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".

The curves of thermal radiation

Thermal radiation is another name for blackbody radiation.  Both terms refer to the emission of light and other non-ionizing radiation from a heated object based solely on the object's temperature.  So sometimes it's also called "temperature radiation."  Also, we're not concerned with the perfect blackbody here, except as a standard for comparision.

So, how else does light get produced other than the heat-it-till-it-glows method? Well, there's fluorescence,  the process that goes on inside a fluorescent bulb, where ultraviolet radiation is absorbed by an atom and visible light is emitted. Fluorescent lights use high voltage to stimulate mercury atoms to emit ultraviolet photons which hit the visible-light emitting phosphor coating on the inside of the tube.  (A related, delayed type of light emission is called phosphorescence.)  LEDs emit light by electroluminescence, where the "bandgap" energy in a semiconductor connected to a DC voltage is turned into light. And then there's the chemiluminescence of fireflies and green-light glow-in-the-dark thingies, but you can click the link to read about that.

For thermal radiation to be visible, the heated body must be hot enough to produce light in the visible spectrum (390 to 700 nanometers, violet wavelength to red wavelength).  Incandescent light bulbs are one example of that, producing light by being heated by an electric current to temperatures around 4500° F.  Stars, including our Sun, are also examples, with surface temperatures in the range of  10,000° F.  But thermal radiation is emitted by any object that's not at absolute zero of temperature--so all objects emit thermal radiation, mostly at very long wavelengths. The Earth, for instance, emits thermal radiation in the infrared region of the electromagnetic spectrum , which goes from 750 nanometers on up to about a million nanometers, or one millimeter. This emission is due to Earth's own internal heat, and from light it absorbs from the sun and re-emits as heat. Our bodies also emit thermal radiation in the infrared region. Our average power output is about 100 watts, another way of saying 2,065 Calories/day.  We are visible not because of radiation we emit but because, like the things around us, we reflect most of the visible light hitting us. This is called diffuse reflection rather than mirror-type (specular) reflection.

The hotter an object is, the shorter the wavelengths of light emitted by the object. (Shorter wavelength means higher frequency.) The webpage I just linked to shows the blackbody curves (spectra) of energy emitted versus wavelength. Optical pyrometers use this spectral relationship--the relationship Max Planck explained by assuming quantized emission of thermal radiation--to measure the temperatures of hot objects such as molten steel.

An aside: Lighting techniques in stage lighting and photography make use of something called "color temperature," which uses the idea of thermal or blackbody radiation as a reference standard, although it associates "warm" colors with what are actually cool objects and "cool" colors with what are actually hot objects.
 

Impulsive waves superposed incessantly part I

In my recent post about the quantum vacuum, I said that nobody can picture a photon.  But I do think we should try to imagine what a photon might look like.

Lately I've been studying three books in particular: Intro to the Theory of Coherence and Polarization, Probability and Stochastic Processes: With A View Toward Applications, and Intro to Laser Physics. Some of the common subjects in these books are stationary random processes, the auto-correlation function and power spectra, and in the case of the two "intro" books, the characteristics of different types of light, such as the Planck spectrum of light from a thermal source (incandescent bulb, the Sun) versus the quasi-monochromatic light output of a laser.

Here is a general description from Intro to Laser Physics of light waves:  "... the light waves emerging from the light source are not a long continuation of harmonic waves, but rather a series of waves of shorter duration.  The reason why light from a light source appears steadily bright, on the other hand, is because these short impulsive waves appear one after the other and are superposed incessantly. The energy of a light wave emitted from an excited atom is a constant (ħω), and the corresponding light waves may be considered a damped oscillation, with the amplitude decreasing in time..."

I like that description, but it does have a problem: if the amplitude is decreasing in time, the light wave ought to be completely extinguished after a certain time period.  The constant energy would not really be constant!  So, instead of saying the light waves are damped oscillations, I would say the motion of an electron in an atom as it emits light is a damped oscillation. 

You can imagine a weight hanging on a spring as an analogy.  Start it oscillating up and down, and it will sooner or later stop oscillating.  That's what a damped oscillation is.  It is interesting (and useful) that for not-too-large initial amplitudes, the frequency of the spring-and-weight system will not decrease. It remains a single frequency wave (sine wave), decreasing in amplitude. The same is true of a pendulum undergoing small oscillations, and this is why a pendulum can be used as a timing device: Its frequency (and therefore its period, or time of a complete oscillation) is constant. Approximately. Also, it does lose energy, so it needs a slight kick occasionally to keep it going.

In the quantum mechanical world, however, energy (E) is essentially the same thing as frequency (ω). They are related by E = ħω, where ħ is Planck's constant. So here is the conundrum: To decrease the energy, doesn't the frequency of the light wave need to decrease, not just the amplitude, since a constant frequency means a constant energy?

The answer is that the energy in a beam of light is dependent on both the frequency and the amplitude of the beam.  A greater amplitude can be interpreted as a greater number of photons.  This means a more intense beam, which means more energy per surface area. One way to achieve this is merely by focusing the beam, as when you use a magnifying glass to focus sunlight. The original beam has a diameter or width equal to that of the magnifying glass. The focused beam brings all those photons together in a very small area, almost just a point of light.  Which is enough concentrated energy to ignite a dry leaf or a piece of paper, or burn your skin.

This kind of increase of amplitude by focusing takes the given beam of light and increases its energy-per-area.  The total energy in the beam is not increased.  Now consider a beam whose energy you can increase at the source.  You can think of doing this with a dimmer switch on an incandescent light bulb, or you can imagine doing it with a beam of red laser light from a helium-neon laser with some kind of brightness control on it.  This would be controlling the number of photons leaving the source.

What about controlling the frequency of photons leaving the source?  This is the other way of increasing or decreasing the energy in the beam at its source.  And a dimmer switch on an incandescent bulb is one way of doing this.  The initial dim red glow from the bulb has fewer photons and each photon has a lower energy than when the bulb is glowing brightly.  If you could tune a red laser’s frequency up into the blue part of the visible spectrum, you could increase the energy in the beam without increasing the number of photons.  There are such things as tunable lasers. 

Next time: incandescent bulbs and Planck’s discovery in 1900 of the quantum harmonic oscillator relation E = ħω, and Einstein’s discovery of photons in 1905 by applying Planck’s relation to light as a way of explaining the photoelectric effect.

Roland Omnès weighs in, not for the last time

"Nothing could be more arid than the principles of quantum mechanics. Its concepts and laws are cast in a blunt, inescapable mathematical form, without a trace of anything intuitive, a total absence of the obviousness we see in the things around us. And yet, this theory penetrates reality to a depth our senses cannot take us. Its laws are universal, and they rule over the world of objects so familiar to us. We, who inhabit this world, cannot make our own vision prevail over those arrogant laws, whose concepts seem to flow from an order higher than the one inspired by the things we can touch, see, and say with ordinary words."

--from page 163 of Quantum Philosophy: Understanding and Interpreting Contemporary Science, by Roland Omnès, professor of physics at the University of Paris XI. Copyright 1999 by Princeton University Press. English translation by Arturo Sangalli.

Illuminating quantum vacuum quotes

Quote One

What is the physical significance of these vacuum-state expectation values ... ?  One thing they indicate is that the electromagnetic vacuum is a stationary state of the field with statistical fluctuations of the electric and magnetic fields.  As far as measurements are concerned, however, it is often argued that the entire universe is evidently bathed in a zero-point electromagnetic field, which can add only some constant amount to [non-vacuum] expectation values … .  Physical measurements will therefore reveal only deviations from the vacuum state.  …  This is especially reasonable in the case of the field Hamiltonian [an “operator” representing the total energy of the field], since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition of the zero of energy.  Moreover, this constant energy in the Hamiltonian obviously commutes with a and a^† [the annihilation and creation operators for photons] and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.         

                

So the argument goes.  However, things are not quite that simple, for in general relativity the zero of energy is not arbitrary.  Furthermore we shall see that it is possible to attribute measurable effects, such as the Casimir force and the Lamb shift, to changes in zero-point energy.  And finally, as discussed in Section 2.6, the zero-point field is not eliminated by dropping its energy from the Hamiltonian.

--Peter Milonni, The Quantum Vacuum (1994), from Section 2.4, titled “Quantization of a Field Mode.”

Quote Two

This energy may be attributed to unavoidable fluctuations in the electromagnetic field. … it is not possible for the vector potential at any point in space to vanish (or take any definite fixed value) for a finite time interval; if the field vanishes at one moment, then its rate of change at that moment cannot take any definite value, including zero.  The energy density has no effect in ordinary laboratory experiments, as it inheres in space itself and space cannot normally be  created or destroyed, but it does affect gravitation, and hence influences the expansion of the universe and the formation of large bodies like galaxy clusters.  Needless to say, an infinite result is not allowed by observation.  Even if we cut off the integral at the highest wave number probed in laboratory experiments, say 10¹⁵ cm^-1, the result is larger than allowed by observation by a factor roughly 10⁵⁶.  The energy due to fluctuations in the electromagnetic field and other bosonic fields can be cancelled by the negative energy of the fluctuations in fermionic fields, but we know of no reason why this cancellation should be exact, or even precise enough to bring the vacuum energy down to a value in line with observation.  Since E₀/Ω [the vacuum energy density] was known to be vastly smaller than the value estimated from vacuum fluctuations at accessible scales, for decades most physicists who thought at all about this problem simply assumed that some fundamental principle would be discovered that imposes on any theory the condition that makes E₀/Ω vanish.  This possibility was ruled out by the discovery in 1998 that the expansion of the universe is accelerating, in a way that indicates a value of E₀/Ω about three times larger than the energy density in matter.  This remains a fundamental problem for modern physics, but can be ignored as long as we do not deal with effects of gravitation.

--Steven Weinberg, Lectures on Quantum Mechanics (2013),  from Section 11.6, titled “Photons.”

A “photon,” by the way, is the same thing as the “quantization of a field mode” in QED (Quantum ElectroDynamics).  Our final illuminating quote compares photons with phonons, which result from quantizing the acoustical field modes in a solid. "Modes" means "normal modes," normally. See the last few paragraphs of my November 19 post.  The theory of the acoustic and electronic behavior of electrons and atoms in solids is known as condensed matter physics or solid state physics.  In contrast to the vacuum, interactions in a solid take place in a medium: the atoms and fields of the material. Which leads us up to quote three ...  –DT

Quote Three

In QED there is, apparently, no medium to provide the interactions which, for the electron in a solid, alter its effective charge.  In relativistic quantum mechanics, however, the remarkable possibility exists of quantum fluctuations in the vacuum.  The process we are considering [an electron interacting with a muon, both of which are leptons] is a typical example of such a vacuum fluctuation.  The photon exchanged between the two leptons can create, from the vacuum, a virtual electron-positron pair.  This pair has only a fleeting existence; it lives on borrowed time, in the sense that the energy necessary to call it into existence, ΔE ≈ 2mc², is only available for a time of order Δt ≈ ħ/2mc²  (we are using the so-called ‘energy-time uncertainty relation,’ which states that there is a spread ΔE in the energy of a quantum system given by ΔE ≥ ħ/Δt, where Δt is the time over which the system’s energy is measured; over short times quite large energy fluctuations are possible).  In relativistic quantum mechanics, therefore, we must beware of thinking of the vacuum, too naively, as being ‘nothing’.  Even when a particle is supposedly free and propagating in the vacuum, the parameters describing it (such as the charge) are changed, from the values appearing in the original one-particle Hamiltonian, to ones which include the effect of its interactions with the virtual particles produced from the vacuum.

--I. J. R. Aitchison and A. J. G. Hey, Gauge Theories in Particle Physics, 2nd ed. (1989), from Section 6.10, titled “Higher-order calculations: renormalization and tests of QED.”