Planck's law in wavenumber vs. wavelength

[davidwinth]

TL;DR Summary

Why does Planck's law give inconsistent results depending on which form is used?

I have asked this question elsewhere. I have gotten no clear answer.

What I already know:

• The interval (differential) sizes (areas) are different in terms of wavelength and wavenumber.
• The total energy is the same when the curves are integrated over all wavelengths or wavenumbers

So please don't re-state these if you attempt to answer my question, as I already know these things and they do not answer my question.

(Since others have had difficulty understanding that my question is not about the size of intervals, I will post it by way of a thought experiment.)

Imagine I have a body at 500 K. I have an instrument that can measure either the energy of a particular wavelength of light or the energy of the wavenumber by a simple switch. I point the instrument at the body when set in wavelength mode, and scan across all wavelengths until I see where the maximum energy value is located. It turns out that the maximum is at a wavelength of 5.8E-6 meters. This is wave has a wavenumber of 1/5.8E-6 = 1.7E5 m^-1. This is definitionally true. So, to recap... the maximum energy is at a certain wavelength, and that same physical wave can be described in terms of wavelength or wavenumber. The physical wave does not change whether it is described by wavelength or wavenumber. This physical result agrees with the predicted value given by Planck's law in wavelength form.

The problem comes when I switch the machine to wavenumber mode. Since the machine is still pointed at the body, and nothing has changed at all, it should now read 1.7E5 m^-1 as the wavenumber with the maximum energy. The physical setup is identical, it is only the mode that has changed. We know for certain that any wave with wavelength X has wavenumber 1/X. That is a fact. Yet Planck's law in wavenumber form predicts we will measure 9.8E4 m^-1. This is nowhere near the value that corresponds to the maximum measured wavelength!

So I hope it is clear that my question is not about intervals or how the differential areas under the curves are different sizes. I am speaking of a single value on the curve - a single, specific wavelength/wavenumber and not an interval. How come Planck's law in wavenumber form predicts a different value from that already measured? To make it clearer, imagine that the machine has a display. When it wavelength mode, it displays the measured value of the wavelength and its corresponding wavenumber. Like this:

And when in wavenumber mode, the display looks like this (according to Planck's law but not according to a simple conversion):

The problem seems to be that if the peak is at wavelength X, and a wave of wavelength X has a wavenumber of 1/X by definition, then Planck's law predicts the same exact physical thing has a different value depending on if it is measured one way or the other. But that cannot be right. The same exact wave has a fixed wavenumber no matter what Planck's law says! The readout from the machine should simply have the numbers reversed.

If anyone can explain this, I would be most grateful. I would like an answer to be very logical, like a logical argument. It should start with some premises and end with a logically-derived conclusion: Therefore, the measured value will change from 1/X when we switch the machine to wavenumber mode.

Thank you!

 

[PeterDonis]

You need to write down the actual mathematical laws that you think are "Planck's law in wavelength form" and "Planck's law in wavenumber form". Without that we have no way of checking your calculations.

 

[hutchphd]

You do understand that the frequency (or wavelength or wavenumber) with maximum energy is not the same as the frequency (or wavelength or wavenumber) with maximum occupation number (i.e. photons) yes?

 

[davidwinth]

You need to write down the actual mathematical laws that you think are "Planck's law in wavelength form" and "Planck's law in wavenumber form". Without that we have no way of checking your calculations.

Actually, converting from wavelength to wavenumber is a very simple equation. If a physical wave has a wavelength X, then it has a wavenumber 1/X. So if I measure a physical wave and that wave has a wavelength Y, then it must have a wavelength 1/Y. There can be no way to get the same physical wave to have a different wavenumber just because Planck's law in one form says it should.

 

[davidwinth]

You do understand that the frequency (or wavelength or wavenumber) with maximum energy is not the same as the frequency (or wavelength or wavenumber) with maximum occupation number (i.e. photons) yes?

I am not sure what this means. You seem to understand that if I have a physical wave with a given wavelength X, then it must have a fixed wavenumber 1/X. This is the point. If I measure it as X, then I must also measure it as 1/X. The same physical wave doesn't change just because I set the machine to measure 1/X instead of X - the relation still holds between them.

Thanks.

 

[Vanadium 50]

You need to write down the actual mathematical laws that you think are "Planck's law in wavelength form" and "Planck's law in wavenumber form".

Actually, converting from wavelength to wavenumber is a very simple equation.

That's not what he asked.

For someone who is so demanding on how we respond to you (:So please don't re-state these if you attempt to answer my question ") I would think you'd be more receptive to answering our questions.

So, at risk of repetition, you need to write down the actual mathematical laws that you think are "Planck's law in wavelength form" and "Planck's law in wavenumber form". Without that we have no way of checking your calculations.

 

[davidwinth]

That's not what he asked.

For someone who is so demanding on how we respond to you (:So please don't re-state these if you attempt to answer my question ") I would think you'd be more receptive to answering our questions.

So, at risk of repetition, you need to write down the actual mathematical laws that you think are "Planck's law in wavelength form" and "Planck's law in wavenumber form". Without that we have no way of checking your calculations.

Sorry, I don't mean to be demanding beyond reason. Honestly. I am a bit frustrated by people talking about intervals when I am speaking of a point, not an interval. I checked my calculations here:

https://www.spectralcalc.com/blackbody_calculator/

They are correct.

 

[hutchphd]

Consider the 1 dimensional case. Suppose that we draw the graph as a discrete bar graph for thermal occupation expected in each allowed photon state (very skinny bars!). Then we choose to number each bar according to wavenumber and plot the bar graph of occupation vs wavenumber. There will be a bar at each integer value of k (units of 2pi/a) and they will be uniformly distributed in 1D.

What if we instead plot those bars (same damned bars) vs wavelength. They will now no longer be uniformly distributed and the shape will be very different.

Suppose now that I wish to plot a continuous curve that represents the number density for each bar graph. Do I just connect the tops of the bars? NO! ... I need to account for their density (along the axis). And so I must include the relative scale factor for each infinitesimal interval. This is true for any such transformation from discrete to continuous variables...it is a mathematical issue not a physical one. And it can very easily move the position of maxima.

 

[Vanadium 50]

First, hutchphd is right: a distribution flat in x is not flat in 1/x. Furthermore, f(x) does not equal 1/f(1/x) in general,.

If that doesn't satisfy you, let me ask once again - please show your calculations.

 

[PeterDonis]

converting from wavelength to wavenumber is a very simple equation.

Then it should be a simple matter for you to write down the explicit equations that you think are "Planck's law in wavelength form" and "Planck's law in wavenumber form". Then we can check your calculations. If you are unwilling to do that, then this thread will be closed as we can't have a productive discussion on this topic if you are unwilling to provide explicit math to back up your claims.

 

[PeterDonis]

I don't mean to be demanding beyond reason.

If you are unwilling to provide explicit equations when asked, then you are being demanding beyond reason, whether you want to be or not.

If you are unable to provide explicit equations because you don't know what they are, then you should go fix that before continuing this discussion.

Honestly. I am a bit frustrated by people talking about intervals when I am speaking of a point, not an interval.

You can say you are speaking of a point, not an interval, but that doesn't make it true. The physical meaning of the point values you are calculating cannot be understood without understanding what intervals are involved in the limiting process that produces that point value. (Note that no actual physical detector measures a point value; every detector has a finite bandwidth. So every real detector measures over an interval, not a single point value.)

I checked my calculations here:

That site doesn't give the equations it's using, so it's useless for this discussion.

 

[tech99]

Summary:: Why does Planck's law give inconsistent results depending on which form is used?

I have asked this question elsewhere. I have gotten no clear answer.

What I already know:

• The interval (differential) sizes (areas) are different in terms of wavelength and wavenumber.
• The total energy is the same when the curves are integrated over all wavelengths or wavenumbers

So please don't re-state these if you attempt to answer my question, as I already know these things and they do not answer my question.

(Since others have had difficulty understanding that my question is not about the size of intervals, I will post it by way of a thought experiment.)

Imagine I have a body at 500 K. I have an instrument that can measure either the energy of a particular wavelength of light or the energy of the wavenumber by a simple switch. I point the instrument at the body when set in wavelength mode, and scan across all wavelengths until I see where the maximum energy value is located. It turns out that the maximum is at a wavelength of 5.8E-6 meters. This is wave has a wavenumber of 1/5.8E-6 = 1.7E5 m^-1. This is definitionally true. So, to recap... the maximum energy is at a certain wavelength, and that same physical wave can be described in terms of wavelength or wavenumber. The physical wave does not change whether it is described by wavelength or wavenumber. This physical result agrees with the predicted value given by Planck's law in wavelength form.

The problem comes when I switch the machine to wavenumber mode. Since the machine is still pointed at the body, and nothing has changed at all, it should now read 1.7E5 m^-1 as the wavenumber with the maximum energy. The physical setup is identical, it is only the mode that has changed. We know for certain that any wave with wavelength X has wavenumber 1/X. That is a fact. Yet Planck's law in wavenumber form predicts we will measure 9.8E4 m^-1. This is nowhere near the value that corresponds to the maximum measured wavelength!

So I hope it is clear that my question is not about intervals or how the differential areas under the curves are different sizes. I am speaking of a single value on the curve - a single, specific wavelength/wavenumber and not an interval. How come Planck's law in wavenumber form predicts a different value from that already measured? To make it clearer, imagine that the machine has a display. When it wavelength mode, it displays the measured value of the wavelength and its corresponding wavenumber. Like this:

View attachment 257918

And when in wavenumber mode, the display looks like this (according to Planck's law but not according to a simple conversion):

View attachment 257919

The problem seems to be that if the peak is at wavelength X, and a wave of wavelength X has a wavenumber of 1/X by definition, then Planck's law predicts the same exact physical thing has a different value depending on if it is measured one way or the other. But that cannot be right. The same exact wave has a fixed wavenumber no matter what Planck's law says! The readout from the machine should simply have the numbers reversed.

If anyone can explain this, I would be most grateful. I would like an answer to be very logical, like a logical argument. It should start with some premises and end with a logically-derived conclusion: Therefore, the measured value will change from 1/X when we switch the machine to wavenumber mode.

Thank you!

I think others may have already stated the following in another way. As the emitted radiation is noise-like, the energy in a bandwidth of zero will be zero. Therefore I presume your machine measures over a finite bandwidth. I am thinking that there is a difference in how the machine is defining unit bandwidth when it is in wavelength or wave number mode. In the first case bandwidth might be defined in nanometres, whilst in the second it might be defined in Hertz. A bandwidth of one nanometre is not equivalent to one Hertz exept at a single frequency.

 

[PeterDonis]

In the first case bandwidth might be defined in nanometres, whilst in the second it might be defined in Hertz.

No, in the second case it would be defined as inverse nanometers. Wavenumber is not the same as frequency.

 

[vela]

Sorry, I don't mean to be demanding beyond reason. Honestly. I am a bit frustrated by people talking about intervals when I am speaking of a point, not an interval.

You need to consider the possibility that answering your question in terms of a single point is wrong, that the answer to your question fundamentally involves talking about intervals. In other words, your interpretation of the various forms of the Planck law is incorrect.

 

[tech99]

No, in the second case it would be defined as inverse nanometers. Wavenumber is not the same as frequency.

Yes, I agree with you, although wavenumber is analogous to frequency. An interval of, say, 10^-9 metres in wavelength does not necessarily contain the same energy as an interval in wave number of 1/10^-9 = 10^9 m-1.

 

[davidwinth]

If you are unwilling to provide explicit equations when asked, then you are being demanding beyond reason, whether you want to be or not.

If you are unable to provide explicit equations because you don't know what they are, then you should go fix that before continuing this discussion.

You can say you are speaking of a point, not an interval, but that doesn't make it true. The physical meaning of the point values you are calculating cannot be understood without understanding what intervals are involved in the limiting process that produces that point value. (Note that no actual physical detector measures a point value; every detector has a finite bandwidth. So every real detector measures over an interval, not a single point value.)

That site doesn't give the equations it's using, so it's useless for this discussion.

For those unfamiliar with Planck's law in different forms, here are the equations. These are the equations that Spectracalc uses.

I am speaking of a point, namely, the peak of the curve. It only occurs for one value, meaning the curve has a global maximum and it is that maximum value, a point on the curve, that I am addressing.

Again: I hope someone can explain physically, not mathematically, why the machine would say the peak is at one wavenumber when wavelength is measured and the simple conversion is used, but a (supposedly) different wavenumber when measured directly in wavenumbers.

The same physical thing, a wave in this case, must have the same value for its characteristic description. It should not matter if we say the wave is described as X or 1/X. It is the same physical thing so how could it have two different wavenumbers?!

If nobody can explain what is going on physically here, then I would be open to suggestions as to where else I might go to get a physical explanation. I am not intending to be a bother. Are there other forums that are more for understanding and less for just doing math? I simply want to understand why the equation would (supposedly) give a different result than that which is physically measured. Does the body know what setting the instrument has?

Here are some Spectracalc results.

 

[davidwinth]

You need to consider the possibility that answering your question in terms of a single point is wrong, that the answer to your question fundamentally involves talking about intervals. In other words, your interpretation of the various forms of the Planck law is incorrect.

But the global maximum of the curve is a single point. That is the only point of interest to me. If the global maximum is a value of X wavelengths, then this is definitionally equal to 1/X wavenumbers. But not according to Planck's law in wavenumber form.

 

[PeterDonis]

For those unfamiliar with Planck's law in different forms

We didn't ask you to give us the equations because we are not familiar with them. We asked to find out whether or not you were familiar with them.

Since you apparently are familiar with the equations, you should be able to use them to calculate the global peaks for yourself, instead of using a website to do your calculations for you. Have you tried that? What do you get when you try it?

the curve has a global maximum and it is that maximum that I am addressing.

What does the global maximum of each curve mean, physically? If you can't answer that, you have no basis for any conclusion about how you should expect the global maximum values of the two curves to be related.

Others have been talking about intervals because that is how we understand, physically, what the two curves mean.

The same physical thing, a wave in this case, must have the same value for its characteristic description.

What does "the same value" mean, physically? How does that translate into the values for the two curves? Again, if you can't answer that, you have no basis for any conclusion about how you should expect the two curves to be related.

Note that neither of these curves is a curve of "the amplitude of the wave", which is what a "characteristic description" of a wave is normally taken to be. So your logic falls apart right from the start, since it assumes that the amplitude of the wave is what the curves are describing.

I simply want to understand why the math gives a different result (supposedly) than that which is physically measured.

You don't know what is physically measured unless you have actually done the experiment. Have you? Or have you looked to see if someone has, and what results they got?

the global maximum of the curve is a single point. That is the only point of interest to me.

And that is your problem: you are only interested in a single point of each curve, but understanding the physical meaning of the value of each curve at even a single point requires you to look at more than just a single point.

 

[PeterDonis]

What does the global maximum of each curve mean, physically?

understanding the physical meaning of the value of each curve at even a single point requires you to look at more than just a single point.

Perhaps this will help: what are the units of each curve? (They're given on the spectralcalc plots you have posted.) What do those units mean, physically?

 

[vanhees71]

It's simply that many people forget that a distribution transforms under change of the variable as a distribution, which is why it is called a distribution. Thinking in terms of "photon number" (though it's a delicate subject concerning Lorentz-invariance properties, but that's not the point here) it's clear that
$$\mathrm{d}N = \mathrm{d} \omega f(\omega) = \mathrm{d} \lambda \tilde{f}(\lambda).$$
Now you need the relation between $\omega$ and $\lambda$, which is
$$\omega = c k =\frac{2 \pi c}{\lambda}$$
and thus
$$\mathrm{d} \omega = \left |\frac{\mathrm{d} \omega}{\mathrm{d} \lambda} \right| \mathrm{d} \lambda =\frac{2 \pi c}{\lambda^2} \mathrm{d} \lambda.$$
So you get
$$\tilde{f}(\lambda) =\frac{\mathrm{d} N}{\mathrm{d} \lambda} = f(\omega) \frac{2 \pi c}{\lambda^2} \mathrm{d} \lambda = f \left (\frac{2 \pi c}{\lambda} \right) \frac{2 \pi c}{\lambda^2}$$

 

[vela]

But the global maximum of the curve is a single point. That is the only point of interest to me. If the global maximum is a value of X wavelengths, then this is definitionally equal to 1/X wavenumbers. But not according to Planck's law in wavenumber form.

There is not one curve here, but two. The two curves don't mean exactly the same thing. I think @PeterDonis's suggestion to look at the units is a good one to see why.