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Max Planck’s Quantum Theory

In December of 1900, a conservative German scientist named Max Planck wrote a paper that included, according to the Nobel Prize winning Physicist Max Born, “the most revolutionary idea which ever has shaken physics.[i]”  For in this paper Planck assumed that light came in little energy packets.  Why did he do this and why was it so revolutionary?  Good questions!  Let’s ask Planck.  No, not with a Ouija board silly.  Why don’t we just read his autobiography and his other papers and see what Planck himself said?  And that is exactly what I did.



Table of Contents

Black Body

The Blackbody Radiation Equation

Planck’s Paper

References


Black Body

From left to right: Nernst, Einstein, Planck, Millikan, and von Laue in 1931.
From left to right: Nernst, Einstein, Planck, Millikan, and von Laue in 1931.

It all started when Planck was just 2 years old and a German scientist named Gustav Kirchhoff came up with a theoretical puzzle about heat and light.  Kirchhoff had just accidentally discovered that elements will absorb exactly the same frequency light that they emit.  In 1860, Kirchhoff published a paper where he imagined a “perfect” object that could completely absorb every frequency of radiation that hit it, and therefore, could emit all frequencies. 

Kirchhoff called this object a black body, and predicted that the amount of light that a black body will emit does not depend on the material but only on the temperature and the frequency of the light.  By the way, don’t get too distracted by the word black in blackbody, it just means that if it is cool it will be perfectly black, not that it can’t emit any light. 

The blackbody radiation question became one of the great-unanswered questions of science and almost every great Physicist of the late 1800s attempted to solve it.  But no one could crack the code for over 30 years.  Then, in 1894, a friend of Planck’s named Wilhelm Wien came up with an expression for blackbody radiation that seemed like it was a pretty good fit and was generally accepted as true. 

The Blackbody Radiation Equation

Max Planck in 1918 by AB Lagrelius & Westphal
Max Planck in 1918 by AB Lagrelius & Westphal

The only problem with Wien’s equation is that it was derived from experiment not theory.  By this time Max Planck was the head of theoretical physics at the University of Berlin and as Germany’s sole theoretician he decided it was important to derive Wien’s equation from basic theories of Physics. 

This was not an easy task but 5 years later, in 1899 Planck triumphantly published his work. Planck was convinced that the case was closed and, for a while, this blackbody radiation equation was called the Planck-Wien law.  However, nature had another idea.  For another friend of Planck’s had found a way to measure low energy infrared waves and found that at relatively low energies, the Planck-Wien equation didn’t work. 

This friend gave Planck a heads up and he quickly made up a new equation, “which, as far as I can see at the moment, fits the observational data.[1]” Planck politely titled this “an improvement of Wien’s equation” although it was a totally new equation (although it looks like Wien’s equation at high frequencies). 

This new Planck’s formula seemed to work perfectly, but Planck wasn’t happy, it wasn’t a law derived from basic laws; it was just “lucky intuition[2]” just like Wien had done 6 years earlier that had no basis in physics principles.  “For this reason, on the very day when I formulated this law, I began to devote myself to the task of investing it with true meaning.[3]” 

Planck recalled, “For six years I had struggled with the blackbody theory.  I knew the problem was fundamental and I knew the answer.  I had to find a theoretical explanation at any cost[4].”  So, in “an act of desperation” Planck used an idea he hated, statistical mechanics.

Statistical mechanics had been around since 1859 when James Clerk Maxwell (of Maxwell’s equations) decided to study gasses by using statistics.  The general idea is that gasses are filled with a ridiculous number of atoms moving in different directions at different speeds.  In order to study them you need to determine what they are doing on average, which is why you need statistics and probability. 

Five years after Maxwell, a 20-year-old physicist named Ludwig Boltzmann read Maxwell’s paper and devoted his life to this research and became Germany’s main champion of the existence of atoms. Max Planck said that for most of his career up to this point he had been, “hostile to the atomic theory which was the foundation of [Boltzmann’s] entire research.[5]

Why did Planck hate statistical mechanics?  It wasn’t the math, it was what it meant about Physics, specifically what it meant about the second law of thermodynamics and entropy. 

See, Planck had gotten his Ph.D (at age 19!) in the second law of thermodynamics.  The second law can be written in many, many, forms but the basic idea is that things cannot become more ordered (or less messy) by themselves.  Physicists call the messiness of a system, the entropy, which is given the letter S for no reason I can see (maybe it was a convenient letter not used for other things).

However, if you believe in statistical mechanics then all of those atoms could randomly become more ordered, it was just really, really, unlikely.  In other words, the second law wasn’t a law as much a statistical certainty. Maxwell elegantly put it this way, “The second law of Thermodynamics has the same degree of truth as the statement that if you throw a cup of water into the sea, you cannot get the same cup of water out again.[6]” To Planck, that statistical certainty didn’t seem good enough.

Planck’s Paper

Max Planck's Quantum Theory
Max Planck was awarded the Nobel Prize for his noted physics researches.

However, desperate times called for desperate measures, and, holding his nose, Planck dived into, “the interrelation of entropy and probability – in other words, to pursue the line of thought inaugurated by Boltzmann.[7]”  Boltzmann had been writing about an idea that atoms in gasses could have different arrangements to produce their average energies, and had given the letter W (for the German word for probability) for the measure of how many different arrangements there could be. 

It made sense that the entropy, or the amount of messiness, S, would relate to the number of ways that one could arrange your materials, W, but how?  Planck recalled, “Since the entropy S is an additive magnitude but the probability W is a multiplicative one, I simply postulated that S = k log W, where k is a universal constant.[8]

This became known as Boltzmann’s entropy equation and k is known as Boltzmann’s constant, despite the fact that it was created and defined by Planck.  In fact, this equation “S=k log W” is actually written on Boltzmann’s tombstone!  Ironically, Boltzmann did not determine “his” constant.  Planck did.

But then Planck had a problem.  He couldn’t seem to get the equation to work without another constraint because if the energy is, “considered to be continuously divisible quantity, this distribution is possible in infinitely many ways.[9]”  He, therefore, imagined that light was made of little bundles of  “energy elements” with an energy that equaled a constant h times the frequency of the light. 

In his paper, Planck stated it clearly, “the most essential point of the whole calculation [is to consider the energy], E to be composed of a very definite number of equal parts[10].” This is the delineation between classical and modern physics.  Let’s take a moment to recognize what a radical and strange idea this is. 

Think of a water wave, or a sound wave.  They do not come in little packages.  They are vibrations of water or molecules where the more the water or molecules vibrate, the more energy the wave has.  Planck was creating a brand new thing in physics, a wave packet!  Five years later Einstein called these packets “quanta of energy” although they are currently called photons.  This is the birth of the quantum.  It is a really, really big deal.

However, Planck did not realize what he had done to the whole nature of Physics.  Quantizing energy was, according to Planck, “a purely formal assumption, and I did not give it much thought.[11]”  Instead, he assumed that he could massage this new theory into classical theory and spent over a decade in the futile attempt to distance himself from his own idea. 

The rest of the scientific world politely ignored Planck’s startling claim and most believed a competing theory that used statistical mechanics but didn’t require energy to be in little packages.  The only problem with the other method is that as the frequency went up you got more and more possible states with no quantum limitations, so that as you went to the ultraviolet range, the radiation went up exponentially, a situation poetically called the ultraviolet catastrophe! 

Some people tried to solve this problem with a fudge factor or by just saying it didn’t work at high frequencies for some unknown reason.  One of the inventors of this theory said it was fine (this is fine) as the catastrophe was happening but just unevenly and very, very slowly.

By the way, Planck thought that man was an idiot, writing a friend that he, “is the model of the theorist as he should not be… [he just] ignores the facts if they don’t fit.[12]”However, at the time a theory that didn’t fit the facts seemed far preferable to a theory with an odd energy packets that didn’t follow the laws of physics.

Even a young college graduate named Albert Einstein had “mixed feelings” about Planck’s paper.  Then in May of 1901, Einstein read a paper about something called the photoelectric effect by Phillip Lenard and it all made sense.  Planck’s equation was not a “formal assumption” it was a fundamental discovery about the nature of light. 

He wrote to his girlfriend that, “I have just read a wonderful paper by Lenard …[and]under the influence of this beautiful piece I am filled with such happiness and joy that I absolutely must share it with you.”  By 1905, Einstein and his “dear kitten” became the first people to actually use Planck’s idea of energy packets to explain what light is. 

Well over a hundred years later, this paper is still considered to be correct and Einstein actually won his Nobel Prize due to the photoelectric effect.   And I will get to that.  But first I want to answer another question. What is the photoelectric effect, why did Lenard study it, and what did Lenard think he was seeing?  That is next time on the Lightning tamers!

References

[i] Born, M “Obituary Notices of Fellows of the Royal Society” Trans of the Roy Soc (1947) Issue 17,p. 167

[1]Planck, M “On an Improvement of Wien’s Equation for the Spectrum” English translation from Haar, D.The Old Quantum Theory (1967) p. 81

[2] Planck, MScientific Autobiography and Other Papers p 41

[3]p 41 “Scientific Autobiography and Other Papers” Max Planck

[4] 1931 letter to R. Wood recounted in p 76 “From X-rays to Quarks: Modern Physicists and Their Discoveries” Emilio Segre.

[5]p 32 “Scientific Autobiography and Other Papers” Max Planck

[6]recalled on p 55 “Einstein and the Quantum” Stone

[7] Planck, M Scientific Autobiography and Other Papers (1947) p. 41

[8] Planck, M Scientific Autobiography and Other Papers (1947) p. 41

[9] Planck, M “On the Theory of the Energy Distribution Law of the Normal Spectrum” English translation by D. HaarThe Old Quantum Theory (1967) p. 85

[10] Planck, M “On the Theory of the Energy Distribution Law of the Normal Spectrum” English translation by D. HaarThe Old Quantum Theory (1967) p. 85

[11] Planck, M referenced in Kantorovich, A Scientific Discovery: Logic and Tinkering (1993) p. 164

[12] Planck, M to Wein Referenced in Stone, A Einstein and the Quantum (2013) p. 102

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