Max Planck

How to use this course

Max Planck is remembered for one constant, \(h\), and one date, 14 December 1900. Behind both lies twenty years of work on the second law of thermodynamics, which he regarded as an absolute law of nature. He came to the quantum while trying to prove that radiation obeys the second law. This course follows that path, then the later work on relativity and statistical physics, alongside the story of a man who led German science through two world wars and lost most of his family along the way.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 08 follow the historical order of Planck’s route to the quantum. Lessons 09 and 10 cover his work after 1900. Formulas are written in modern notation; where Planck’s own form differs, the text says so.

Timeline

Kiel and Munich, 1858–1877

Max Karl Ernst Ludwig Planck was born in Kiel on 23 April 1858. Kiel then belonged to the Duchy of Holstein, under the Danish crown; it passed to Prussia after the wars of the 1860s. The Plancks were a family of scholars, lawyers and Lutheran theologians. His father, Julius Wilhelm Planck, was professor of constitutional law at Kiel; his mother, Emma Patzig, was his father’s second wife.

In 1867 his father accepted a chair in Munich, and the family moved there. Planck entered the Maximilian Gymnasium that May. A teacher named Hermann Müller introduced him to mechanics and to the principle of conservation of energy, which made a lasting impression: here was a law that held everywhere, independent of human beings. The search for such absolute laws became the theme of his scientific life.

Planck was also a gifted pianist and considered a career in music. He chose physics. He entered the University of Munich in October 1874 and studied physics with Philipp von Jolly and Wilhelm Beetz and mathematics with Ludwig Seidel and Gustav Bauer. Jolly reportedly advised him that physics was nearly complete, with little left to discover. Planck replied that he wanted only to understand the foundations, not to make discoveries.

Berlin, the thesis and the long wait, 1877–1885

In October 1877 Planck went to Berlin for a year. He attended lectures by Hermann von Helmholtz and Gustav Kirchhoff, the two leading physicists in Germany, and by the mathematician Karl Weierstrass. He found Helmholtz’s lectures poorly prepared and Kirchhoff’s dry. His real teacher was a book: the papers of Rudolf Clausius on the mechanical theory of heat. Planck admired the clarity with which Clausius formulated the two laws of thermodynamics, and he decided to work on the second.

Back in Munich he wrote his dissertation, Über den zweiten Hauptsatz der mechanischen Wärmetheorie (On the second law of the mechanical theory of heat). He received the doctorate summa cum laude in July 1879, aged 21. The thesis argued that entropy measures the irreversibility of a process, and that the second law applies to all natural processes, not only to heat engines. Nobody paid attention. Helmholtz did not read it, Kirchhoff disagreed with it, and Clausius did not answer his letters.

His habilitation thesis, on equilibrium states of isotropic bodies at different temperatures, was accepted in June 1880, and he became a Privatdozent in Munich: entitled to lecture, unpaid. He lived with his parents for five years, publishing on thermodynamic equilibria, phase changes and solutions. Much of what he found had already been done by Josiah Willard Gibbs in America, whose papers were unknown in Germany until their translation in 1892.

Kiel, Berlin and the radiation problem, 1885–1900

In 1885 Planck was appointed extraordinary professor of theoretical physics at Kiel. In 1887 he married Marie Merck, the daughter of a Munich banker; they had four children: Karl (1888), the twins Emma and Grete (1889), and Erwin (1893). The same year an essay on the principle of conservation of energy won him a prize from the philosophical faculty at Göttingen. He also published a series of papers on the principle of increase of entropy, applying it to chemical equilibrium and dilute solutions.

Kirchhoff died in 1887. After Ludwig Boltzmann and Heinrich Hertz declined the chair, Berlin offered it to Planck in 1889 as an extraordinary professorship; he became ordinary professor in 1892 and director of the new Institute for Theoretical Physics. Theoretical physics was barely recognized as a discipline, and Planck helped make it one. He was elected to the Prussian Academy of Sciences in 1894.

Why radiation?

Around 1894 he turned to the radiation emitted by a hot cavity. Kirchhoff had shown that its spectrum is universal, independent of the walls. The problem was also practical: the Imperial Physical-Technical Institute (PTR) in Charlottenburg was measuring cavity radiation precisely, partly to set standards for electric lighting. For Planck the attraction was the universality. He hoped to show that the approach of radiation to equilibrium is irreversible for reasons of electrodynamics alone, without Boltzmann’s probabilities. Between 1897 and 1899 he published five papers on irreversible radiation processes. Boltzmann pointed out that Maxwell’s equations are time-reversible, so they cannot by themselves produce irreversibility. Planck had to concede. That concession, and new data from the PTR in 1900, set up the events of Lessons 06 and 07.

Entropy and the second law

For the general reader. Heat flows from hot to cold and never back by itself. A dropped glass shatters and never reassembles. The second law of thermodynamics says that a quantity called entropy never decreases in an isolated system. Planck spent his first twenty years as a physicist making this law precise and applying it to chemistry, and he held it to be exact.

From this one deduces, by Carnot’s argument, that every reversible engine between temperatures \(T_1>T_2\) has efficiency \(1-T_2/T_1\), and then Clausius’s inequality for any cycle,

with equality for reversible cycles. Hence \(dS=\delta Q_{\text{rev}}/T\) defines a state function, the entropy, and for any process \(dS\ge\delta Q/T\). For an isolated system \(\Delta S\ge 0\).

Planck’s emphasis: irreversibility

Planck’s thesis drew a sharp line. A process is reversible if the initial state of all bodies involved can be fully restored by some means; otherwise it is irreversible. The increase of entropy is a quantitative measure of irreversibility. Planck rejected the idea, then common, that the second law was only a rule about heat engines; he applied it to every natural process, including chemical reactions.

Derivation. With the reservoir, the total entropy change is \(dS-\delta Q/T\ge 0\), and \(\delta Q=dU+p\,dV\). At constant \(T,p\) this gives \(d(U-TS+pV)\le0\).

Planck used these principles in 1887–1890 for dilute solutions, deriving the lowering of the freezing point and the laws of electrolytic dissociation of Svante Arrhenius. In 1890 he extended Walther Nernst’s theory of diffusion of ions, giving the Nernst–Planck equation for ion flux \(J=-D\left(\nabla c+\frac{zec}{kT}\nabla\phi\right)\), still used in electrochemistry and membrane biophysics.

Cavity radiation before Planck

For the general reader. A small hole in a hot oven glows with a colour that depends only on the temperature, not on what the oven is made of. Kirchhoff proved this in 1859. The task was to find the universal curve of brightness against colour. Thermodynamics alone pinned down a remarkable amount of it before Planck.

Stefan–Boltzmann

Maxwell’s theory gives radiation pressure \(p=u/3\) for isotropic radiation, where \(u=\int_0^\infty u(\nu,T)\,d\nu\) depends only on \(T\). Treat the radiation as a thermodynamic system with \(U=uV\). The identity \(\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V-p\) gives

Wien’s displacement law (1893)

Wilhelm Wien considered radiation in a cavity with a slowly moving mirror. A reversible adiabatic compression shifts each frequency by the Doppler effect and keeps the spectrum black. Combining this with \(TV^{1/3}=\text{const}\) for adiabatic radiation, he showed:

Thermodynamics stops here. Finding \(\varphi\) requires a model of matter and radiation. Wien proposed \(\varphi(x)=\alpha e^{-\beta x}\) in 1896, which fit the measurements of Friedrich Paschen and of Otto Lummer and Ernst Pringsheim at short wavelengths.

October 1900: the radiation formula

For the general reader. By 1900 one formula described the blue end of the spectrum and another the red end, and neither fit the whole. On a Sunday in October, after hearing new infrared data, Planck found a single formula that fit everything. He found it by thinking about entropy, not about light.

Oscillators in a cavity

Planck modelled the walls as charged harmonic oscillators (he called them resonators) of frequency \(\nu\). From Maxwell’s equations he had derived in 1899 the equilibrium relation between the mean oscillator energy \(U\) and the field:

So the problem reduces to the mean energy of one oscillator at temperature \(T\), and thermodynamics links \(U\) to the oscillator entropy \(S(U)\) through \(dS/dU=1/T\).

The second derivative

Planck had found that Wien’s law corresponds to an entropy with a particularly simple second derivative. If \(U=b\nu\,e^{-a\nu/T}\), then \(1/T=-\frac{1}{a\nu}\ln\frac{U}{b\nu}\), and

In early October 1900 Heinrich Rubens told him that his measurements with Ferdinand Kurlbaum at long wavelengths showed \(U\) proportional to \(T\), as John William Strutt, Lord Rayleigh, had proposed that June. If \(U=kT\), then \(dS/dU=k/U\) and \(d^2S/dU^2=-k/U^2\). Planck chose the simplest expression reducing to \(-1/U\) for small \(U\) and \(-1/U^2\) for large \(U\):

Integrate with partial fractions: \(\frac{dS}{dU}=\frac{\alpha}{\beta}\ln\frac{U+\beta}{U}=\frac1T\), so

Wien’s displacement law requires \(U=\nu\,f(\nu/T)\), so \(\beta\propto\nu\). Writing \(\beta=h\nu\) and \(\alpha=k\) gives the modern form.

December 1900: the quantum of action

For the general reader. A good formula was not enough for Planck; he wanted to know why it was true. In eight weeks of what he later called an act of despair, he adopted Boltzmann’s statistical view of entropy, which he had long resisted, and found that the formula follows if energy is handed out in indivisible packets proportional to frequency.

Counting complexions

Take \(N\) oscillators of frequency \(\nu\) with total energy \(U_N=P\varepsilon\), divided into \(P\) equal elements \(\varepsilon\). Planck counted the number of distinct distributions (complexions): the ways to place \(P\) identical elements into \(N\) distinguishable boxes,

With Boltzmann’s relation \(S_N=k\ln W\) and Stirling’s formula for large \(N,P\),

Per oscillator, with \(u=P/N=U/\varepsilon\), this is \(S=k[(1+u)\ln(1+u)-u\ln u]\), exactly the entropy of Exercise 6.2 provided \(\varepsilon=h\nu\).

Derivation. \(\frac1T=\frac{dS}{dU}=\frac{1}{\varepsilon}\frac{dS}{du}=\frac{k}{\varepsilon}\ln\frac{1+u}{u}\). Solving, \(u=1/(e^{\varepsilon/kT}-1)\).

What Planck claimed

Planck presented this to the German Physical Society on 14 December 1900. The constant \(k\), which he introduced and evaluated, is now named after Boltzmann; Boltzmann himself never wrote it. Planck treated the energy elements as a counting device for the resonators. He did not claim that light consists of quanta. That step was taken by Einstein in 1905, and Planck resisted it for years. Einstein also pointed out in 1909 that Planck’s count is not what a consistent classical treatment would give: classical statistics applied to the resonators yields the Rayleigh–Jeans law. The modern derivation uses the Boltzmann factor directly (Exercise 7.2), and Planck’s count is the Bose–Einstein count of indistinguishable quanta.

Constants of nature and natural units

For the general reader. Planck’s formula contained two new constants. From measured radiation he computed both, and from them the number of atoms in a mole and the charge of the electron, more accurately than anyone had before. A year earlier he had noticed that his constant, with gravity and the speed of light, defines units of length, time and mass valid for any civilization in the universe.

From two measurements to two constants

Integrating Planck’s law with \(\int_0^\infty\frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}\):

Kurlbaum’s value of \(\sigma\) and Lummer and Pringsheim’s value of \(b\) gave two equations for \(h\) and \(k\). Planck obtained \(h=6.55\times10^{-27}\) erg s and \(k=1.346\times10^{-16}\) erg/K. Then Avogadro’s number follows from the gas constant, \(N_A=R/k\approx6.175\times10^{23}\), and the elementary charge from the Faraday constant, \(e=F/N_A\approx4.69\times10^{-10}\) esu. The modern value of \(e\) is \(4.803\times10^{-10}\) esu; contemporary estimates from other methods were much less accurate. Planck regarded this agreement, more than the spectral fit, as evidence that his theory was right.

Natural units (1899)

In 1899, before the quantum, Planck noted that his radiation constant together with the gravitational constant \(G\) and the speed of light \(c\) (and \(k\) for temperature) fixes units that do not depend on human conventions. In today’s form, with \(\hbar=h/2\pi\):

Planck wrote his units with \(h\) rather than \(\hbar\), which makes them larger by \(\sqrt{2\pi}\). They are now read as the scale where quantum gravity must matter. Since the redefinition of the SI in 2019, \(h\) is fixed at exactly \(6.626\,070\,15\times10^{-34}\ \mathrm{J\,s}\) and defines the kilogram.

Planck and relativity

For the general reader. When an unknown patent clerk sent Einstein’s relativity paper to the Annalen der Physik in 1905, Planck was one of the first to take it seriously. He lectured on it at once and, in 1906, wrote down the correct equations of motion for a fast particle. He saw in relativity another absolute law of the kind he valued most.

The canonical momentum is \(\mathbf p=\partial L/\partial\mathbf v=\gamma m\mathbf v\), and the Hamiltonian is \(H=\mathbf p\cdot\mathbf v-L=\gamma mc^2+V\). Planck used this form to show that relativity is compatible with the principle of least action, which he regarded as the deepest principle of physics. In 1906 he also gave the theory an early name, Relativtheorie. In 1907 he extended thermodynamics to moving bodies and argued that entropy is the same in all inertial frames; he also showed that absorbed heat adds to inertial mass, consistent with Einstein’s \(E=mc^2\). His doctoral student Max von Laue wrote the first textbook on relativity in 1911.

Phase cells, zero-point energy, Fokker–Planck

For the general reader. For a decade Planck tried to fit the quantum into classical physics with as little damage as possible. These attempts produced two ideas that outlived their purpose: phase space divided into cells of size \(h\), and an energy that remains even at absolute zero. Later, in 1917, he gave the general equation for the probability of a randomly driven system.

Elementary regions of phase space

In his 1906 lectures on heat radiation, Planck reinterpreted the energy elements geometrically. The curves of constant energy of an oscillator \(H=\frac{p^2}{2m}+\frac12m\omega^2x^2\) are ellipses. Planck divided the phase plane into rings of equal area \(h\). This idea, that \(h\) measures a cell in phase space, became the semiclassical quantization rule \(\oint p\,dx=nh\) of Bohr and Arnold Sommerfeld, and it explains the factor \(h^{3N}\) in the classical partition function.

The second theory (1911–1912)

In 1911 Planck proposed that oscillators absorb radiation continuously but emit only in whole quanta. The mean energy then becomes

The theory was abandoned, but zero-point energy was not. Quantum mechanics gives the oscillator levels \((n+\tfrac12)h\nu\), and zero-point motion is measured in the isotope shifts of molecular spectra and in the Casimir effect.

The Fokker–Planck equation (1917)

Adriaan Fokker, in 1914, derived an equation for the distribution of energies of dipoles in a radiation field. Planck gave the general form in 1917. For a probability density \(p(x,t)\) of a process with drift \(A(x)\) and diffusion \(B(x)\):

It governs Brownian motion, laser noise, chemical kinetics and stochastic finance.

The leader of German physics, 1905–1932

After 1900 Planck became the senior figure of German physics. He was president of the German Physical Society from 1905 to 1909 and permanent secretary of the mathematics and natural science section of the Prussian Academy from 1912. He recognized Einstein early, and in 1911 he and Walther Nernst organized the first Solvay Conference in Brussels, on radiation and the quanta. In 1913 the two travelled to Zurich to recruit Einstein, and from 1914 Einstein was in Berlin with a research post free of teaching. The two men became close friends and played chamber music together, Planck at the piano.

His first wife Marie died in 1909. In 1911 he married her niece, Marga von Hösslin; their son Hermann was born that year. At the outbreak of war in 1914 Planck shared the patriotic mood and signed the Manifesto of the 93, which denied German war crimes in Belgium; he later distanced himself from it. The war and its aftermath took three of his children. Karl was killed at Verdun in 1916. Grete died in childbirth in 1917; her twin Emma married the widower and died in childbirth in 1919. Erwin was taken prisoner by the French.

The Nobel Prize in Physics for 1918, awarded in 1919, recognized his discovery of energy quanta. In the 1920s he worked to keep German science alive through inflation and international boycott. Erwin Schrödinger succeeded him in the Berlin chair in 1927. He received the Copley Medal of the Royal Society in 1929, and the German Physical Society created the Max Planck Medal, which it gave first to Planck and Einstein. In 1930 he became president of the Kaiser Wilhelm Society, Germany’s network of research institutes.

Planck was also known for a remark on how new ideas win acceptance: a new scientific truth does not triumph by convincing its opponents, but because they eventually die. Historians of science call this Planck’s principle.

Dictatorship, war and legacy, 1933–1947

When the Nazis took power in 1933, Planck chose to stay and to protect what he could from inside. In May 1933 he met Hitler and argued that forcing Jewish scientists to emigrate would ruin German science. Hitler answered with a tirade, and the meeting achieved nothing. Planck did not resign or protest publicly. In 1935, a year after Fritz Haber died in exile, he held a memorial meeting for Haber at the Kaiser Wilhelm Society, against the wishes of the government, which had forbidden state employees to attend. Supporters of the “Deutsche Physik” movement, led by Johannes Stark and Philipp Lenard, attacked him, Arnold Sommerfeld and Werner Heisenberg as “white Jews” for teaching Einstein’s physics. Under political pressure he left the presidency of the Kaiser Wilhelm Society in 1937.

The war destroyed the rest of his world. In 1943 he and Marga moved to Rogätz, near Magdeburg, to escape the bombing. In February 1944 an air raid destroyed their house in Berlin-Grunewald with his library and papers. His son Erwin, a former state secretary, was arrested after the attempt on Hitler’s life of 20 July 1944, sentenced to death by the People’s Court, and hanged at Plötzensee on 23 January 1945. Planck’s appeals for clemency failed. After the fighting ended in 1945, Allied troops brought the Plancks to Göttingen. There he served as acting president of the Kaiser Wilhelm Society, and he died on 4 October 1947, aged 89. He is buried in the Göttingen city cemetery.

Legacy

• Quantum theory: \(E=h\nu\) is the first equation of quantum physics. Einstein’s photons, Niels Bohr’s atom, Louis de Broglie’s waves and all of quantum mechanics descend from it.
• Units: since 2019 the kilogram is defined by fixing \(h\). The Planck units set the scale of quantum gravity.
• Statistical physics: the Fokker–Planck equation, the Nernst–Planck equation, zero-point energy, phase-space cells.
• Institutions: the Kaiser Wilhelm Society was refounded in 1948 as the Max Planck Society, now with over eighty institutes. The German Physical Society awards the Max Planck Medal annually. The European Space Agency named its cosmic microwave background satellite (launched 2009) after him; it measured the most precise black-body spectrum in the sky.

Sources and further reading

Biography

1. J. J. O’Connor, E. F. Robertson, Max Planck, MacTutor History of Mathematics.
2. Max Planck, Wikipedia.
3. Max Planck Society, Max Planck and the Kaiser Wilhelm Society.
4. Max Planck, biographical, Nobel Prize website.
5. J. L. Heilbron, The Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science, Harvard University Press, 1986; new edition 2000.
6. M. Planck, Wissenschaftliche Selbstbiographie, 1948; English translation Scientific Autobiography and Other Papers, 1949.

Physics

1. M. Planck, Über den zweiten Hauptsatz der mechanischen Wärmetheorie, dissertation, Munich, 1879.
2. M. Planck, Über eine Verbesserung der Wienschen Spektralgleichung, Verh. Dtsch. Phys. Ges. 2 (1900) 202–204.
3. M. Planck, Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum, Verh. Dtsch. Phys. Ges. 2 (1900) 237–245.
4. M. Planck, Über das Gesetz der Energieverteilung im Normalspectrum, Ann. Phys. 4 (1901) 553–563.
5. M. Planck, Das Prinzip der Relativität und die Grundgleichungen der Mechanik, Verh. Dtsch. Phys. Ges. 8 (1906) 136–141.
6. M. Planck, Vorlesungen über die Theorie der Wärmestrahlung, Barth, 1906; 2nd ed. 1913. English translation The Theory of Heat Radiation, Dover.
7. M. Planck, Vorlesungen über Thermodynamik, 1897; English translation Treatise on Thermodynamics, Dover.
8. T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912, Oxford University Press, 1978.
9. R. Baierlein, “Long live effrontery!” Albert Einstein and the birth of quantum theory, arXiv:physics/0512034.
10. Planck’s law and Planck constant, Wikipedia.