Max Born

How to use this course

Max Born’s name is attached to more everyday tools of physics than almost anyone else’s: the Born rule, the Born approximation, the Born–Oppenheimer approximation, Born–von Kármán boundary conditions, the Born–Haber cycle, Born–Infeld theory. He coined the phrase “quantum mechanics”, recognized Heisenberg’s multiplication rule as matrix algebra, and gave the wave function its meaning as a probability amplitude. He also trained or hosted a remarkable share of the physicists who built twentieth-century physics. This course presents the physics alongside a life that ran from imperial Germany through exile to the West German peace movement.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations in modern notation and exercises with folded solutions. Lessons 04, 05 and 10 cover the crystal physics that Born regarded as his main field. Lessons 06 to 08 cover his part in quantum mechanics. Lesson 09 covers his attempt at a finite classical electron. The Heisenberg course on this site treats the Umdeutung paper and the oscillator in detail; Lesson 06 here concentrates on what Born and Pascual Jordan added.

Timeline

Breslau, 1882–1904

Max Born was born in Breslau, the capital of Prussian Silesia, on 11 December 1882, into an assimilated Jewish family. His father, Gustav Born, was an anatomist and embryologist at the University of Breslau, known for experimental work on embryonic development. His mother, Margarethe Kauffmann, came from a family of Silesian textile industrialists; she died in August 1886, when Max was not yet four. His father remarried in 1890. The household was comfortable and intellectual, and the boy grew up around scientists and musicians.

He attended the König-Wilhelm-Gymnasium in Breslau, where the curriculum was classical. He was not a prodigy and later described his school years without enthusiasm. His father advised him to sample widely at university before choosing a subject, and Born followed that advice.

Wandering student

He entered the University of Breslau in 1901 and took courses in physics, chemistry, zoology, philosophy, logic, astronomy and mathematics. The mathematician Jakob Rosanes introduced him to matrix calculus, a detail he recalled when matrices returned to his life in 1925. German students commonly spent semesters at other universities. Born went to Heidelberg in the summer of 1902 and to Zurich in the summer of 1903, where Adolf Hurwitz’s lectures on elliptic functions gave him his first taste of advanced analysis. Two Breslau friends, Otto Toeplitz and Ernst Hellinger, told him that Göttingen was the place for mathematics. He arrived there in April 1904.

Göttingen and the rigid electron, 1904–1914

Göttingen in 1904 had David Hilbert, Hermann Minkowski and Felix Klein in mathematics, Carl Runge in applied mathematics, Karl Schwarzschild in astronomy and Woldemar Voigt in physics. Born became Hilbert’s assistant, writing up lecture notes, and joined Hilbert and Minkowski’s seminar on the electron theory of Hendrik Lorentz. Mathematics and physics in his mind were never separate subjects after this.

He had a difficult relationship with Klein, whom he annoyed by attending his seminar unprepared. To avoid Klein in the oral examination he took astronomy, with Schwarzschild, as one of his subjects. In 1906 Born nonetheless won the faculty’s prize for a study of the stability of elastic wires and tapes, which combined variational calculus with careful experiments. The essay became his doctoral thesis, supervised by Runge.

Cambridge, Breslau and Minkowski

After a short period of military service, he spent six months in 1907 at Gonville and Caius College, Cambridge, attending the lectures of Joseph Larmor and the laboratory of J. J. Thomson. Back in Breslau he worked with the experimentalists Otto Lummer and Ernst Pringsheim. Reading Einstein’s papers on relativity, he worked out his own approach to the relativistic electron and sent it to Minkowski, who invited him to Göttingen as a collaborator. Minkowski died of appendicitis on 12 January 1909, only weeks after Born’s arrival. Born edited his physics papers for publication.

Born’s 1909 paper on the rigid electron defined what is now called Born rigidity: a motion of an extended body is rigid if the distance between neighbouring points, measured in their momentary rest frame, never changes. Paul Ehrenfest soon showed that a disk cannot be set spinning in a Born-rigid way, the Ehrenfest paradox. On this work Born habilitated in 1909, with Voigt as sponsor, and became a Privatdozent. His trial lecture of 23 October 1909 was on J. J. Thomson’s model of the atom.

Lattices, Chicago and marriage

In 1912 Born and Theodore von Kármán published the theory of lattice vibrations that is the subject of Lesson 04. The same year Albert Michelson invited him to lecture on relativity in Chicago. On 2 August 1913 he married Hedwig (Hedi) Ehrenberg, daughter of the Göttingen jurist Victor Ehrenberg. They had three children: Irene (1914), Margarethe (1915) and Gustav (1921).

Berlin, Frankfurt and the return, 1914–1921

In 1915 Born became extraordinary professor in Berlin, appointed to take some of Max Planck’s teaching. He arrived as war began to dominate everything. He served in the army’s artillery testing commission, working on sound ranging, the location of enemy guns from the arrival times of their reports at several microphones. His book Dynamik der Kristallgitter appeared in 1915.

Berlin gave him a lifelong friendship with Albert Einstein. The two played music together, Einstein on the violin and Born at the piano, and argued about physics and politics. Their letters, from 1916 to 1955, were published with Born’s commentary as The Born–Einstein Letters. With Alfred Landé, Born computed the energy and compressibility of ionic crystals (Lesson 05), and in 1919, after conversations with Fritz Haber, he linked lattice energy to thermochemistry.

Frankfurt

In 1919 Max von Laue, who held the chair of theoretical physics at Frankfurt am Main, wanted to be in Berlin near Planck. The two men exchanged positions, and from April 1919 Born was ordinary professor and director of the Frankfurt institute. His assistant was Otto Stern, who began there the molecular-beam experiments that led to the Stern–Gerlach experiment. Born, a theorist, found himself running a laboratory and measuring the mean free path of silver atoms.

Back to Göttingen

In 1921 Göttingen offered Born the chair of theoretical physics, succeeding Peter Debye. Born made his acceptance conditional on the appointment of his friend James Franck as professor of experimental physics. Together with Robert Pohl, they made Göttingen one of the three centres of the new atomic physics, with Copenhagen and Munich. Born was 38.

Lattice dynamics: Born and von Kármán

For the general reader. A crystal is a regular array of atoms held by springs. When it is warm, the atoms vibrate, and their vibrations travel through the crystal as waves. In 1912 Born and Theodore von Kármán worked out exactly which waves a lattice can carry and how many there are. Their method is now the opening chapter of every solid-state physics course.

Einstein’s model of 1907 gave every atom the same frequency. It explained why heat capacities fall at low temperature but predicted an exponential fall, while measurements by Walther Nernst’s group showed a slower one. In 1912 two papers appeared within months. Peter Debye treated the crystal as an elastic continuum with a frequency cutoff. Born and von Kármán, in Über Schwingungen in Raumgittern (Physikalische Zeitschrift, 1912), solved the discrete lattice exactly.

The monatomic chain

Take atoms of mass \(M\) at positions \(na\), joined to their neighbours by springs of constant \(K\). If \(u_n\) is the displacement of atom \(n\),

Try \(u_n=A\,e^{i(kna-\omega t)}\). Then \(-M\omega^2=K(e^{ika}+e^{-ika}-2)=-4K\sin^2(ka/2)\), so

Wavenumbers \(k\) and \(k+2\pi/a\) give the same motion of the atoms, so all distinct waves have \(-\pi/a<k\le\pi/a\), the first Brillouin zone.

Periodic boundary conditions

A real crystal has surfaces, which make exact solution awkward. Born and von Kármán imposed instead \(u_{n+N}=u_n\), as if the chain were closed into a ring. This allows only \(k=2\pi m/(Na)\) with \(N\) distinct values of \(m\). For a large crystal, bulk properties do not depend on the choice of boundary condition, and the periodic one keeps the full translation symmetry. Felix Bloch used the same device for electrons in 1928. It is now universally called the Born–von Kármán boundary condition.

At high temperature each term tends to \(k_B\), giving the Dulong–Petit value \(3rNk_B\). At low temperature only acoustic modes with \(\hbar\omega\lesssim k_BT\) contribute, their number grows as \(T^3\), and \(C_V\propto T^3\), the same law that Debye found from the continuum model.

Two atoms per cell

For a chain alternating masses \(M_1\) and \(M_2\), all springs \(K\), with cell length \(a\), the same substitution gives a \(2\times2\) eigenvalue problem and

The lower branch is acoustic. In the upper, optical branch the two sublattices move against each other; in an ionic crystal this motion carries an oscillating dipole and absorbs infrared light, which is why it is called optical. Born’s theory of the infrared “Reststrahlen” of salts rests on it.

Lattice energy and the Born–Haber cycle

For the general reader. How strongly is a grain of salt held together? The energy needed to pull a crystal apart into separate ions cannot be measured directly. Born calculated it from electrostatics and the crystal’s resistance to squeezing. Then, with Fritz Haber, he showed how to obtain the same number from heats of reaction measured by chemists. The agreement between the two was early proof that salts really are built from ions.

The Born–Landé model

In an ionic crystal of the rock-salt type, with ions of charges \(\pm ze\) and nearest-neighbour distance \(r\), the electrostatic energy per ion pair is \(-\mathcal M z^2e^2/(4\pi\varepsilon_0 r)\). The Madelung constant \(\mathcal M\) is a lattice sum depending only on geometry; for rock salt \(\mathcal M=1.7476\). Alone, this attraction would collapse the crystal. Born and Alfred Landé (1918) added a short-range repulsion \(B/r^n\) to represent the resistance of the closed electron shells to overlap.

Derivation

Write \(A=\mathcal Mz^2e^2/4\pi\varepsilon_0\). The condition \(U'(r_0)=0\) gives \(A/r_0^2=nB/r_0^{n+1}\), so \(B=Ar_0^{n-1}/n\) and \(U(r_0)=-\frac{A}{r_0}\big(1-\frac1n\big)\). For the bulk modulus, the volume per ion pair in rock salt is \(V=2r^3\), so \(dV=6r^2dr\). At equilibrium \(K=V\,d^2U/dV^2=V\,U''(r_0)/(6r_0^2)^2\). Since \(U''(r_0)=-2A/r_0^3+n(n+1)B/r_0^{n+2}=(n-1)A/r_0^3\), we get \(K=(n-1)A/(18r_0^4)\). Measured compressibilities give \(n\) between about 6 and 12 for the alkali halides.

The cycle

In 1919 Born and Haber, and independently Kasimir Fajans, observed that the lattice energy closes a loop of measurable steps. For NaCl, form the solid from the elements by two routes: directly, or via gaseous atoms and ions.

Matrix mechanics: pq − qp

For the general reader. In July 1925 Werner Heisenberg handed Born a strange paper. Its central rule for multiplying tables of numbers reminded Born of something from his student days in Breslau: matrix multiplication. Within weeks Born and his student Pascual Jordan rebuilt the paper as an algebra of matrices and found its one fundamental law, that position times momentum differs from momentum times position by a fixed amount. The formula is carved on Born’s gravestone.

The 1924 programme

In Über Quantenmechanik (1924) Born set out to replace the orbit rules of the old quantum theory with a systematic “quantum mechanics”. He noticed that in Hendrik Kramers’s dispersion theory classical derivatives with respect to the action variable \(J\) turn into differences between neighbouring states:

This discrete calculus of transitions, with two state indices, is exactly what Heisenberg’s array \(X(n,m)\) of 1925 implemented.

Born and Jordan, September 1925

Born recognized Heisenberg’s rule \((xy)(n,k)=\sum_mX(n,m)Y(m,k)\) as matrix multiplication. He also noticed that the diagonal elements of Heisenberg’s quantum condition are those of \(pq-qp\), and guessed that its off-diagonal elements vanish. Unable to prove it, he enlisted Jordan. Their paper Zur Quantenmechanik reached the Zeitschrift für Physik on 27 September 1925.

Their argument that the right side is diagonal runs as follows. For \(H=p^2/2m+V(q)\), the equations of motion give \(\frac{d}{dt}(pq-qp)=0\) (Exercise 6.2). In the energy basis every matrix element oscillates as \(e^{2\pi i\nu(n,m)t}\), so a constant matrix can only have non-zero elements where \(\nu(n,m)=0\), that is, on the diagonal for a non-degenerate spectrum. The diagonal elements equal \(h/2\pi i\) by Heisenberg’s sum rule. The Born–Heisenberg–Jordan paper (received 16 November 1925) extended the theory to many degrees of freedom, with \([p_k,q_l]=\frac{h}{2\pi i}\delta_{kl}\), perturbation theory and angular momentum, and Jordan’s section quantized the radiation field in a cavity.

No finite matrices

Part (a) is one line: \(\operatorname{tr}(qp)=\operatorname{tr}(pq)\), so the left side has trace 0 and the right side \(i\hbar N\). Born and Jordan saw that their matrices had to be infinite. Part (b) is Exercise 6.3. This is why quantum mechanics needs unbounded operators, a fact Born confronted in the winter of 1925–26 at MIT. With Norbert Wiener he rewrote the theory in terms of linear operators acting on functions, the first step toward the operator formalism that John von Neumann completed in 1927–32.

Collisions and the statistical interpretation

For the general reader. In 1926 Erwin Schrödinger gave quantum mechanics the form of a wave equation, and he hoped the wave was a real smeared-out electron. Born, studying what happens when an electron hits an atom, saw that this could not be right: the wave spreads in all directions, but each electron is found in one place. He proposed that the wave tells only the probability of finding the electron. This idea earned him the Nobel Prize in 1954 and remains the bridge between the theory and every experiment.

Born’s short paper Zur Quantenmechanik der Stoßvorgänge (“On the quantum mechanics of collision processes”) appeared in the Zeitschrift für Physik in July 1926, followed by a long paper with the same theme. In the text he wrote that the amplitude of the outgoing wave determines the probability of scattering into a given direction. A footnote added in proof corrected this: the probability is proportional to the square of the amplitude. Born later credited Einstein’s idea of a “ghost field” guiding light quanta as his inspiration.

Stationary scattering

A particle of mass \(m\) and energy \(E=\hbar^2k^2/2m\) meets a potential \(V(\mathbf r)\) of short range. Far away, the solution of the Schrödinger equation has the form

The second relation is the Born rule applied to scattering: the outgoing flux into a solid angle, divided by the incident flux, is \(|f|^2\). Rewriting the Schrödinger equation with the free Green’s function gives the integral equation

Born solved it by iteration, the Born series. Keeping only the first term inside the integral gives:

The scattering amplitude is thus the Fourier transform of the potential at the momentum transfer. This makes electron, neutron and X-ray scattering a direct probe of the structure of atoms, nuclei and crystals.

Molecules: Born–Oppenheimer and adiabaticity

For the general reader. A molecule is a crowd of light, fast electrons and heavy, slow nuclei. In 1927 Born and his student J. Robert Oppenheimer showed how to separate the two: first solve for the electrons with the nuclei frozen, then let the nuclei move in the energy landscape the electrons create. Almost all of chemistry, from bond lengths to reaction paths, rests on this separation.

Let electrons have mass \(m\) and coordinates \(\mathbf r\), and nuclei mass \(M\) and coordinates \(\mathbf R\). The Hamiltonian is \(H=T_N+T_e+V(\mathbf r,\mathbf R)\). For each fixed \(\mathbf R\), solve the electronic problem

Then try \(\Psi=\chi(\mathbf R)\,\phi_k(\mathbf r;\mathbf R)\) and neglect derivatives of \(\phi_k\) with respect to \(\mathbf R\). The nuclei obey

The paper, Zur Quantentheorie der Molekeln, appeared in the Annalen der Physik in 1927. The approximation fails where electronic surfaces cross or nearly touch, as at conical intersections in photochemistry. The modern form, with the nonadiabatic couplings written explicitly, comes from the book of Born and Kun Huang (1954).

The adiabatic theorem

The idea behind the separation is adiabatic following: the electrons adjust instantly to the slow motion of the nuclei. In 1928 Born and Vladimir Fock proved the quantum version.

The phase contains a dynamical part \(\int E_k\,dt/\hbar\) and a geometric part identified by Michael Berry in 1984.

Born–Infeld electrodynamics

For the general reader. In Maxwell’s theory the field of a point electron grows without limit as one approaches it, and its energy is infinite. In exile at Cambridge in 1934, Born and Leopold Infeld changed Maxwell’s equations so that no electric field can exceed a maximum value, much as relativity forbids speeds above that of light. The electron’s energy became finite. The theory did not solve the problem of the electron, but sixty years later the same equations reappeared in string theory.

The analogy is with the relativistic free particle, whose Lagrangian \(-mc^2\sqrt{1-v^2/c^2}\) caps the speed at \(c\). Born and Infeld, in Foundations of the new field theory (Proceedings of the Royal Society A, 1934), proposed the same square-root structure for fields. In Gaussian units:

The point charge

For static fields with \(\mathbf B=0\), \(\mathcal L=\frac{b^2}{4\pi}\big(1-\sqrt{1-E^2/b^2}\big)\). The displacement is \(\mathbf D=4\pi\,\partial\mathcal L/\partial\mathbf E=\mathbf E/\sqrt{1-E^2/b^2}\), and Gauss’s law \(\nabla\cdot\mathbf D=4\pi e\,\delta^3(\mathbf r)\) is unchanged, so \(D=e/r^2\) as usual. Inverting,

Far away \(E\approx e/r^2\). At the charge, \(E\to e/r_0^2=b\). The energy density is \(\frac{b^2}{4\pi}\big(\sqrt{1+D^2/b^2}-1\big)\), and integrating with \(r=r_0x\) gives \(U=\frac{e^2}{r_0}\int_0^\infty x^2\big(\sqrt{1+x^{-4}}-1\big)dx\), whose value is the constant in the box. Born and Infeld set \(U=m_ec^2\) to fix \(b\), finding a field of order \(10^{20}\) V/m.

Paul Dirac returned to the theory in 1962. In 1985 Efim Fradkin and Arkady Tseytlin showed that the effective action of the gauge field on open strings has Born–Infeld form, and it now describes the gauge fields living on D-branes.

Edinburgh: stability, liquids, optics

For the general reader. In Edinburgh Born returned to the physics of matter. He asked when a crystal stops being stable and melts, how the atoms of a liquid arrange themselves, and he wrote, with Emil Wolf, the book on optics that has been the standard reference for more than sixty years.

Mechanical stability

A crystal at equilibrium is stable only if every small homogeneous strain raises its energy. In the Voigt notation the elastic energy density of a cubic crystal is

The second condition is positivity of the bulk modulus \((C_{11}+2C_{12})/3\); the other two are positivity of the two shear moduli. In 1939 Born proposed that melting occurs when a shear modulus of the heated lattice falls to zero, the Born criterion of melting. It is now understood as a limit of mechanical stability, which real crystals usually do not reach because they melt first by a thermodynamic transition; it remains an important idea for superheated and strained solids.

The Born–Green hierarchy

For a classical fluid of particles with pair potential \(u(r)\), let \(\rho^{(s)}\) be the \(s\)-particle density. Differentiating the Gibbs distribution with respect to the position of particle 1 gives, in equilibrium,

Born and Herbert S. Green published this in 1946, with the time-dependent hierarchy for non-equilibrium. Jacques Yvon (1935), Nikolay Bogoliubov (1946) and John Kirkwood (1946) reached the same structure, now called the BBGKY hierarchy.

Principles of Optics

Born’s Optik of 1933 became, after a long rewriting with Emil Wolf in Edinburgh and later, Principles of Optics (1959). It treats electromagnetic theory of propagation, diffraction, interference, coherence and the optics of crystals and metals from Maxwell’s equations, and through many editions it remains a standard reference.

The Göttingen school and exile, 1921–1936

Born’s twelve years as director in Göttingen were the most productive of his life. His assistants included Wolfgang Pauli (1921–22), Werner Heisenberg (1923–25) and Friedrich Hund. Among his doctoral students were Pascual Jordan, Lothar Nordheim, J. Robert Oppenheimer (1927), Max Delbrück (1930), Maria Goeppert (later Goeppert Mayer) and Victor Weisskopf. Enrico Fermi, Paul Dirac, Egil Hylleraas and many others came as visitors. Born’s lecture courses became books, among them Vorlesungen über Atommechanik (1925). Göttingen, with Franck’s experiments next door, was where matrix mechanics was made and where the collision theory was tested.

The credit for quantum mechanics was a lasting grief. The Nobel Prize in Physics for 1932, announced in 1933, went to Heisenberg alone for the creation of quantum mechanics. Heisenberg wrote to Born that he felt uneasy at being honoured alone for work done jointly with Born and Jordan. Born would wait more than twenty years for his own prize.

1933

In April 1933 the new civil-service law removed Jewish officials from their posts. Franck resigned in protest. Born and the mathematician Richard Courant were placed on leave that spring. Born left Germany with his family, spent the summer in the South Tyrol in northern Italy, and in the autumn took up a temporary post as Stokes Lecturer at Cambridge, where he was a member of St John’s College. There he worked with the Polish physicist Leopold Infeld (Lesson 09) and wrote popular books, among them The Restless Universe (1935).

In the winter of 1935–36 he spent six months at the Indian Institute of Science in Bangalore, at the invitation of C. V. Raman, who hoped to give him a permanent chair. Faculty politics prevented it. In November 1935 Germany revoked his citizenship, and the family became stateless. In 1936 Born accepted the Tait Chair of Natural Philosophy at the University of Edinburgh, which Charles Galton Darwin had left.

Edinburgh, the Nobel Prize and conscience, 1936–1970

Born held the Tait Chair from October 1936 until his retirement in 1953. He built a school of theoretical physics in a department that had none. His collaborators and students there included Klaus Fuchs, Kun Huang, Huanwu Peng, Herbert S. Green, Sheila Power and Emil Wolf. He was elected to the Royal Society of Edinburgh in 1937 and the Royal Society in 1939, and became a British subject on 31 August 1939.

Fuchs, a refugee from Germany, came to Edinburgh in 1937 and wrote several papers with Born. When the war began he was interned as an enemy alien and shipped to Canada in 1940; Born petitioned for his release, which came in December 1940. Fuchs went on to the British and American atomic bomb projects, and in 1950 confessed to having passed atomic secrets to the Soviet Union. There is no evidence that Born knew of it.

Born refused all work on the atomic bomb. Hiroshima turned his long-standing dislike of military uses of science into a public cause.

Bad Pyrmont and the Nobel Prize

In 1953 Göttingen made him an honorary citizen, and on retirement he and Hedi settled in the spa town of Bad Pyrmont, near Göttingen. In 1954 he received the Nobel Prize in Physics, shared with Walther Bothe, for his fundamental research in quantum mechanics and “especially for his statistical interpretation of the wavefunction”. The same year he published Dynamical Theory of Crystal Lattices with Kun Huang, the definitive treatment of the field he had founded in 1912.

Born used his standing for peace. In 1955 he and Otto Hahn initiated the Mainau Declaration, signed at first by eighteen Nobel laureates and within a year by 52, warning that nations must renounce force or cease to exist. The same year he signed the Russell–Einstein Manifesto. On 12 April 1957 he was one of the eighteen signatories of the Göttingen Manifesto, which opposed equipping the West German army with nuclear weapons and declared that none of them would take part in making, testing or deploying them. He continued to argue in essays and lectures for scientists’ responsibility and against atomic armament.

Death and legacy

Born died in a hospital in Göttingen on 5 January 1970, aged 87. He is buried in the Göttingen Stadtfriedhof, and the stone he shares with Hedi carries the formula \(pq-qp=h/2\pi i\). His granddaughter, through Irene, was the singer Olivia Newton-John; his son Gustav became a professor of pharmacology.

• Quantum theory: the matrix formulation, the commutation relation, the probability interpretation, scattering theory, molecular structure and the adiabatic theorem.
• Condensed matter: lattice dynamics, lattice energies, stability of crystals and the statistical theory of liquids, codified with Kun Huang.
• Optics: Principles of Optics with Emil Wolf.
• Teaching: a Göttingen school that included Heisenberg, Pauli, Jordan, Oppenheimer, Goeppert Mayer, Delbrück and Weisskopf.
• Honours: Max Planck Medal (1948), Hughes Medal (1950), Nobel Prize (1954), Grand Cross of Merit of the Federal Republic of Germany (1959). The German Physical Society and the Institute of Physics award the Max Born Medal and Prize, established in 1972; the Max Born Institute in Berlin and a lunar crater bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Max Born, MacTutor History of Mathematics.
2. Max Born, Wikipedia (with extensive references).
3. Max Born, Biographical, and Nobel Lecture: The statistical interpretation of quantum mechanics, NobelPrize.org.
4. N. Kemmer and R. Schlapp, Max Born 1882–1970, Biographical Memoirs of Fellows of the Royal Society 17 (1971).
5. M. Born, My Life: Recollections of a Nobel Laureate, Taylor & Francis, 1978.
6. N. T. Greenspan, The End of the Certain World: The Life and Science of Max Born, Basic Books, 2005.
7. M. Born (ed.), The Born–Einstein Letters, Macmillan, 1971.
8. J. L. Heilbron and C. Rovelli, Matrix mechanics mis-prized: Max Born’s belated Nobelization, arXiv:2306.00842.
9. Mainau Declaration and Göttingen Manifesto, Wikipedia.
10. J. J. O’Connor and E. F. Robertson, Klaus Fuchs, MacTutor.
11. COMSOL blog, A tour of the famous scientists laid to rest in Göttingen city cemetery.

Physics

1. M. Born and Th. von Kármán, Über Schwingungen in Raumgittern, Physikalische Zeitschrift 13 (1912).
2. M. Born, Dynamik der Kristallgitter, Teubner, 1915.
3. M. Born and A. Landé, papers on ionic lattices, Verh. Dtsch. Phys. Ges. 20 (1918); M. Born, Verh. Dtsch. Phys. Ges. 21 (1919) 679–685. Born–Haber cycle, Wikipedia.
4. M. Born, Über Quantenmechanik, Z. Phys. 26 (1924).
5. M. Born and P. Jordan, Zur Quantenmechanik, Z. Phys. 34 (1925) 858–888; M. Born, W. Heisenberg and P. Jordan, Zur Quantenmechanik II, Z. Phys. 35 (1926) 557–615. English translations in B. L. van der Waerden (ed.), Sources of Quantum Mechanics, North-Holland, 1967.
6. M. Born, Zur Quantenmechanik der Stoßvorgänge, Z. Phys. 37 (1926) 863–867; Quantenmechanik der Stoßvorgänge, Z. Phys. 38 (1926). Born rule, Wikipedia.
7. M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys. 84 (1927) 457–484.
8. M. Born and V. Fock, Beweis des Adiabatensatzes, Z. Phys. 51 (1928).
9. M. Born and L. Infeld, Foundations of the new field theory, Proc. R. Soc. A 144 (1934). Born–Infeld model, Wikipedia.
10. M. Born, On the stability of crystal lattices I, Proc. Camb. Phil. Soc. 36 (1940).
11. M. Born and H. S. Green, A general kinetic theory of liquids I, Proc. R. Soc. A 188 (1946). BBGKY hierarchy, Wikipedia.
12. M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford, 1954.
13. M. Born and E. Wolf, Principles of Optics, Pergamon, 1959; 7th ed., Cambridge, 1999.