A black body is an idealized object, not found in the physical world: it has zero reflectivity. Happily for physicists, actual objects in the real world which approximate the behavior of a black body can be found. These real-world objects approach the ideal specifications of a black body close enough that they can be used in experiments.
When such objects absorb energy, e.g. in the form of light or heat or radio waves, they radiate this energy outward, and so return to an initial, lower-energy state. The question was when, how, and at which rate, the black bodies will emit such radiation.
A simple example is an incandescent lightbulb: its filament absorbs energy, in particular electromagnetic energy created by an electric current; it emits energy in the form of light and heat. The task was to find a mathematical expression of a descriptive law which could predict how, how much, and which type of energy a blackbody would emit, given the amount of energy it absorbed.
Another example is a piece of iron in a blacksmith’s shop. As the iron is heated, it begins to glow. Depending on the temperature, the iron can emit light of different colors. How much light will it emit at different temperatures? Which color of light will it emit?
Wilhelm Wien proposed one early attempt to answer these questions. He published his hypothesis in 1897. Wien’s hypothesis seemed to work for short wavelengths, but its results for longer wavelengths diverged from observational results.
Following Wein’s work, the next, or next significant, attempt to answer this question was the Rayleigh-Jeans law. Developed by Lord Rayleigh and Sir James Jeans and published in 1900, this formula seemed to predict energy emissions not well, but well enough, for longer wavelengths, but not at all for shorter wavelengths: The law’s predictions for the the visible segment of the electromagnetic spectrum were disastrous, and, looking back years later, physicists refer to those results as the “ultraviolet catastrophe.” This problem was the troubling divergence between what was predicted by the Raleigh-Jeans Law and what was observed.
The Rayleigh-Jeans law seemed to have a problem which was the opposite of the problem which Wien’s law had.
Readers detected the failure of the Rayleigh-Jeans law within a few months of its original publication. The failure of Rayleigh and Jeans, combined with the failure of Wilhelm Wien, hinted at a broader problem. Not only did these two particular attempts fail; they pointed to the possibility that Newtonian physics itself was failing.
Newton’s physics was known to be incomplete, and had given way to Newtonian physics, which added other concepts, like Michael Faraday’s Field Theory, to Newton’s original system. But now even the expanded Newtonian system seemed inadequate.
All of this happened quickly. By the end of the year 1900, physicists were at work to find a law to replace the Rayleigh-Jeans law.
Two questions were now on the table: Which laws predict the rate, amount, and timing of blackbody radiation? And what system of physics will reveal those laws if Newtonian physics do not apply in this context?
Before Wilhelm Wien published his work on the question of blackbody radiation, and before the Rayleigh-Jeans Law was published, Max Planck had been working on the same problem. By December 1900, Planck would solve the problem of blackbody radiation.
Planck’s achievement in solving the blackbody problem, however, was overshadowed by the implications of the method he used in solving it. Sometime during the second half of the year 1900, Max Planck discovered quantum mechanics, and he did so more-or-less singlehandedly. At the time, he had a premonition that his solution would have much broader implications than a specific question about blackbody radiation. He was correct.
Of course, Planck’s discovery was not purely independent or unassisted. He used the discoveries of several other physicists, notably Ludwig Boltzmann, and had discussed an early draft of the paper with the DPG (Deutsche Physikalische Gesellschaft: The German Physical Society). So Planck’s breakthrough was almost but not quite singlehanded.
What was this discovery? Planck reexamined the blackbody problem using a different concept of energy. Newton himself didn’t write much about energy, but the Newtonian concept of energy, as developed by later authors, was such that in an equation, the value for energy could be any real number, rational or irrational.
No discussion, however superficial, of quantum mechanics can be simple, easy, or intuitive. Even in Newtonian physics, there is a distinction between work and energy. For the present purposes, that distinction will be largely ignored, because work and energy are measured in the same unit: joules, BTUs, or foot-pounds. For example:
Work = Force * Distance
Or, by substitution:
Work = Mass * Acceleration * Distance
Planck introduced a constant into the Newtonian framework. For obvious reasons, this number became known as the Planck Constant, or Planck’s Constant. It represents the smallest possible quantity of energy in certain situations. No smaller amount of energy can be in those situations, and any larger amount of energy must be a multiple of this number.
In which situations does this quantizing occur? The energy of an electron in an orbit is quantized. But, e.g., a free electron, which is not orbiting a nucleus and is not part of some larger system, is not quantized, and can have energy at any arbitrary level, i.e., the coefficient can be any real number.
In the black body problem, even though the energy released from the black body was usually in the form of light or heat, the energy emission was the result of an electron changing from a higher energy state to a lower energy state, and it is this change which is quantized. Therefore the emission of light or heat from a black body is quantized, because the emission was occasioned by a quantized change in an electron’s energy level.
To say that an electron’s change in energy state is the same as an electron changing its orbit is somewhat misleading, because these orbits cannot be envisioned as circular or elliptical, like the orbits of plants and satellites. For this reason, physicists refer to the electrons as having “orbitals” instead of “orbits.” These orbitals have all kinds of strange shapes. A change in an electron’s energy state is a change in an electron’s orbital.
An electron’s energy level is related not only to an electron’s orbital, but also to the electron’s other variables, e.g., its spin, angular momentum, or nodal structure. It is therefore possible for two electrons with the same orbital to have different energy levels if they differ in these other variables.
Planck’s innovation was to hypothesize that energy couldn’t be delivered in any arbitrary quantity, but rather only in multiples of some unit which represents the smallest possible quantity of energy.
Yet the Newtonian equations for energy and work are structured algebraically so that between any two values for work or energy, a third value can be found. This is so because the values for force, mass, acceleration, and distance are real numbers, whether rational or irrational. Therefore, energy in a Newtonian framework is not quantized, but rather can have an uncountably infinite number of values, and therefore form a smooth curve, and not the step-like curve which would result from quantizing.
How did Planck alter the Newtonian equations so that they would yield quantized results? Mathematicians use the phrase “boundary conditions” to describe the placing of limits on an equation. Many algebraic equations and many differential equations have infinitely many solutions. Boundary conditions limit the domain of an equation. Boundary conditions are not arbitrary, but represent real-world conditions in which an equation is to be applied. A more precise description and explanation of boundary conditions is beyond the scope of the present discussion. It will be left as an exercise for the reader to spend a semester or two in advanced mathematics in a nearby university.
Suffice it to say that Planck introduced his constant and formulated the equations in such a way that the resulting values for energy were discrete and quantized. Planck’s method here was not arbitrary, because it, unlike the work of Wilhelm Weil and Rayleigh and Jeans, was capable of predicting the energy radiated from a blackbody.
The use of boundary conditions might seem like cheating, or like rigging the equation, but such use is justified because this is not a situation of pure mathematics, but rather of applied mathematics. Given confirmed precise data-points, the stipulation of domains is the way to find an equation which describes and predicts the data. It is intrinsic to the nature of the observational sciences, or the natural sciences, that an investigator looks for, or creates, a best-fit line for the known data points, which is a hypothesis which fits with the collected evidence. Such a line of best fit expressed mathematically is an equation.
Planck was reluctant to settle on the idea of quantized energy, knowing that it would end unchallenged universal application of Newtonian physics. Werner Heisenberg writes that Planck was “a conservative personality in all his views.” Eventually, however, Planck persuaded himself that he had enough evidence, had read the evidence correctly, and was ready to publish his results.
Heisenberg reports that Planck continued to try to find some way to harmonize or integrate his results into the Newtonian system, but finally had to abandon this attempt:
Der Gedanke, daß Energie nur in diskreten Energiequanten emittiert und absorbiert werden könnte, war so neu, daß er nicht in den überlieferten Rahmen der Physik eingefügt werden konnte. Ein Versuch Plancks, seine neue Hypothese mit den älteren Vorstellungen der Strahlungslehre zu versöhnen, scheiterte an den wesentlichen Punkten. Es dauerte etwa fünf Jahre, bis der nächste Schritt in der neuen Richtung erfolgen konnte.
By 1901, Planck had addressed the black body problem, and in the process discovered quantum mechanics. There was more to be done: the concept of quantum mechanics would be applied to other problems. Planck had started the ball rolling, but others would continue to expand the realm to which quantum mechanics would be applied.
Moving on from the black body problem, the next area of investigation was the photoelectric effect. Philipp Lenard discovered that the energy of individual emitted electrons was independent of the applied light intensity. Lenard writes:
Es sind aber die Grössen der Anfangsgeschwindigkeiten unabhängig von der Intensität des Lichtes
Writing about Lenard’s work, Bruce Wheaton reports:
Philipp Lenard discovered in 1902 that the maximum velocity with which electrons leave a metal plate after it is illuminated with ultra violet light is independent of the intensity of the light.
While Lenard was able to identify and give evidence of the counterintuitive behavior of light and electricity in the photoelectric effect, he was unable to explain it; he suggested some vague hypotheses which in hindsight were not helpful. Werner Heisenberg writes:
Dieses Mal war es der junge Albert Einstein, ein revolutionärer Genius unter den Physikern, der sich nicht scheute, noch mehr von den alten Begriffen aufzugeben. Einstein fand zwei neue Probleme, bei denen er die Planckschen Vorstellungen mit Erfolg anwenden konnte. Das eine war der sogenannte photoelektrische Effekt, die Aussendung von Elektronen aus Metallen unter dem Einfluß von Licht. Die Experimente, die besonders sorgfältig von Lenard ausgeführt worden waren, hatten gezeigt, daß die Energie der ausgesandten Elektronen nicht von der Intensität des Lichtes abhängt, sondern nur von der Farbe oder, genauer gesagt, von der Frequenz oder der Wellenl&aunl;nge des Lichtes. Dies konnte auf der Grundlage der früheren Strahlungstheorie nicht gedeutet werden. Einstein konnte aber die Beobachtungen erklären, indem er die Plancksche Hypothese durch die Annahme interpretierte, daß das Licht aus sogenannten Lichtquanten, d.h. aus Quanten von Energie bestehe, die sich wie kleine Korpuskeln durch den Raum bewegen. Die Energie des einzelnen Lichtquantums sollte, in Übereinstimmung mit Plancks Annahmen, gleich sein dem Produkt aus der Frequenz des Lichtes und der Planckschen Konstante.
So it turned out that not only the energy of orbiting electrons was subject to quantization. Photon energy is also quantized: the energy of a photon can only occur at certain discrete levels. What Planck had done for electrons in the blackbody problem, Einstein had done for photons in the photoelectric effect.
The third problem concerned specific heat (sometimes called ‘specific heat capacity’) which is the amount of energy required to raise the temperature of a specified mass of a substance by a specified number of degrees, for example, the number of joules needed to raise the temperature of 1 gram of iron by 1 degree Kelvin, or the number of BTUs needed to raise 1 pound of water by 1 degree Fahrenheit. Note that ‘specific heat capacity’ is different from ‘thermal capacity’ or ‘heat capacity’ and only ‘specific heat capacity’ is relevant to the present discussion.
Counterintuitive results, or at least results not consistent with Newtonian systems, resulted when the specific heat of the same substance was varied depending on the starting temperature. Newtonian thought called for the specific heat to be the same for any substance no matter what the starting point might be.
For example, the number of joules needed to raise the temperature of 1 gram of iron 1 degree Kelvin will be different depending on the starting temperature. A certain number of joules is required to raise 1 gram of iron if the iron starts at 3 degrees Kelvin and is raised to 4 degrees Kelvin. A different number of joules is required if the iron starts at 300 degrees Kelvin and is raised to 301 degrees.
Like the blackbody problem and like the photoelectric effect, the problem of specific heat pointed toward quantization, as Werner Heisenberg explains:
Das andere Problem war die spezifische Wärme fester Körper. Die übliche Theorie führte zu Werten für die spezifische Wärme, die zwar mit den Experimenten im Bereich hoher Temperaturen gut übereinstimmten, die aber bei sehr tiefen Temperaturen viel höher als die beobachteten Werte waren. Wieder konnte Einstein zeigen, daß man dieses Verhalten der festen Körper verstehen konnte, indem man die Plancksche Quantentheorie auf die elastischen Schwingungen der Atome im festen Körper anwandte. Diese beiden Ergebnisse stellten einen sehr wichtigen Fortschritt dar, denn sie zeigten die Wirksamkeit der Planckschen Konstante in verschiedenen Erfahrungsbereichen, die gar nicht unmittelbar mit dem Problem der Wärmestrahlung zu tun hatten. Auch enthüllten sie den zutiefst revolutionären Charakter der neuen Hypothese; denn die Einsteinsche Fassung der Quantentheorie hatte zu einer Beschreibung des Lichtes geführt, die völlig verschieden war von der seit Huyghens üblichen Wellenvorstellung. Licht konnte also entweder als eine elektromagnetische Wellenbewegung gedeutet werden, so wie es seit Maxwells Arbeiten und Hertz’ Experimenten angenommen wurde, oder als bestehend aus einzelnen »Lichtquanten« oder »Energiepaketen«, die sich mit hoher Geschwindigkeit durch den Raum bewegen. Aber konnte das Licht beides sein? Einstein wußte natürlich, daß die bekannten Erscheinungen der Beugung und Interferenz nur auf der Grundlage der Wellenvorstellung erklärt werden können. Er konnte auch nicht bestreiten, daß ein zunächst unauflösbarer Widerspruch bestand zwischen der Wellenvorstellung und seiner Lichtquantenhypothese. Einstein versuchte auch gar nicht, den inneren Widerspruch dieser Deutung zu beseitigen. Er nahm den Widerspruch hin als etwas, das vielleicht sehr viel später durch ganz neue Gedankengänge verstanden werden könnte.
Together, these three problems, and their solutions, launched the project of quantum mechanics. All three dealt with situations of matter absorbing energy and emitting it again; in all three cases, the pattern of these energy emissions could not be described or predicted using Newtonian physics; and in all three cases, the solution pointed to the quantized nature of energy in certain circumstances. It is significant that not only was energy quantized in these three situations, but its quantization proceeded by means of Planck’s constant.
Max Planck had discovered not only a solution for the rather narrow niche of the blackbody problem, but rather he had discovered a constant which applied to the entire electromagnetic spectrum, which called into question the universal hegemony of Newtonian physics, which required a re-thinking of how energy is conceptualized, and which eventually required a major change in the concept of light.
Many students will be familiar with a formulation like “sometimes it’s useful to think of light as waves, and sometimes it’s useful to think of light as particles.” This statement, and the various rephrasings of this same thought in different words, applies not only to light, but rather to the entire electromagnetic spectrum: radio waves, thermal radiation, and even heat conduction can justifiably be conceptualized sometimes as waves and sometimes as particles.
The fact that heat conduction is brought under the umbrella of quantum mechanics shows that Planck’s constant and the concept of quantization apply to phenomena which are not purely electromagnetic. This understanding of heat, already indirectly indicated in Ludwig Boltzmann’s work — although Boltzmann himself might not have understood its import — , reveals the broad scope of the eventual applications of quantum mechanics.
The approach in the first decade of the twentieth century was a case-by-case heuristic approach, introducing the needed repairs to the Newtonian system as each set of non-conforming results appeared out of the experiments and measurements made by various physicists.
The quantized solutions to these three problems — black body emissions, the photoelectric effect, and the specific heat of solid bodies — do not properly constitute a complete system of quantum mechanics but rather are merely a sort of “patch” — like a software patch — on Newtonian physics, which allowed the Newtonian conceptual framework to survive, and yet which pointed the way to a thoroughly quantized physics which would be built “from the ground up” as its own discipline and not merely as a revised version of Newtonian physics.
