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Max Planck and the Foundation of Quantum Mechanics

My writing interests are general, with expertise in science, history, biographies, and how-to topics. I have written over 70 books.

Max Planck circa 1919 sitting at his desk.

Max Planck circa 1919 sitting at his desk.

What Is Quantum Mechanics?

Quantum mechanics is a branch of physics that deals with the submicroscopic world on the atomic and molecular level. Throughout the 1800s, most physicists believed without question the laws of motion put forth by Isaac Newton, but as scientists started looking more closely at phenomena on a very small scale, such as the interaction of atoms and radiation, classical physics began to show its limitations. To bridge the gap between the classical physics of Newton and the submicroscopic world, physicists began to experiment and develop fundamental theories during the first thirty years of the 20th century.

Quantum mechanics attempts to describe the properties of molecules and atoms and their constituents—electrons, protons, neutrons, and other more esoteric particles. It also explains the interaction with atomic particles with radiation, such as, light, X-rays, and gamma rays. The behavior of matter and radiation at the atomic level often seems contrary to our day-to-day experiences, making the theory difficult and abstruse. However, there is no reason we should suppose the behavior of the atomic world should conform to our observations of the familiar large-scale world. For example, one of the fundamental tenets of quantum theory is that matter and radiation can have both a wave and a particle nature—the concept of wave-particle duality.

At the turn of the 20th century a German physicist named Max Planck investigated the radiation that comes from a hot object. Experimentalists were able to measure the energy spectrum of the light radiating from these hot objects, called black bodies; however, theoretical models based on classical physics failed to predict the energy spectrum as the temperature changed. Max Planck made the bold assumption in his theoretical model that the energy emitted was not continuous, but came in small packets, or quantum. This revolution, though not fully understood by Planck at the time, started a revolution in physics which today we call quantum mechanics.

Electromagnetic spectrum. Note the portion of the spectrum we can see with our eyes is a very small part of the total spectrum.

Electromagnetic spectrum. Note the portion of the spectrum we can see with our eyes is a very small part of the total spectrum.

Who Was Max Planck?

Max Planck was born in Kiel, Germany, in 1858, and later his family moved to Munich. He attended Maximilian Gymnasium (high school) in Munich and after graduation attended the University of Munich, where he studied physics and math for three years. After Munich, he transferred to the University of Berlin to study under the physicists Hermann von Helmholtz and Gustav Kirchhoff. According to Planck, “I must confess that the lectures of these men netted me no perceptible gain. It was obvious that Helmholtz never prepared his lectures properly. He spoke haltingly, and would interrupt his discourse to look for the necessary data in his small notebook; moreover, he repeatedly made mistakes in his calculations at the blackboard…” Unhappy with the teaching style of his two major professors, he spent much of his time in personal study, particularly of the work of Rudolf Clausius on thermodynamics.

At age 21 during the summer of 1879, he received his doctoral degree with a thesis titled, “On the Second Principles of Mechanical Heat Theory.” The following year he completed his qualifying dissertation at Munich. His doctoral research lay at the core of his understanding that led him to discover the quantum of action, now known as Planck’s constant, h, in 1900.

Professor at the University of Berlin

With the help of his father, who was a professor of law at Kiel, he was appointed associate professor at the University of Kiel. After the death of his old teacher Kirchhoff, Planck received an appointment at the University of Berlin. In 1892 he was promoted to full professor, a position he held until his retirement in 1928. At Berlin few students clamored to his lectures; theoretical physics was considered inferior to laboratory work. As a result, during his long tenure at Berlin, he had only nine doctoral students in theoretical physics.

The Black Body Radiation Problem

A major problem in physics and in industry at the end of the 19th century was the lack of understanding of how a hot object emits radiation. Experimentalists had measured the energy spectrum that originated from a heated object and found that the intensity of the radiation increased with wavelength up to a maximum value and then fell off as the wavelength increased. For example, the filament in an incandescent lightbulb starts out cold before electricity is applied, but when the switch is flipped, it warms up rapidly to “white hot” where it emits the light we see. However, we only see a portion of the light emitted by the bulb because much of it is emitted in the form of ultraviolet light (higher frequencies, shorter wavelengths) and in infrared light (lower frequencies, longer wavelengths), both regions of the electromagnetic spectrum that our eyes cannot perceive.

Another example of the radiation that comes off a warm body is a campfire. On a camping trip, when you stand facing the fire the front of your body feels warm, while your back feels cold. This is due to the infrared radiation emitted by the hot fire only falling on your front side.

The problem was of practical interest since electric lighting was starting to appear in the larger cities at the end of the 19th century. Scientists and engineers needed to understand the properties of the light emitted by different types of lightbulbs; this was key to building more efficient light sources. From the perspective of pure physics, knowing the mechanisms responsible for the generation of light and the characteristics of that light was fundamental to understanding the nature of matter.

Thomas Edison’s 1880 patent for an electric-lamp (lightbulb).

Thomas Edison’s 1880 patent for an electric-lamp (lightbulb).

What Is a “Black Body?”

To attempt to understand the basic physics of radiation emission from warm bodies, scientists had developed an ideal model of an object they called a “black body,” which was a perfect emitter and perfect receptor of radiation. Though perfect black bodies don’t exist in the real world, they can be approximated in the laboratory, and the concept is useful for theoretical analysis.

To investigate the nature of black body radiation, experimenters constructed ovens with a single exit hole where they placed photoelectric detectors and colored filters to measure the spectral properties of the radiation. As the oven heated up, they were able to measure the amount of energy that emerged over a range of wavelengths for each temperature. To their amazement, what they found was that the type of material used to construct the oven and the geometry of the oven didn’t matter; only a change in temperature of the oven resulted in a change in the energy distribution of the light. So, a black box made of charcoal and shaped like a cylinder produced the same light as a black body shaped like a basketball made of steel, provided both were at the same temperature. This told the researchers that the process of emitting radiation from heated matter is a fundamental property on the atomic level, even though little was known of the inner workings of the atom at the time.

Schematic of an apparatus to measure black radiation from an oven.

Schematic of an apparatus to measure black radiation from an oven.

Classical Physics Failed to Explain Black Body Radiation

At the beginning of the 20th century the laws of physics could not explain the properties of the light that emerged from the black body. The internal structure of the atom was only beginning to be investigated and understood. To make matters worse, the classical theories of light and of heat predicted that a blackened box held at constant temperature would create infinite amounts of luminous energy—this was clearly not a valid description of reality. Perhaps a little overly dramatic, this problem with the theory became known as the “ultraviolet catastrophe.”

In 1896 Wilhelm Wien and his collaborators in the physics department of the German Reichsanstalt constructed an expansive empty cylinder of porcelain and platinum and recorded the color distribution of radiation that escaped from the tube as it was heated. Their research was concentrated at the shorter wavelengths, from violet to the near infrared. At the Technische Hochschule, in Berlin, another of Planck’s associates, Heinrich Rubens, took measurements of the radiation from the oven into the deep infrared part of the spectrum—light with a much longer wavelength than measured by Wien. Several empirical formulas had been concocted that approximately matched the experimental measurements.

Planck’s black body radiation curves for temperatures of 3000K, 4000K, and 5000K. Also shown is the curve derived from classical physics, which is incorrect.

Planck’s black body radiation curves for temperatures of 3000K, 4000K, and 5000K. Also shown is the curve derived from classical physics, which is incorrect.

Planck’s Search for a Solution to the Black Body Problem

Planck reasoned that it should have been possible to use the laws of classical thermodynamics and electromagnetics to give a mathematical expression for the frequency dependence of black body radiation. Planck built a mathematical model of the black body cavity walls as a collection of “monochromatic vibrating resonators,” consisting of a massless spring with an electric charge at its end. Each resonator could emit or absorb light of a particular frequency. The theoretical springs had an infinite range of stiffnesses, allowing them to oscillate at all possible frequencies. The black body radiation is then hypothetically produced by a large number of these bouncing electrons at a wide range of different frequencies. Planck was following a tried-and-true method of analysis in science: build a simple theoretical model of a complex system and use it to determine the properties of the system; in this case, the energy distribution of light from a black box.

Planck’s Equation

As a result of his model analysis, in 1900 Planck developed a relatively simple equation that described the distribution of radiation accurately over the entire range from short wavelengths, ultraviolet light, to the region of long wavelengths of radiation, infrared light. Though Planck’s equation fit the experimentally observed radiation energy density as it varied with the wavelength of the emitted radiation, he could not derive the equation from first principles. Initially, he thought his equation was just a mathematical fluke that fit the data, but he didn’t understand the underlying physics of why the equation matched what was observed. His mathematical “trick” of allowing energy to only be in discrete quantities, rather than continuous, made everything work. Over the next few years, with the help of other scientists like Albert Einstein and Niels Bohr, he developed an understanding of the nature of the emission of radiation from a black body.

As part of developing his formula for the frequency dependance of the emitted radiation, Planck had to introduce a “fudge factor” to make the equation fit the curve. This constant is now called Planck’s constant, symbolized h, and is now recognized as one of the constants of the universe. Planck found the h = 6.55x10-27 erg-sec for the size of the elementary quantum. Planck’s constant is not a pure number. It has units of energy (erg) and time (sec). To illustrate just how small a unit of energy an erg is, consider a penny dropped from a height of five feet—it strikes the floor with an energy of 400,000 ergs!

Planck’s equation is E = nhf, where

  • n is an integer (later called a quantum number);
  • h is Planck’s constant;
  • f is the frequency in Hertz.

Since “n” in Planck’s equation is an integer, this implies that the allowable energy levels of the theoretical oscillators in the black body cavity can only have discrete values rather than vary continuously.

Introduction of the Quantum

On December 14, 1900, Planck gave a lecture to the German Physical Society in Berlin. The 42-year-old scientist presented to his colleagues his results on research into black body radiation. In the lecture he proposed the astonishing idea of the quantum: energy does not exist as a continuous stream; rather, there is a smallest amount of energy that can be divided no further, which he called a quantum, from Latin meaning “how much?” He also assumed that the size of the quantum for any form of electromagnet radiation was directly proportional to the frequency. This meant a quantum of ultraviolet light with higher frequency carries more energy than a quantum of lower frequency infrared light.

He further supposed that energy can only be absorbed or emitted in a single quantum (quanta is plural of quantum). Therefore, when a black body radiates energy, it is not likely to radiate at all wavelengths with equal probability. Since the energy associated with a quantum depends on frequency, at low frequencies less energy is required than at high frequencies.

Planck’s Theory Was Not Immediately Accepted

When Planck published his findings in 1900 in an article titled, "On the Theory of the Energy Distribution Law of the Normal Spectrum,” it was met with little fanfare. Though his theory fit the radiation curve nicely, it introduced the new idea of the energy quantum, but he offered little rationale for its existence in his paper. In the closing paragraph of the paper Planck alludes to his own trepidation regarding the results, promising more work in the future. He wrote: “This hypothesis I have expressed before only in the form that the energy of the radiation is completely ‘randomly’ distributed over the various partial vibrations present in the radiation. I plan to communicate elsewhere in detail the consideration, which have only been sketched here, with all calculations and with a survey of the development of the theory up to the present…” Planck himself struggled with apprehension about his own theory. Was it just a lucky mathematical coincidence or was there some radical new concept at work?

1918 Nobel Prize in Physics

After several other scientists, like Einstein and Niels Bohr, conducted research showing the reality of quantized energy levels at the atomic level, the larger scientific community started to accept the idea. The Nobel Prize in Physics 1918 was awarded to Max Planck "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta." Due to the interruptions caused by World War I, Plank was not awarded the Nobel prize until 1919 and was not able to receive his prize until 1920.

Legacy of Planck’s Constant

Planck’s idea that energy, like matter itself, comes in small packets or quanta eventually was accepted into the world of science. Planck’s constant has become accepted as a fundamental constant of nature, not just on earth, but into the far reaches of the universe. Scientists, such as Niels Bohr and Werner Heisenberg, showed that Planck’s quantum constant, h, determines the sizes of all atomic and subatomic domain interactions.

Today Planck’s constant appears in a wide variety of places in the physical sciences, including: Heisenberg’s Uncertainty Principle, the smallest possible size of transistors that go into thousands of modern electronic devices, the theoretical density of matter at the birth of the universe, and the smallest increment of time in which time has a meaning. When Max Planck died in 1948, Albert Einstein eulogized his old friend, writing, “He showed convincingly that in addition to the atomistic structure of matter there is a kind of atomistic structure of energy…This discovery became the basis of all twentieth century research in physics.”


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This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Doug West