Blackbody Radiation
Discussion
introduction
Newton's laws of motion and universal gravitation, the laws of conservation of energy and momentum, the laws of thermodynamics, and Maxwell's equations for electricity and magnetism were all more or less nearly complete at the end of the 19th century. They describe a universe consisting of bodies moving with clockwork predictability on a stage of absolute space and time. They were used to create the machines that launched two waves of industrial revolution — the first one powered by steam and the second one powered by electric current. They can be used to deliver spacecraft to the ends of the solar system with hyperpinpoint accuracy. They are mathematically consistent in the sense that no one rule would ever violate another. They agree with reality to a high degree of accuracy as tested in experiment after experiment.
At the end of the 19th century, physics appeared to be at an apex. Several people are reported to have said something like this
There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.
This has been attributed to William Thomson, Lord Kelvin (1824–1907) in an address to the British Association for the Advancement of Science in 1900, but I haven't been able to find the primary source. A similar statement was made twice by the GermanAmerican scientist Albert Michelson (1852–1931) as was discussed earlier in this book. It is often reported that Michelson got the idea from Kelvin, but there is little evidence to back this claim up.
At the turn of the century, Kelvin wasn't saying that physics was finished. In fact, I think he was saying quite the opposite. There were two clouds hanging over 19th century physics.
The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds. I. The first came into existence with the undulatory theory of light, and was dealt with by Fresnel and Dr Thomas Young; it involved the question, How could the Earth move through an elastic solid, such as essentially is the luminiferous ether? II. The second is the MaxwellBoltzmann doctrine regarding the partition of energy.
Kelvin is describing two problems with the physics of his time. They are highly technical in nature and not something you could easily describe to your grandmother (unless she had some training in physics). The first one refers to the now discredited theory of the luminiferous ether. The second one describes the inability of electromagnetic theory to adequately predict the characteristics of thermal radiation.
In essence, the first argument went like this. Light is a wave. Waves require a medium. The medium for light was called the luminiferous ether. It must be extremely rigid (since light travels so quickly) and extremely tenuous (since we can't detect its drag). Rigid and tenuous are adjectives that are incompatible (strong yet soft). 19th century physics cannot handle this, therefore 19th century physics is in trouble.
The ray of sunshine that dispersed this dark cloud was the theory of relativity devised by Albert Einstein. The major revelations of this theory were that there is no ether, there is no absolute space, there is no absolute time, mass is not conserved, energy is not conserved, and nothing travels faster than light. For a while, this was the most revolutionary theory in all of physics.
The second dark cloud identified by Kelvin is the subject of the rest of this section and (essentially) most of the rest of this book. I won't breeze through it like I did with the first dark cloud, but I will tell you this, the solution to the problem Kelvin called "the MaxwellBoltzmann doctrine" lead to the most revolutionary theory in all of physics — quantum mechanics. The major revelations of this theory are that all things are both particles and waves at the same time and that nothing can be predicted or known with absolute certainty.
The arrival of these two revolutionary theories divided physics up into two domains. All theories developed before the arrival of relativity and quantum mechanics and any work derived from them are called classical physics. All theories derived from the basic principles of relativity and quantum mechanics are called modern physics. The word modern was chosen since the foundations of these theories were laid in the first three decades of the 20th century. This the era of modern architecture, modern dance, modern jazz, and modern literature. Modern technologies were starting to appear like electric lights, toasters, refrigerators, sewing machines, radios, telephones, movies, phonograph records, airplanes, automobiles, subways, elevators, skyscrapers, synthetic dyes, nylon, celluloid, machine guns, dynamite, aspirin, and psychology. The early 20th century was filled with revolutionary ideas and inventions. Life now seems unimaginable without them. Modern physics was just one aspect of the modern era.
the failures of classical physics
Frayed edges on the tapestry of classical physics leading to modern physics…
 relativity
 no apparent motion through the ether
 precession of the perihelion of mercury
 quantum mechanics
 blackbody radiation and the ultraviolet catastrophe
 photoelectric effect
 discrete atomic spectra and the problem of how atoms manage to exist
 radioactive decay
What we know about blackbody radiation
 the shape of the distribution
 the peak shifts according to Wien's law
 the total power output is described by the StefanBoltzmann law
John Strutt, Lord Rayleigh and James Jeans Ultraviolet Catastrophe
A blackbody is an idealized object which absorbs and emits all frequencies. Classical physics can be used to derive an equation which describes the intensity of blackbody radiation as a function of frequency for a fixed temperature — the result is known as the RayleighJeans law. Although the RayleighJeans law works for low frequencies, it diverges as f^{2}; this divergence for high frequencies is called the ultraviolet catastrophe.
Wilhelm Wien Infrared Catastrophe
In 1896 Wien derived a distribution law of radiation. Planck, who was a colleague of Wien's when he was carrying out this work, later, in 1900, based quantum theory on the fact that Wien's law, while valid at high frequencies, broke down completely at low frequencies.
energy is quantized
Max Planck (1858–1947) Germany. On the Law of Distribution of Energy in the Normal Spectrum. Max Planck. Annalen der Physik 4 (1901): 553.
Proposition…
E = hf
And also (from Einstein later, I think)…
p =  h 
λ 
Let's try to derive the blackbody spectrum.
Planck's law is a formula for the spectral radiance of an object at a given temperature as a function of frequency (L_{f}) or wavelength (L_{λ}). It has dimensions of power per solid angle per area per frequency or power per solid angle per area per wavelength. (Yuck!)
L_{f} =  2hf^{3}  1  ⎡ ⎢ ⎣ 
W  ⎤ ⎥ ⎦ 

c^{2}  e^{hf/kT} − 1  sr m^{2} Hz  
L_{λ} =  2hc^{2}  1  ⎡ ⎢ ⎣ 
W  ⎤ ⎥ ⎦ 

λ^{5}  e^{hc/λkT} − 1  sr m^{2} m 
When these functions are multiplied by the total solid angle of a sphere (4π steradian) we get the spectral irradiance (E_{f} or E_{λ}). This function describes the power per area per frequency or the power per area per wavelength.
E_{f} =  8πhf^{3}  1  ⎡ ⎢ ⎣ 
W  ⎤ ⎥ ⎦ 

c^{2}  e^{hf/kT} − 1  m^{2} Hz  
E_{λ} =  8πhc^{2}  1  ⎡ ⎢ ⎣ 
W  ⎤ ⎥ ⎦ 

λ^{5}  e^{hc/λkT} − 1  m^{2} m 
When either of these functions is integrated over all possible values from zero to infinity, the result is the irradiance or the power per area.
∞  ∞  
E =  ⌠ ⎮ ⌡ 
E_{λ} dλ =  ⌠ ⎮ ⌡ 
E_{f} dλ  =  P 
A  
0  0 
Trust me, the solution looks like this…
P  =  2π^{5}k^{4}  T^{4} 
A  15h^{3}c^{2} 
The pile of constants in front of the temperature is known as Stefan's constant.
σ = 


σ =  5.670 374 419 × 10^{−8} W/m^{2}K^{4}  
Multiplying the irradiance by the area gives us the essence of the StefanBoltzmann law.
P  = σT^{4}  ⇒  P = σAT^{4} 
A 
Apply the first derivative test to the wavelength form of Planck's law to determine the peak wavelength as a function of temperature.
d  E_{λ}(λ_{max}) = 0 
dλ 
Trust me, the solution looks like this…
λ_{max} =  hc  1  
kx  T 
where x is the solution of the transcendental equation…
xe^{x}  − 5 = 0 
e^{x} − 1 
x = 4.965 114 231 744 276 303 698 759 131 322
Combine all the constants together into Wien's constant (or Wien's wavelength constant to distinguish it from the next constant)…
hc  =  (6.626 07 × 10^{−34} J s)(2.997 92 × 10^{8} m/s)  
kx  (1.380 65 × 10^{−23} J/K)(4.965 11)  
b =  2.897 771 955 mm K  
and we get Wien's displacement law (or Wien's wavelength displacement law to distinguish it from the variant that follows it)…
λ_{max} =  b 
T 
Repeat the first derivative test but this time use the frequency form of Planck's law to determine the peak frequency as a function of temperature.
d  E_{f}(f_{max}) = 0 
df 
Trust me again, the solution looks like this…
f_{max} = −  kx  T 
h 
where x is the solution of the transcendental equation…
xe^{x}  − 3 = 0 
e^{x} − 1 
x = 2.821 439 372 122 078 893 403 191 330
Combine all the constants together into Wien's frequency constant…
kx  =  (1.380 65 × 10^{−23} J/K)(2.821 43)  
h  (6.626 07 × 10^{−34} J s)  
b′ =  58.789 257 57 GHz/K  
and we get Wien's frequency displacement law…
f_{max} = b′T
Discuss effective temperature. No object emits a mathematically perfect blackbody radiation spectrum. There will always be lumps in the curve. Set the area under intensitywavelength curve for a real source of radiation equal to the area under the intensitywavelength curve for an ideal blackbody and solve for temperature. The effective temperature of an object is the temperature of an ideal blackbody that would radiate energy at the same rate as the real body. Different parts of the Sun are at different temperatures. When combined, the Sun has an effective temperature of 5772 K.
biographical resource
Hierzu ist es notwendig, U_{N} nicht als eine stetige, unbeschränkt teilbare, sondern als eine discrete, aus einer ganzen Zahl von endlichen gleichen Teilen zusammengesetzte Grösse aufzufassen. Moreover, it is necessary to interpret U_{N} [the total energy of a blackbody radiator] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts.
the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be.
Nobel Prize in Physics 1918 Presentation Speech by A.G. Ekstrand, President of the Royal Swedish Academy of Sciences
Ladies and Gentlemen. The Royal Academy of Sciences has decided to award the Nobel Prize for Physics, for the year 1918, to Geheimrat Dr. Max Planck, professor at Berlin University, for his work on the establishment and development of the theory of elementary quanta. From the time that Kirchhoff enunciated the principle "that the intensity of radiation from a black body is dependent only upon the wavelength of the radiation and the temperature of the radiating body, a relationship worth while investigation", the theoretical treatment of the radiation problem has provided a rich, fertile source of fresh discoveries. It is only necessary here to recall the fertile Doppler principle, and further, the transformation of our  concept of the nature of light as seen now in the electromagnetic theory of light formulated by that great man, Maxwell, the deduction of Stefan's Law by Boltzmann, and Wien's Law of Radiation. Since it was clear, however, that this did not correspond exactly with the reality, but was rather, like a radiation law propounded by Lord Rayleigh, only a special case of the general radiation law, Planck sought for, and in 1900 found, a mathematical formula for the latter, which he derived theoretically later on. The formula contained two constants; one, as was demonstrated, gave the number of molecules in a gram molecule of matter. Planck was also the first to succeed in getting, by means of the said relation, a highly accurate value for the number in question, the socalled Avogadro constant. The other constant, the socalled Planck constant, proved, as it turned out, to be of still greater significance, perhaps, than the first. The product hν, where ν is the frequency of vibration of a radiation, is actually the smallest amount of heat which can be radiated at the vibration frequency ν. This theoretical conclusion stands in very sharp opposition to our earlier concept of the radiation phenomenon. Experience had to provide powerful confirmation, therefore, before Planck's radiation theory could be accepted. In the meantime this theory has had unheardof success….
Source?
 Using statistical mechanics, Planck derived an equation similar to the RayleighJeans equation, but with the adjustable parameter h. Planck found that for 6.63 × 10^{−34} J s, the experimental data could be reproduced. Nevertheless, Planck could not offer a good justification for his assumption of energy quantization. Physicists did not take this energy quantization idea seriously until Einstein invoked a similar assumption to explain the photoelectric effect.
symbology
There's h and then there's ℏ.
symbol  name(s)  values  

h  Planck's constant  6.626 070 15 4.135 667 696 9 
× 10^{−34} J s* × 10^{−15} eV s 

hc  "h c"  1.986 445 857 1 239.841 984 
× 10^{−25} J m eV nm 


"h bar", Dirac's constant, reduced Planck constant 
1.054 571 817 6.582 119 570 
× 10^{−34} J s × 10^{−16} eV s 
Planck units
Here we are near the end of this book and we're talking about the subject that most teachers start a basic physics course with — units. In 1899, at the time when Max Planck first proposed his radical theory of energy quantization, he also proposed building a system of "natural units" (natürliche Maasseinheiten) from a few of the more important constants in physics: the speed of light, the universal gravitational constant, and the two recently identified constants that later came to be known by their discoverers: Planck's constant and Boltzmann's constant. The significance of these quantities is now know to be more than just a way to get the units to work out. The big four fundamental physical constants each tell us something different about the nature of reality.
c = 299 792 458 m/s
The speed of light in a vacuum is a value dictated by nature and thus is a natural unit for speed. It is the universal "speed limit". Nothing may travel faster than the speed of light in a vacuum — not even light itself. Even before we entered the information age, it was recognized that material objects and the photons of electromagnetic radiation are, in essence, carriers of information. The speed of light is then a restriction on the speed at which information may travel. More on information theory later.
G = 6.674 30 × 10^{−11} N m^{2}/kg^{2}
The universal gravitational constant relates massenergy to spacetime curvature. (Although, since general relativity was 15 years away, Planck would not have known this.) It contains in it the natural units for length, mass, and time — the fundamental quantities of mechanics (which, of course, he would have known in 1899). Gravity is obviously an essential characteristic of the universe, which makes the gravitational constant an obvious candidate for one of the fundamental descriptors of reality.
ℏ = 1.054 571 817 × 10^{−34} J s
Planck's constant plays two roles. In its traditional form, h is the proportionality constant that relates frequency and energy for electromagnetic radiation. It is sometimes called the quantum of action. In its reduced form, ℏ is the quantum of angular momentum. The second form is now considered by many to be the more fundamental of the two, but it did not appear until 1930. Whereas the previous two constants had a long and distinguished history. Planck's constant had never been seen before. His revolutionary paper on blackbody radiation wasn't published until 1901 — two years after he proposed this system of natural units. (Can you say "foresight"?)
k = 1.380 649 × 10^{−23} J/K
Boltzmann constant relates energy and temperature. It has the same unit as entropy and determines the quantum of this quantity. Entropy and information are related. The smallest amount of information is the bit — a choice between one of two things (1 or 0, yes or no, true or false, guns or butter, stay together or break up). The quantum of entropy is thus the entropy of a bit S = k ln 2. Surprisingly, Boltzmann himself never tried to determine the constant that now bears his name. Planck needed the value to complete his model of blackbody radiation and had to determine it himself. (Actually, the constant he used was the ratio h/k, but this fact is not so important.) Adding the last value to the list meant that a natural unit for temperature was now available. Again, the amazing thing about this work is that Planck could see its importance in the first place. Ludwig Boltzmann's work on statistical thermodynamics was based on the assumption that atoms exist. In 1899, this still wasn't widely accepted.
The procedure for generating the Planck units is to combine these four fundamental constants in a way that gives an answer with the right unit. If the unit corresponds to the quantity you desire, you've just made a Planck unit. For example, if it ends in meters it must be the Planck length…





This is small beyond comprehension. The next biggest material thing is a proton, the diameter of which is on the order of 10^{−15} m. That's a full 20 orders of magnitude bigger. Think of something that's about 10^{5} m across (100 km). The big island of Hawaii comes to mind. If a proton was blown up to the size of the island of Hawaii, the Planck length would be as big as the original proton.
Next up, the Planck time…





How long does this last? Think of something very fast — a photon. Think of something very small — a proton. How long does it take a photon to cross the diameter of a proton?





We're 20 orders of magnitude short. The universe is 13.8 billion years old. That's about…
t = 13.8 × 10^{9} × 365.25 × 24 × 60 × 60 s 
t = 4.35 × 10^{17} s 
Twenty orders of magnitude smaller than that gives you a millisecond. If the time it took a photon to cross the diameter of a proton was slowed to the point where the photon needed the entirety of time itself to complete its task, the Planck time would still only last a thousandth of a second.
On to the Planck mass…





This one always strikes me as a let down. We're talking 22 μg. That's like a speck of dust. Compare it to an atom of uranium, the heaviest naturally occurring atom…
m = 238 u = 3.95 × 10^{−25} kg
or the heaviest known subatomic particle, the top quark…
m = 173 GeV/c^{2} = 185 u = 3.08 × 10^{−25} kg
Both of these values are about 17 orders of magnitude smaller than the Planck mass. Whereas the Planck length and Planck time seem to represent some lower limit on how finely space and time can be divided, the Planck mass seems to be an upper limit on how big the small things in nature can be. No elementary particle will ever be more massive than the Planck mass.
With what we've got so far, we can create a whole coherent set of units for mechanics: Planck acceleration, Planck force, Planck pressure, Planck density, and so on. We'll do one more fully described calculation — the Planck temperature — and just summarize the rest in a table.





How hot is this? Nothing humans or nature has done recently comes close. The interiors of the hottest stars are close to a billion kelvin (10^{9} K) — 24 orders of magnitude short. The hottest laboratory experiments take place inside large particle accelerators like the Tevatron at Fermilab near Chicago and the Large Hadron Collider (LHC) at CERN near Geneva. Here we're looking at quadrillions of kelvins (10^{15} K) and we're still 18 orders of magnitude short. In contrast, the coldest temperatures ever achieved in the lab are a few hundred picokelvins (10^{−10} K). The entire range of laboratory temperatures achieved so far is an astounding 25 orders of magnitude, but we're still short eight zeros. The Planck temperature is so hot as to be meaningless. As we shall soon see, that's the point.
For the next 50 years or so, Planck's notion of a natural unit system — one derived from physical laws, not accidents of human history — was considered an interesting diversion with little or no meaning. The primary reason for this was probably that quantum theory and general relativity were just too new and unfamiliar. (Relativity did not even exist at the time of Planck's publication.) The physics of the Modern era was a strange world that few understood at first.
Die Quantenmechanik ist sehr Achtung gebietend. Aber eine innere Stimme sagt mir, dass das noch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, dass der Alte nicht würfelt.
Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one". I, at any rate, am convinced that He is not playing dice.
Denn wenn man nicht zunächst über die Quantentheorie entsetzt ist, kann man sie doch unmöglich verstanden haben.
Anyone who is not shocked by quantum theory does not understand it.
There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper, a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics.
Eine neue wissenschaftliche Wahrheit pflegt sich nicht in der Weise durchzusetzen, daß ihre Gegner überzeugt werden und sich als belehrt erklären, sondern vielmehr dadurch, daß die Gegner allmählich aussterben und daß die heranwachsende Generation von vornherein mit der Wahrheit vertraut gemacht ist.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.
The last quote gives you an idea of what eventually happened. People who grew up with the theory applied it in situation after situation and found that it worked. We will end this chapter by addressing the meaning of all of this.
The Planck units have no practical application. No car odometer will be calibrated in Planck lengths, no stopwatch will tick off Planck times, and no thermometer will ever give temperatures as a teeny, tiny fraction of the Planck value. These numbers tell us the limits of physics as we currently know it and maybe even the limit of physics as it could ever be known. That's why it's an important theory.
year  quantity  interpretation  principal scientist 

1954  length  gravitational limit of quantum theory  Oskar Klein 
1955  length  quantum limit of general relativity  John Wheeler 
1965  mass  upper limit on the mass of elementary particles  Moisey Markov 
1966  temperature  upper limit of temperature (absolute hot)  Andrei Sakharov 
1971  mass  lower limit on the mass of a black hole  Stephen Hawking 
1982  density  limiting density of matter  Moisey Markov 
Space and time are generally regarded as smooth and continuous. The number places between any two points is apparently infinite. We pass from one place to another with no sensation of granularity. There is no "screen resolution" to the video game of reality. There is no apparent "frame rate" either. One moment is followed by another with no perceivable jerkiness. Existence does not play itself out like a turn of the century nickelodeon movie. If the universe is some sort of computer simulation (as some have suggested), it is rendered with an apparently infinite level of detail.
"Apparently" is the key word, however. The Planck length is now generally regarded as the lower limit of space. Distances less than this are meaningless. Likewise, the Planck time is the lower limit of time. No detectable change will occur in a period shorter than this. You cannot cut space and time up into infinitely small parts. Eventually, you will get to the point where the notion of subdividing space and time any further becomes meaningless. Eventually there will be found an "atom" of space and an "atom" of time. Recall that atom comes from the Greek ἄ τομος (a tomos) meaning uncuttable.
That matter is quantized should be evident to everyone with even the tiniest bit of education. Who doesn't know of atoms? It is less likely that the average person would know that energy was quantized, but such knowledge isn't considered exotic. Many people know of photons. Matter and energy are quantized, and as a consequence, so too is the stage on which matter and energy act. Space and time are quantized. This is perhaps the greatest meaning that one could extract from Max Planck's little excursion into units.
quantity  symbol  value 

speed of light^{1}  c  3.00 × 10^{8} m/s 
gravitational constant  G  6.67 × 10^{−11} N m^{2}/kg^{2} 
reduced planck constant^{2}  ℏ  1.05 × 10^{−34} J s 
boltzmann constant^{3}  k  1.38 × 10^{−23} J/K 
quantity  expression  value 

length  √(ℏG/c^{3})  1.62 × 10^{−35} m 
mass  √(ℏc/G)  2.18 × 10^{−8} kg 
time  √(ℏG/c^{5})  5.39 × 10^{−44} s 
temperature  √(ℏc^{5}/Gk^{2})  1.42 × 10^{32} K 
quantity  expression  value 

acceleration  √(c^{7}/ℏG)  5.56 × 10^{51} m/s^{2} 
force  c^{4}/G  1.21 × 10^{44} N 
momentum  √(ℏc^{3}/G)  6.52 kg m/s 
energy  √(ℏc^{5}/G)  1.96 × 10^{9} J 
power  c^{5}/G  3.63 × 10^{52} W 
pressure  c^{7}/ℏG^{2}  4.63 × 10^{113} Pa 
density  c^{5}/ℏG^{2}  5.16 × 10^{96} kg/m^{3} 
angular frequency  √(c^{5}/ℏG)  1.85 × 10^{43} rad/s 
Electromagnetic Planck units
What about the natural units of electricity and magnetism? Planck never dealt with the subject that I know of. Your natural choice for a natural unit of electric charge might be the elementary charge…
e = 1.602 176 634 × 10^{−19} C
but this would not be in keeping with the spirit of Planck's work. After all, the Planck mass isn't related to the mass of an electron, proton, or any other physical thing. The Planck quantities are derived from the laws of nature. To that end some have suggested using the coulomb law constant to extend the original system since it's analogous to the universal gravitational constant G.
1  = 8.987 551 792 × 10^{9} N m^{2}/C^{2} 
4πε_{0} 
Including this unwieldy symbol pile gives us the following electromagnetic Planck units. The values for current and voltage look like they could be upper limits. The value for magnetic flux looks like it could be a lower limit. That's nice, I suppose. The value for resistance means… what? Resistance is a bulk property of an object. Subatomic particles or black holes — the kind of things we've been talking about in this section — don't really have a resistance. The value for charge is slightly larger than the elementary charge. Once again, I'm lost. These quantities aren't as easy to interpret as the original Planck units. I don't think anyone is really working on them as a subject of theoretical study.
quantity  expression  value 

electrostatic constant  1/4πε_{0}  8.99 × 10^{9} N m^{2}/C^{2} 
charge  √(4πε_{0}ℏc)  1.88 × 10^{−18} C 
current  √(4πε_{0}c^{6}/G)  3.48 × 10^{25} A 
voltage  √(c^{4}/4πε_{0}G)  1.04 × 10^{27} V 
resistance  1/4πε_{0}c  30.0 Ω 
magnetic flux  √(ℏ/4πε_{0}c)  5.62 × 10^{−17} Wb 