Offbeat thoughts on the Planck box

One thought is that a cubic-meter box such as I'm currently discussing would support resonant sound waves within human hearing range. The fundamental frequency would have a wavelength twice the length (or width or height, all the same) of the box. That wavelength is two meters, which corresponds to a frequency of 173.5 Hz, which is close to the F below middle C on a piano keyboard.  The fundamental frequency of this F note is 174.6 Hz.  And the wave we are talking about would be a pure sine wave, a single frequency wave.  This is what "normal modes" are: single frequency waves.  If higher-frequency sine waves (with lower "volume" or amplitude as physicists say) called harmonics are added to the fundamental, we can get the sound of a piano or of other instruments.   For a good discussion with simple graphics, see The Physics Classroom website. It also shows visually why the fundamental standing wave or lowest acoustic normal-mode wave is one-half a wavelength instead of a full wavelength.  And it should give you enough info to figure out what harmonics of this fundamental 173.5 Hz sound wave are supported in our cubic meter box.

If we raised the height of the box so that you could stand in it, it would be like a small shower stall, and you could find resonant frequencies by singing different frequencies and listening for the increase in sound volume when you hit the right notes.  This brings up one difference in sound waves and electromagnetic waves in the box: the sound waves need to have a source inside the box (or a nearby source that resonates with the walls of the box). Heating the box doesn't make the whole thing vibrate acoustically, at least not so you could hear it.*  Heating the box does make it produce more electromagnetic waves of higher frequency than it was producing at room temperature.

That leads me to think about processes in the box that are normally not considered when discussing the Planck radiation spectrum.  (I almost wrote "blackbox" radiation, which is actually a good description, too.) So, this will be an example of thinking outside the box about what's inside the box! 

One thing to consider is thermionic emission,  the emission of electrons from the walls of the box due to heating of the walls. Another thing to wonder about is the photoelectric effect:  light hitting a clean metal surface and knocking electrons out of the surface.  If the walls of our platinum box are covered with a soot-like material, then we don't have a clean metal surface.  But if we think of both the thermionic effect and the photoelectric effect working together, then we can well imagine at high temperatures some electrons being ejected from the walls of the box into the open space inside the box. Free electrons!  Yeeha! Come and get your free electrons!

Usually, for both thermionic emission and photoelectric emission, the emitted electrons are not free, because an applied electric field is used to cause the emission in the first place, so that once electrons are emitted they're in the presence of an electric field that makes them accelerate toward a positively charged plate (anode) connected to an external circuit for the electrons to flow through. The negative side of this circuit, the cathode, could be the heated metallic surface from which the electrons are emitted.

Emission of electrons from a hot cathode is the principle behind the old vacuum tubes that have mostly been replaced by transistors, and of old-style TV picture tubes. They're called cathode-ray tubes, CRTs, remember?  I still have one. The cathode-ray technique is also used to generate X-rays for research and medical testing. Using an accelerating voltage in the 10,000 to 100,000 volt range, high-speed electrons hitting atoms in the anode generate X-rays.

Another of my thoughts about the Planck black-body radiation box is related to the type of object it is supposed to simulate: an object, or a surface, or in the usual parlance a body capable of absorbing all the radiation that hits it. This perfect absorber is, by Kirchhoff's law of thermal radiation, also a perfect emitter.  Contrast such a body with "dark matter".  Dark matter seems to be an example of a perfect non-emitter of radiation, which, if Kirchhoff's law applies to it, means it's also a perfect non-absorber.  Opposite of black-body!

And what about a comparison to that other true-black object in physics, the black hole? It does absorb all the radiation impinging on it, and, except for particle-antiparticle Hawking radiation, emits none of the absorbed radiation

Dark matter is postulated to exist in a halo near the borders of some galaxies, and black holes are postulated to be, and experimentally indicated to be, but not positively proven to be, at the centers of some galaxies. Because of this, the mass of a black hole at the center a galaxy can't account for the rotational speed "profile" of the galaxy like dark matter can.  Some scientists still argue against the existence of black holes, and even more are skeptical about the existence of dark matter.

*well, heating does increase the internal vibrations--which are acoustic vibrations or sound waves--in the walls of the box, a related subject I'll discuss later!  These acoustic vibrations in the solid walls are similar to what's happening with the confined electromagnetic waves in the empty space of the box.

EM modes of one-cubic-meter metallic box

Let’s look at a Planck box that has sides equal to one meter and ask what frequencies of “light” constitute the standing waves in such a box. With L = B = H = 1 meter in the frequency formula of my previous post, and of course c = the speed of light = 300,000,000 meters per second, which is divided by two, we have

f  = 150,000,00(n²_x + n²_y + n²_z)^1/2 hertz,

= (n²_x + n²_y + n²_z)^1/2 (1.5 X 10⁸) Hz.

This is meant to be a picture in mathematical form that shows the meaning of the word mode: put in integers for n_x, n_y, and n_z and you have chosen a particular mode, which you can see also corresponds to a particular frequency. But there are several modes for each frequency. For instance, the 1,1,0 mode has the same frequency as the 1,0,1 and 0,1,1 modes. This is the lowest frequency mode that can be supported by this box (for the 1,0,0, and 0,1,0 and 0,0,1 modes, wave motion doesn’t happen).

As you can see, this lowest frequency is 1.5 X 10⁸Hz.  My bad... That's gotta be multiplied by the square root of two, as you can see in the equation when you put in ones for two of the n's. The square root of two is 1.414. Putting that in gives the famous Manhattan area code, formerly the only one for Manhattan, in megahertz

f  = 2.12 X 10⁸ Hz = 212 X 10⁶ Hz = 212 MHz.

This is in the VHF (very high frequency) region where FM radio and lower-frequency TV stations broadcast (in fact this is almost exactly the carrier frequency of channel 13). UHF stations broadcast at higher frequencies, and cell phones transmit and receive at microwave frequencies which are even higher in frequency. All these are far below the frequencies of visible light.

This equation for frequencies in the one-cubic-meter box (approximately a one-cubic-yard box, for those of you not familiar with the meter) shows the normal modes of electromagnetic waves that can exist in the box.  What we really want to know is how such a box can produce electromagnetic (EM) waves in the first place, and what higher-frequency waves--even into the visible spectrum--can be produced.

You remember how, right?  The box is heated to a certain temperature and a spectrum of EM radiation is produced that is uniquely associated with that temperature:  the Planck spectrum. The Planck spectrum is observed when the walls of the box have equal rates of absorption and emission of electromagnetic energy, in other words when thermodynamic equilibrium is established. We’re getting back to thermodynamics now, and will be looking next at how temperature affects the frequencies produced by our cubic meter box.

Good description of a Planck radiation box

“The simplest way to model a radiating body is to regard it as a large number of linear oscillators (on the order of 10²³) performing simple harmonic motion. Since the particles undergoing the oscillations are, in general, charged particles, they will radiate electromagnetic waves. In the case of a cavity in thermal equilibrium, the electromagnetic energy density inside the cavity will equal the energy density of the atomic oscillators situated in the cavity walls. When the walls are raised to a higher temperature, the following events take place: more energy is put into existing oscillator modes by increasing their amplitudes, new modes corresponding to stiffer spring constants (higher frequencies) are excited, and the radiation density in the cavity is increased until a new equilibrium point is reached.”

--Elmer E. Anderson, from page 45 of Modern Physics and Quantum Mechanics, published in 1971 by W. B. Saunders Company.  It’s a good textbook, but I must confess I slightly edited this quotation. I didn't edit the last sentence, and for me that's the main explanatory part of the quote. My copy of the book is a complimentary copy sent by the publisher to Oswald F. “Mike” Schuette of the Physics & Astronomy Department at the U. of South Carolina. Professor Schuette died in 2000, thus allowing graduate students like me to obtain books from his office. We were given permission by Schuette’s family and the department, and it was suggested that we take any books we wanted, first come first served.  I got five or six books, including the transcript of J. Robert Oppenheimer's security hearings of 1954. 

What I like about Elmer's description is that he says increasing the temperature of the box puts more energy into existing modes and creates new modes at higher frequencies.  

I mentioned  in my previous post that a mode is almost the same thing as a single frequency, but not quite. The standing waves in the box, which are also called resonant frequencies or normal modes, are determined by the dimensions of the box.  The same idea in one dimension is shown by standing waves on a piece of string tied at both ends--a guitar or violin string for instance. These are the normal modes of the string and are determined by the length of the string. A single integer, usually expressed by our old friend n, multiplied by the speed of the waves on the string and divided by the twice the length of the string, determines the normal mode frequencies: f = n(v/2L).  

To put it in the language of music, when n=1 we have the fundamental frequency, and for n > 1 we have the harmonics. All the different strangely shaped waveforms produced by musical instruments and other sound sources are made of a linear superposition of pure tones--the fundamental and its harmonics--having different amplitudes. Mathematically, we're talking about Fourier analysis.

The same idea applies to light waves in the Planck box, but we have three integers because we have three dimensions, and a mode is determined by the different values these integers can have, ranging from zero to infinity. And the speed v in the equation above is now c, the speed of light. Our equation for the allowed frequencies is then a little more complicated:



where L, B and H are the box's dimensions in the x, y and z directions, respectively. For a light wave there are two independent polarizations that result in twice as many possible modes.

All of the above is classical physics, not quantum physics.  Planck's quantization of the energy of the waves is where the quantum of quantum mechanics comes in:  E = nhf, where E is energy, f is frequency, h is Planck's constant, and n is an integer that is now called "the number of photons in the mode." So when we say more energy is put into a mode, increasing its amplitude, that means n increases but f remains the same.

In addition, as Mr. Dr. Anderson says, new modes corresponding to higher frequencies are "excited" when the box is heated to a higher temperature.

Can there be different modes with the same energy?  Yes. To use easy-to-write frequencies, let's say we have two photons in a mode of frequency 100,000,000 Hz and one photon in a mode of frequency 200,000,000 Hz in our box. These modes of oscillation of the EM field in the box have the same energy but different frequencies. 

The quantum world would not be so mysterious if it only meant energy is found to exist in discrete quantities rather than as a continuum. What makes the quantum world mysterious is that these discrete-energy entities that we would normally call particles interfere with each other like waves, and this interference is a result of there being a certain probability of finding, say, a photon in a particular place.  The interference of probability amplitudes is what makes quantum mechanics weird!  More on that later, since it's a bit off our current subject of the Planck radiation box.

Fires in Washington, air in Planck box, color of fire

This is off the subject a little, but is still about thermal radiation, although not "thermal" in physicist's sense of the word:  The fires in Washington state.  I heard indirectly through a friend in  Little Rock that a couple named Patty and Freddie have read some of my blog posts, and they live in Washington in one of the areas experiencing the wildfires.  I hope the fires are decreasing in intensity and will soon be under control and extinguished!  Glad to hear the alpacas and other animals survived and came back home, and that your home is still there.

Back on the subject of a box of light, I haven't discussed whether there is air in the box or if it's a vacuum. I just have not even thought about that until recently. It isn't discussed in the books and articles I've read.  But its pretty clear that our Planckian box of radiation does contain air, based on the difficulty and extra expense and lack of need to pump air out of the box.  Oh, yeh, and the box has a hole in it!  For viewing the spectrum of the light inside! So to be precise I must say that the Planck thermal spectrum box is not "empty" because it does contain air at atmospheric pressure.

Air itself can become incandescent when it's sufficiently heated.  This happens with powerful bombs, both of the chemical type and of the nuclear type.  The temperature of air surrounding the detonation briefly reaches thousands of degrees for chemical-based explosives and millions of degrees for nuclear detonations. For comparison, the temperature of the hydrogen and helium in the sun's photosphere is about 10,000 degrees Fahrenheit.  These are all sources of thermal radiation, having approximately the Planck spectrum.

Getting back to the subject of fire, the flames we see from a fire are not due to incandescence.  They're due to combustion of the material involved, and the type of material is what determines the the color of the flames.

Cat plus thermal radiation in a box, etc

Did somebody say this is an obvious but yet unmentioned connection: the Planck box of radiation and the Schrodinger box with a cat in it? Well nobody has mentioned it to me yet.  I thought of it yesterday after editing my previous post.  A cat in a closed box is a source of thermal radiation, the average body temperature of cats being about 101.5 Fahrenheit. The box would have to be insulated so the walls don't radiate away the heat and it just bounces around as thermal radiation inside the box. That's the necessary condition of the Planck box of radiation anyway.

This idea might lead somewhere if the cat is idealized to be a hot body composed of electromagnetic harmonic oscillators: an ideal, thermal, macroscopic solid.  Planck's analysis was for a box containing nothing but thermal radiation. Einstein's analysis in 1916-17 put atoms inside the box, so it was not just the walls of the box producing the radiation but also an enclosed gas of particles with two energy levels that emitted and absorbed radiation. Einstein derived in a simpler way the expression that Planck found in 1900, plus he predicted the existence of stimulated emission of light from an atom, which became the basis for the laser many years later.

So, first "nothing" in the box (Planck), then a gas of atoms (Einstein), and now a cat!  The cat consists of atoms in an amorphous solid state.  Well, now that I think about it, it's not all that exciting, sorry.  The effect is just to change the shape of the box which itself is a hollowed-out solid.  The Planck thermal spectrum and its derivation are independent of the shape of the box.

The question of entropy is something to keep thinking about though. Actually, it was the entropy of the box of radiation that Planck calculated, while Lord Rayleigh and others were trying to calculate the energy spectrum directly.  Planck calculated the energy spectrum by calculating entropy then using the thermodynamic relation between energy and entropy to find the average energy, then in the last step he derived the energy spectrum of box of thermal radiation.

And I'll just mention one source of black-body or thermal radiation that seems pretty nearly ideal and also unpleasant and menacing under the midday August sun: the new asphalt paving I've driven on in several places around Pine Bluff recently.  Some hellacious heat absorption and emission going on there!

But! the iron manhole covers are even better heat absorbers and radiators than black asphalt. In fact, the emissivity of asphalt is only 0.88.  This is a number between zero and one that is the ratio of the thermal radiation the material is emitting at a particular temperature to the thermal radiation an ideal black-body would emit. See the table of emissivities and the infrared photo of the cube with sides made of different materials at the Wikipedia entry on emissivity.  Iron isn't listed but some other substances with surprising values of emissivity higher than asphalt are listed.

Finally, different materials at the same temperature will feel hotter or colder when you touch them--even though they are at the same temp!--because of their different thermal conduction properties.  The iron manhole cover under direct sun in the summer will feel hotter and thus burn you quicker than the asphalt next to it.  The thermal conductivity of iron is about 100 times that of asphalt.
 

Schrödinger’s cat Inside Llewyn Davis

Before getting back to the Planck thermal spectrum and "normal modes" discussion, I'd like to consider the Schrödinger’s cat aspect of Inside Llewyn Davis. Parts of the movie are a superposition of "cat gets out of Gorfein's apt" and "cat doesn't get out of Gorfein's apartment."

We all know how tricky and fun-loving those movie-making Coen Brothers are! They like to throw in references from their earlier movies, and A Serious Man features a classroom scene in which Schrödinger’s cat is discussed. The Coens also like to mess with the time frame of a movie, such as making references in A Serious Man to Santana's Abraxas and Creedence's Cosmo's Factory--albums released in 1970 are discussed in a movie set in 1967.  A similar anachronistic reference in Inside Llewyn Davis, which is set in February 1961, is the movie poster for The Incredible Journey that Llewyn stops and looks at on the morning of his Gaslight gig.  That movie was released in 1963. Perhaps was playing in February of 1964?  Another 3-year anachronism?

The internal time of Inside Llewyn Davis is messed with also. The first scenes showing Llewyn's Gaslight performance and alley encounter are also the last scenes, and in this sense the movie is made to be circular and never-ending, especially with Llewyn's ending line of "au revior," or "to the seeing again" as my Webster's dictionary translates it. Re-watch the movie and you see the last scenes first.

Schrödinger’s cat  is a quantum superposition, or coherent linear combination, of the states "Live Cat" and "Dead Cat." By the logic of quantum mechanics, the cat is both alive AND dead (not alive OR dead) until an act of observation determines its state. It's in a closed box with a vial of cyanide gas that will be broken by a hammer triggered by the radioactive decay of a nucleus that has a 50% probability of decaying in one hour's time. Once the timing starts, the wavefunction for the nucleus is a quantum superposition of the two states "decayed" and "not decayed." See my discussion of August 2011 for a description of quantum superposition in this context. Since its decay determines whether the cat is alive or dead, the cat is in a superposition of live and dead states during this time period.

 One aspect of using a quantum superposition in a movie is that the movie is continually being observed. We see the cat getting out one day and the cat not getting out on another day.  But everything about those two scenes is the same. Llewyn is wearing same clothes, scarf in same position, guitar in hand, and also same apartment scenes with cat waking him up first then his saying "hello?" then hanging around then leaving a note. Different music is playing in the background during the scenes, first classical music (Mozart's Requiem) which may be part of the movie (coming from apartment above or below), and second "Hang Me, Oh Hang Me" as part of the movie soundtrack but not part of the scene. It becomes part of the scene when the scene switches to the Gaslight, and bam! we're back to what we saw at the beginning, plus more than we saw, but we know it's a repeat. And we learn the reason for Llewyn's getting punched and kicked in the alley, although it's still strange that he has no bleeding nose or busted lip after this rather heavy punching.

There are several points in the movie where the timeline could intentionally be thrown off--the video segue near the beginning, the moment of blackout between Llewyn going to bed and being waked up by the cat near the end, and the following audio/video segue of "Hang Me, Oh Hang Me" being in the background as he looks at movie poster then switching to Llewyn singing it at the Gaslight.

Here's the Coens' method of superposing the cat-gets-out and cat-doesn't-get-out "states" in the movie: the repeating of the first scenes at the end. The movie is a loop of time, self-enclosed not in space but in time.  At the end of the movie, we realize that the movie's first scene in real time was when we saw the cat walking down the hall, a video segue from the scene of the Arkansas good-ol'-boy walking away in the alley  Llewyn's waking up after that segue is the first day of the movie. It's repeated in the last day of the movie, not the cat walking down the hall part, but starting with Llewyn being awakened by the cat.  On the first day the cat gets out. On the last day the cat doesn't get out. Then the beginning of the movie becomes the end and the end becomes the beginning.  A self-enclosing superposition in time. Not for the person in the movie like Groundhog Day, but for the viewer.

Why would I spend time writing this instead of writing something important, like trying to get articles or a book published?  As Mr. Cromartie says to Llewyn in the Columbia recording session on Feb. 18, 1961 (a Saturday, which is another Coen oddity), "Take your time. We're here to have fun." Also I'd like to try to interpret Schrödinger’s cat in a new way, such as setting up a closed cat-in-box with a timer so that the experimenter can't look in the box until one cat-hour has passed. A closed system--unobservable--versus one where the experimenter could open the box at any time.  Then what happens if we consider the entropy increase in the closed cat-box during the cat-hour?  Entropy is the logarithm of the number of accessible micro-states...something to think about. Information philosophy is a good place to start.

Planck and the platinum box

I expected to quickly finish up the blackbody radiation discussion and move on to more important subjects.  But there's so much connected with this subject I can't give it up without trying to get a better understanding of it.  It's more important and relevant than I thought.

For one thing, it's connected with the use of "normal modes" that crop up so often in quantum mechanics and in classical vibration theory.  I discussed the significance of normal modes in quantum electrodynamics in the middle of my recent (12 Dec 2014) post called Illuminating Quantum Vacuum Quotes.  Planck's 2nd paper from 1900 on the blackbody radiation formula he discovered is titled, "On the theory of the Energy Distribution Law of the Normal Spectrum." (He first presented it as a talk or colloquium for his colleagues.)

Normal modes are often called characteristic modes, and both terms refer to the description of a vibrating or oscillating system in terms of its independent or non-interacting vibrations or oscillations. Finding the normal modes means describing the system in the simplest way possible. But what exactly is a "mode"?  It's almost the same thing as a single frequency of oscillation, but not quite. 

Let's look at one of the simplest systems that can be described in terms of normal modes of vibration:

|~~~~~~~~~~~~~~~0~~~~~~~~~~~~~0~~~~~~~~~~~~~~|

The curvy lines are springs, the 0s are masses, and the system is confined between two walls and can oscillate along the horizontal direction only, like it's sitting on a frictionless table. The normal modes of this system are single-frequency motions of the masses in unison: one where both masses are moved the same distance to the right or the left and let go (they move in phase with each other), and one where they are moved toward (or away from) each other and let go (out of phase with each other). More complicated motions are linear superpositions (addition of amplitudes of) these two normal modes.

This system is confined to a one dimensional box. You can imagine making it two dimensional by thinking of springs attached to the top and bottom of each mass. This would be enclosed in a square.  Next, make the leap to three dimensions, with springs also attached to the back and front of each mass, and the system enclosed in a cube.  Finally, imagine more masses and more springs forming a large grid or lattice in this cubic box.

Well, that's pretty much what Planck described in his Energy Distribution Law of the Normal Spectrum, but the only thing in the box was radiant energy: light waves of many different frequencies.  Well, most of the frequencies were not in the visible region of the spectrum, so let's say electromagnetic radiation of many different frequencies. What kind of box was it?  A platinum box!  It was a platinum-walled oven, coated on the inside with soot (or maybe iron oxide).  There was a little hole in the oven for experimenters to be able to measure the frequencies and intensities of light in it as the temperature was gradually changed.

Such a box, more often called a cavity, is a perfect emitter of thermal radiation, or as perfect as can be achieved. Remember that a blackbody is a perfect emitter and a perfect absorber?  With radiation coming from a tiny hole in a heated, insulated, metal box, absorption of light from outside is not something that needs to be considered. No radiation is assumed to be absorbed through the small hole, and the emission and absorption occurring inside the box is due to its temperature only. The absorption part of the required equilibrium between  emission and absorption is produced simply from heating the box.

Two more related comments about boxes containing vibrations. An empty box full of sound waves whose frequencies are multiples of the dimensions of the box is another type of model on which the blackbody-box (cavity radiation) spectrum calculation is based.  The "box of sound" has its acoustical normal modes of vibration, just as the "box of light" has its electromagnetic normal modes. Lord Rayleigh (William Strutt), an older contemporary of Planck's, did both these calculations, the box-of-sound then then the box-of-light, before anybody else, but the formula he found in the case of light didn't fit the experimental data.  Planck came along just months later and got it right by using the quantization of energy of the emitters and absorbers (models for atoms) in the walls of the box. These are the sources of the radiation in the box, and the radiation they produce is due entirely to the temperature of the box.

Also, the 3-dimensional box I  mentioned above, with a lattice of equal masses and equally springy springs, is the simplest model of matter in the solid state. This model was first proposed by Einstein in 1907. He used Planck's idea of quantized emission and absorption of energy, but it was not in this case electromagnetic energy.  It was acoustical or sound wave energy! 

Einstein's model was oversimplified in that he assumed all the atoms vibrated at one frequency. He didn't do a normal-mode calculation!  His model nevertheless predicted that the heat capacity of a solid would go to zero as the temperature of the solid approached zero, something that had been observed experimentally but that classical physics didn't predict. The problem was Einstein's model didn't predict the correct rate at which the heat capacity decreased.  As described in the Wikipedia entry on the Einstein Solid:

In Einstein's model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.

The "normal modes" link is a good one to follow to get educated on that subject.
 

Planck's birthday. He'd be 157.

Well, I didn't know it until I looked at the American Association of Physics Teachers calendar on my refrigerator today, but yes, it is Max Carl Ernst Ludwig Planck's birthday!  He's got more middle names than Paul Adrian Maurice Dirac! I wonder who has the record for most middle names?

Yesterday was J. Robert Oppenheimer's birthday, which I already knew but is also noted on the calendar (as is Earth Day).  He'd be 111. Julius is his first name.  He went by Robert, or for some friends, Oppie.  Maxie and Oppie were temperamental opposites.  They died 20 years apart, MCELP in 1947 and JRO in 1967.

Planck won the 1918 Nobel Prize for his work, but Oppenheimer never did work deemed worthy of a Nobel. Of course, Oppenheimer's career took a sharp turn toward technological physics and administration duties during WWII, even though his talent was in theory.  For instance:  "In 1939, working with graduate student Hartland S. Snyder, Oppenheimer discovered a solution of Einstein's equations of general relativity describing the gravitational collapse of a massive star. This solution shows how the star can end its life as a collapsed object. Such objects were later observed and given the name "Black Holes." They are now known to play an important role in the evolution of the universe." From Institute for Advanced Study website.  Black holes, by the way, were named (1960s) long before their presence was observed indirectly (1990s) from the behavior of stars and gas whose orbital motion can only be explained (so far) by an invisible object with a very strong gravitational field.  Observational evidence is now considered to be sufficient to confirm the existence of black holes.

(On second thought, Oppenheimer might have won a Nobel prize for his black hole prediction if he'd lived 10 or so years longer. But maybe not. Hawking's significant work on black holes has not yet resulted in a Nobel for him.  Many people think  he deserves one, and if he does, Oppenheimer and Snyder would have deserved one even more.)

Back to The Planck. One thing to keep in mind about Planck's derivation in 1900 of the correct black-body spectrum formula is that he didn't assume discrete frequencies, he assumed discrete energies, according to the relation E = nhf, where f is not restricted to integers, but because of that little n in there, E is. Because frequencies are not restricted to integers, there is a continuum of frequencies in black-body or thermal radiation emission and absorption. 

Discrete or integer-related frequencies are emitted by isolated atoms, however, as first described theoretically for the hydrogen atom in 1913 by Niels Bohr.

Planck and Bohr were both trying to understand and mathematically describe the physical interactions responsible for already-known electromagnetic spectra. In Planck's case, it was the spectrum of heated solid objects.  In Bohr's case, it was the "line spectrum" of discrete frequencies of light produced by individual hydrogen atoms. They succeeded where others had failed.

Planck in 1900 assumed a quantization of the energy levels in the material (the energy levels of the abstract oscillators of the material).  He did not allow himself to think of light itself as being quantized. Einstein was the first modern physicist to suggest it was necessary to consider light itself to consist of quanta. He made this intellectual leap in 1905, in his theory of how electrons can be ejected from a clean metal surface by ultraviolet light, a process that was already known experimentally as the photoelectric effect.

Bohr's model of the atom also is based on the idea that light itself is emitted and absorbed as electromagnetic quanta.  Bohr showed how discrete energy levels in the atom result from assuming that an electron's orbital angular momentum is quantized and equals integer multiples of Planck's constant h.  He then applied Planck's formula in a new way to calculate how an electron going from one discrete energy level to another would absorb or emit a single frequency of light.

Bohr's model explains the spectrum of isolated hydrogen atoms, such as atoms in a gas discharge tube, where energy levels, and thus the frequencies of light seen in emission and absorption, are widely spaced.  Planck's model applies to solids (or even near-solids such as molten metal), where atoms are packed together, meaning their collective energy levels are packed together. Arising from transitions between these slightly separated energy levels, the frequencies of emission and absorption of light are "packed together" also, giving a continuous spectrum. Graphs of this spectrum for different black-body temperatures can be seen in the next-to-last link in my previous post. You see light--mainly reflected from your surroundings--with this type of spectrum when you're in sunlight or in a room lit by an incandescent bulb. An electric heating element on a stove also produces this type of spectrum. (By the way, how hot do these get?)

My next post will discuss the equivalence of black-body radiation to radiation inside a certain kind of enclosure or "cavity".