum (at about 5800K) expressed in terms of unit wavelength interval, and the peak of the eye's spectral sensitivity function is just that - a coincidence or accident -- not something other as this article implies. If you express Planck's function in terms of unit frequency interval, you find the peak of that curve corresponds to a wavelength of about 880nm. If instead of power, you express Planck's function in terms of numbers of photons per second per unit wavelength interval (at 5800K), you find a wavelength peak at about 633nm. The problem is that Planck's function is a probability density function and thus doesn't transform simply with a change of units. The eye's spectral sensitivity function, however, is not a density function, so changing units is straightforward. But it is misleading to attempt to compare the two types of functions. This is a common error in the vision literature and a common problem when interpreting density functions.

Moreover, all visual systems plainly reflect several constraints at least as important as the solar spectrum: they can't be too sensitive to the infrared without being subjected to noise created by the warmth of the eye itself; they can't be too sensitive to uv because of molecular instability; and they most likely reflect their phylogenetic origins in water, which itself filters the solar spectrum.

All this is spelled out in "Some paradoxes,errors, and resolutions concerning the spectral optimization of human vision" Sofer and Lynch Amer. J. of Physics 67 (11) 1999

Apologies for all violations of protocol -- first time posting.

Greblams (talk) 08:15, 6 January 2008 (UTC)

• And a fine one it is, too! In the following, click on the Wiki-links to see what's here already. You're probably right that this article makes too certain a connection between the low-light (scotopic = rhodopsin based) visual sensitivity of animals (both land and sea animals) which peaks in sensitivity at 507 nm, and the blackbody spectrum of the sun, which peaks at 503 nm, when expressed as a power/wavelength function. However, I don't know what you're driving at when you say "The eye's spectral sensitivity function, however, is not a density function." It CAN be expressed as a density function, if you like, and usually is. In fact the standard luminosity function is commonly expressed as a fraction/wavelength curve. Or really, the two luminosity functions are-- one of which is photopic and applies to color vision and high light powers, and the other of which is scotopic and applies to rods and rhodopsin and night-vision. But in any case, if you want the "brightness" that the eye SEES, you integrate the appropriate luminosity function with your light power spectrum (using the same units for both, in this case we choose "per unit wavelength") and when you're done, out pops the total luminosity in lumens. To see the brightest light per unit energy, you want to maximize the lumens/watt, or the luminous efficacy of your starlight for your eye (actually, when it comes to low light, all eyes are the same in all animals). Here, we're talking actually about the scotopic luminous efficacy (moonlight lumens/watt), not the photopic one (color vision) which is usually treated for standard lighting (reading and photography). And yes, you could express the blackbody spectrum in the various ways you note (per frequency or per photon per wavelength), but then in order to convert to subjective brightness, you'd need a luminosity function for the eye expressed in those same units, and your integral for the luminosity of your source would come out just the same (of course).

  So the real question is what light spectrum you use to maximize this integral, to get the maximal number of subjective lumens per watt out of your source. You can't use a laser tuned to the rhodopsin peak (which would give you the maximal efficacy)-- you're stuck using only blackbody radiation at various temperatures. Now, since this isn't a totally symmetrical curve (as the scotopic luminosity function is) you obviously cannot do this by merely using a blackbody curve with the same peak, but if the two peaks are close you'll probably be fairly close to the maximal efficiency possible for blackbodies. For example, I know that maximal photopic luminous efficacy occurs at a blackbody color temp of 6500 K, which is considerably hotter than the Sun. But the photopic luminosity function peak is 555 nm, and you're really aiming for the scotopic peak at 507 nm, which should give you a maximally efficient color temp of about 91% of that, or 5900 K. Which is very close to that of the Sun. So, faint moonlight, must be very close to maximally bright PER WATT at the color temp of our Sun, when seen by an eye using rhodopsin; if the Sun was any other temperature, the efficiency (lumens/watt) would drop, and you'd see a less bright light for the same raw total power.

  I'll leave your other arguments for later, but I don't buy them. We could see very far into the infrared without being bothered by background IR, as near-IR night vision (and IR photography) shows. But there's less visual energy down there, and evolution probably had a hard time coming up with a reaction that used a single IR photon, and there wasn't much pressure to do it. As for UV, humans actually see UV when their lenses are removed, and bees see UV also, so molecules already exist that sense this. But evolution simply had too hard a time getting UV through the proteins in a lens, and why bother? Bees had a reason to need UV and they had no lenses, so they got a pigment to do it. But it only works in bright light. Humans too have a photopsin which peaks at 420 nm absorption, so it's not as though we could not have shifted our vision into the UV from that standpoint (and in fact, that’s how we see UV). If Vega was our star, we might all have compound eyes and no protein-containing lenses. Who knows? SBHarris 10:43, 6 January 2008 (UTC)

Let me try to clarify the first point. The standard luminosity efficiency function may look like a probability density function. But that is simply because it is created from instruments with finite bandwidth resolution. The data points are averages obtained from sampling within those intervals. But the function itself is not at all a density distribution. Normalized, it is defined, as you say, as a dimensionless fraction/wavelength (not fraction/wavelength interval). A probability density distribution is different. It's unit is by definition a differential interval. For example the blackbody solar spectrum tells you power/dWavelength.

We frequently omit the differential piece when referring to the power spectrum (as you did), but this is what makes the difference when you transform the function from wavelength to frequency, using wavelength=c/frequency. (Sorry, I haven't figured out a more convenient notation). In that case, you have to also supply the Jacobian weighting factor -- here equal to c/(frequency)² (ignoring the sign since that only affects the order of integration.) And this is what changes the shape of the curve! You don't have to do this with the luminosity function because it doesn't involve differentials. If you plot a luminosity function in terms of frequency or wavelength, the shape of the curve looks the same. That is, the wavelength associated with the peak of the curve remains unchanged. When you plot the solar spectrum in terms of dwavelength or dfrequency, while the information it contains is the same(!) the shape of the curve changes as does the wavelength associated with its peak. This is a property of probability density distributions generally: they may change shape radically with transformation of variables. (Very narrow ones do not do so, however.)

That's why it's not legitimate in principle to attempt to infer anything at all from the peak of the solar spectrum plotted against a certain bandwith and, in this case, the peak of a luminosity function, as this article attempts to do, albeit parenthetically. It's a case of "comparing apples and oranges." Essentially, I can transform the shape of the solar spectrum (and the wavelength associated with its peak) any way I like by some combination of a change of variables or plotting against a different bandwidth. We usually see the solar spectrum plotted against wavelength and people often draw significance from the closeness of its peak to the peak of a luminosity function. My main point was that even apart from the cogency of any particular evolutionary argument, the closeness of the two peaks is itself misleading and essentially meaningless. When plotted against frequency the wavelength associated with the peak of the solar spectrum is about 880nm. So what exactly is the "closeness" to the peak of the luminosity function that we are trying to explain via evolution? It's not that too much is made of this -- it's that it's meaningless to talk about. This misunderstanding crops up a lot in both the technical and popular literature about vision.

While it isn't legitimate to relate the peak of the solar spectrum to the luminosity function, it certainly IS legitimate to multiply the two functions as you describe. What you get is indeed then a density distribution. And yes the integrals will all come out the same for different units, as you say --as long as you mind the Jacobians. And it is also legitimate (and most interesting!) to ask of that integral whether it is in some sense optimal. Your comments about source blackbody temperature are intriguing. That at least strikes me as one plausible way to begin to think about how/if the visual system may be optimal at least in a global sense. You have to be careful about what the limits of integration are taken to be, though. For example, they have to take into account the atmospheric cutoff at about 320 nm.

As for my comments about other constraints, some of your points are well taken. For the infrared, there certainly are plenty of photons in the near infrared for a visual system to potentially sop up. During the day, there is plenty of sunlight above 700nm and at night, I believe there is the OH air glow as well. Why visual systems generally don't exploit these, I'm not sure. You are probably right that it has to do with the low energy of those photons. A photoisomerization system would need ever longer molecules to make the low energy transitions to detect these photons, but such molecules are increasingly unstable in aqueous solutions. There is also a temperature of the eye however that will fog a retina attempting to detect in the infrared. I'm not sure what that temperature is though. As for UV, it is true that losing your cornea or lens will enhance your ability to detect UV. But it can't be simply the case that "evolution simply had too hard a time getting UV through the proteins in a lens." Folks without a cornea or lens also suffer increased rates of UV-related ocular damage. I'm not sure what drives a visual system to flirt with UV radiation at all since UV is hell on organic molecules. I'm also not sure what drove the insect-flower communication system to move into the uv in the first place. And I wonder how many bees and other insects are going blind by the time they die. I believe older Brown pelicans typically suffer from glaucoma as a result of accumulated eye damage from all those high impact dives for fish. Maybe bees, like pelicans, simply gotta do what they gotta do to get as far as they do. I'm a great believer in the power of natural selection to climb adaptive peaks, but sometimes living things reflect quite sub-optimal adaptations.

I think the really under-rated factor is water. If you compare the sunlight transmission curve in water at 1 meter and a luminous efficiency curve (you can do this because the transmission curve is not a density distribution) -- it looks like that that is what the visual system evolved to track. My hunch is that we have been tinkering on the margins with an essentially water-evolved visual system ever since.

Greblams (talk) 23:35, 6 January 2008 (UTC)