Thermal Motion to Blackbody Radiation Derivation

[Steleo]

Good Day,

I was wondering whether anyone has a reference or can point me in the right direction for a mathematical derivation of how the photons in blackbody radiation (for solids) are generated from thermal motion.

Regards

Max

 

[Drakkith]

Maybe http://en.wikipedia.org/wiki/Blackbody_radiation will have the answer? I saw some equations in there which may help you.

 

[maverick_starstrider]

It's actually a little bit of a wishy-washy question. A common explanation is that moving charge must create electromagnetic waves (i.e. light) and thus the more rapidly atoms are vibrating the more this occurs (ignoring the important aspect of frequency and such). The problem is this is ultimately a classical perspective and a classical solution to this problem very famously gives the wrong answer, it was actually one of the main flaws in classical mechanics that drove the development of quantum mechanics (today we call the shortcoming the "ultraviolet catastrophe"). Thus, in reality black body radiation is a quantum effect and can only really be understood fully in that light.

 

[Steleo]

Thanks for the answer Maverick. I understand what your saying, and that's what I'm after. I've never actually come across the mathematical description in undergrad solid state/quantum mechanics books. I don't know if it's because I'm looking in the wrong places or that there is no nice analytic way to look at the problem.

Cheers

Max

 

[jed clampett]

It's actually a little bit of a wishy-washy question. A common explanation is that moving charge must create electromagnetic waves (i.e. light) and thus the more rapidly atoms are vibrating the more this occurs (ignoring the important aspect of frequency and such). The problem is this is ultimately a classical perspective and a classical solution to this problem very famously gives the wrong answer...

But in quantum mechanics there are still moving charges, or at least there are expectation values of charge density which change with time. If you use classical electrodynamics to calculate the radiation from these moving charges, I think you get the quantum-mechanically correct result for radiation spectrum of a real body.

It's not so much that the laws of classical e-m break down as the fact that the actual vibrations of real bodies are not given by the laws governing billiard balls connected by springs.

 

[maverick_starstrider]

But in quantum mechanics there are still moving charges, or at least there are expectation values of charge density which change with time. If you use classical electrodynamics to calculate the radiation from these moving charges, I think you get the quantum-mechanically correct result for radiation spectrum of a real body.

It's not so much that the laws of classical e-m break down as the fact that the actual vibrations of real bodies are not given by the laws governing billiard balls connected by springs.

well the direct derivation is from the so called "photon gas" which essentially applies statistical mechanics (i.e. an exp(-E/kT) ) to quantized electromagnetic waves (they are quantized by the boundary conditions of the cavity). This is how one arrives at Plank's law. One essentially ignores the kinetic motion origin of the waves, allows all modes consistent with boundary conditions and applies statistical mechanics

 

[jed clampett]

I'm not saying you can't do it that way. I just think you get the same spectrum if you track the actual kinetic motion of the charges (more specifically, the time-varying expectation of total charge density) and apply Maxwell's equation. Which is what the OP asked about.

 

[Sybren]

Have you read Einstein's paper where he rederives Planck's blackbody equation? I have attached the PDF here.

 

[andrewr]

Thanks for the answer Maverick. I understand what your saying, and that's what I'm after. I've never actually come across the mathematical description in undergrad solid state/quantum mechanics books. I don't know if it's because I'm looking in the wrong places or that there is no nice analytic way to look at the problem.

Cheers

Max

There is a fairly simple explanation of it in "Introduction to Quantum Mechanics" second edition; David J. Griffiths. Pearson/Prentice Hall. It's undergraduate read-able, with patience...

pp243 has the punch line equation; but the preceeding chapters on quantum statistical mechanics is where the useful information is regarding the derivation of the blackbody law. I'm not much for the "ultraviolet catastrophe" re-writing of history thinking of today; but within quantum mechanics, spin is taken as an axiom whereas with maxwell's equations -- spin is a facet of circular polarization only. Transverse waves (planar) are taken as a basis set of vectors to creating circularly polarized waves -- so approaching the problem with maxwell's equations *can* come to the same answer but the path to it is obscure.

Hyper physics has a good background on the classic vs. quantum density of states reasoning with simple graphics illustrations; (identical values result from QM and from classical physics on DOS). A blackbody radiator is anything that will adsorb (almost?) all radiation thrown at it. The typical high quality approximation being a box with black soot on the walls and a small pinhole.

In Quantum statistical mechanics, the spin of the electron is implicitly taken to be the 'cause' of the Pauli exclusion principle; it forces anti-symmetrization of the wavefunction so that no two electrons can be in the same state (let alone three,four,five...). Since in a "box" like the pinhole one mentioned -- there is a physical implementation of a macroscopic infinite quantum well, it must fill with discreet energy levels (which depend on the box's dimensions) (classically, harmonics). The number of such discrete energy levels below the average (or thermal) energy of the box will all be filled since it is equally probable (fundamental axiom of QSM) that transitions between energy levels are equally probable given that the total energy is conserved in every transition. (The time it takes to have these transitions occur is NOT necessary to solving the problem and is a curious problem in it's own right.).

Particles emit photons in proportion to their energy (walls, atoms, electrons,etc.), so the problem of blackbody radiation is to determine the number of different ways a specific energy can fit in the box in order to determine the relative intensity of each state (eg: the number of them) -- and some relationship between the physical volume of the box and the tightest possible packing of states (density of states) since entropy is going to cause higher energy to dispersively fill the lowest energy levels available statisticly.

If you study this, and the book I cited shows how, you will get the plank formula. In the bottom, of the formula, you will notice that it has the e**..-1 term associated with boson spin (the spin of light).