Blackbody Radiation: Plank's Quantum States & Frequencies

[Manaf]

Hi,
I'm just wondering if Plank suggested that the quantum states and frequencies emitted are quantized, then why do we see a "continuous" spectrum of the BB radiation, not just discrete bands?

 

[lalbatros]

Manaf,

Consider some electromagnetic radiation at a given frequency f .

From the classical point of view, the intensity of this radiation (I in W/m²) can be any positive number.

In quantum mechanics, a EM radiation is a flux of discrete lumps of energy, called photons. Therefore, you can define the photon flux N in photons/s/m². Quantum mechanics tells us also that the energy of these lumps is given by the Planck constant: E = h f . And therefore, the intensity of a radiation is now quantized: I = N h f, where N is the number of photons per second and meter² (photons/s/m²).

When the intensity of radiation is high, like the light in our rooms, the flux of photons (N) is so high that everything looks classical. But if you equip yourself with very sensitive detectors and if you decrease very much the intensity of light (radiation), then it is possible to see the photons individually.

Now concerning the BB 'continuous' spectrum. The spectrum is determined by the possible frequencies that could live in the box. This is related to the size of the box.

In classical physics, assuming the box is a conductor, the electric field must vanish on the wall of the box. The result is that the wavelengths of the radiations must be some integer fraction (i) of the size of the box (L). And therefore, the possible frequencies have a discrete spectrum: f = i c/L (c is the speed of light). This may look a bit more complicated if the full geometry is taken into account.

Consider now a box of 1 meter. Since c = 300 000 000 m/s, you get the following frequency spectrum in the box: f = i * 300 MHz. This means that the classical spectrum is a set of frequencies separated by 300 MHz. This may look like a big frequency gap. But you should compare this frequency gap with the observed frequency range. Looking in the visible spectrum, this is really a very small frequency gap: visible frequencies are between 440 THz and 750 THz. Therefore, the frequency gap plays no role in the visible spectrum (say temperatures above 1000°C), since this represents less than 1/100000000 from the width of the visible spectrum.

If you now consider lower temperatures, then the emission spectrum may shift to much lower frequencies, and then the classical discreteness of the frequency spectrum will be seen. And the size and even the shape of the box will play a big role.

In quantum physics, the spectrum of the BB radiation is exactly the same as in classical physics. By the word 'spectrum', I mean the set of possible frequencies. But the difference is in the intensities. The famous UV catastrophe disappears once the photon assumption and the Planck formula are taken into account. This means that in the ultraviolet part of the spectrum quantum physics explains the observed spectrum, while classical physics gave a divergence. This difference between QP and CP simply reflect the existence of photons.

And there is much more to say. As you know, the Planck formula has been soon applied by Einstein and others to electrons and particles. The energy levels of electrons in the atoms are explained again by the Planck formula. But now, the discreteness of the frequency set is a consequence of the Planck formula. Indeed, electrons also behave as waves (with a certain frequency and energy related by E=hf). And therefore, a "resonnance condition" arises, similar to what I explained for the BB, not all frequencies are possible. And a big difference with the BB is that, for atoms, the frequency gap plays a big role, it explains the emission spectra for example.

Michel

 

[Manaf]

thanks..
1- I've understood from you is that (correct me plese if I am worng) the BB "thermal" radiation is not continuous in the strict sense, but rather a discrete spectrus with too small gaps between emitted frequencies.

2- BB radiation and emission spectra in terms of the production: BB is by oscillation and emission spectra are by transition from higher to lower energy states..are they same thing? if not, what is the difference?

 

[lalbatros]

Manaf,

Concerning your point 1)
Your are right, but additionally, it is important to realize that in classical physics, elecromagnetic wave are waves. Therefore, in a finite box, the radiations do have a discrete spectrum even in classical physics. The news from quantum physics is that the intensity goes by lumps (photons). And this was absolutely needed to understand the BB radiation.

So, the BB spectrum is discrete but the frequencies are -usually- very close to each other. However, for very low frequencies, or very small boxes, or very low temperature, the discreteness can be observed. There is a general rule for the distribution of energy in any system, that always applies, called the Gibbs distribution.

Conversely, the electrons are particles in classical physics. But it was understood that they are also waves, like photons. Therefrom comes the discrete spectra in atoms, for example. For electrons, or particles, this is a consequence of the quantum physics.

Concerning your point 2)
I don't see precisely what you mean.

The thermodynamic equilibrium applies in any circumstance. The fundamental distribution function is the Gibbs distribution. It applies to the BB radiation as well as to electrons distribution in metal or semiconductors, or to the matter in a neutron star, or to molecules in a gas ...

The way energy is exchanged does not matter, as long as it is efficient enough to allow to the system to reach the equilibrium.
Note that what you call 'oscillations' could be translated in emission and absorption processes with the wall of the box, in quantum description. Therefore, I would say that transitions are always involved.

This is a point that was not clear at the time of Max Planck (as I understood): was the BB radiation related to the electrons in the wall of the the "gas" of photons, ... Interrestingly, you asked a related question: learning is partly walking through history.

Michel