Werner Heisenberg

How to use this course

In the summer of 1925 a 23-year-old assistant in Göttingen wrote a paper that most of his elders could not follow, and that within six months had become quantum mechanics. Two years later he showed that the new theory forbids sharp values of position and momentum at once. He went on to found the theory of ferromagnetism, the neutron–proton model of the nucleus and the S-matrix. He also stayed in Germany under Hitler and led the scientific side of its wartime nuclear programme. This course treats the physics in depth and the history from the documentary record.

Each technical lesson starts with a paragraph for the general reader, then gives precise statements, derivations in modern notation and exercises with folded solutions. Lesson 10 covers the reactor physics behind the German uranium project; Lesson 11 covers its history, the 1941 Copenhagen visit and Farm Hall, with sources cited.

Timeline

Würzburg and Munich, 1901–1920

Werner Karl Heisenberg was born in Würzburg on 5 December 1901. His father, August Heisenberg, taught classical languages at a gymnasium and lectured at the university; his mother, Annie Wecklein, was the daughter of the headmaster of the Maximiliansgymnasium in Munich. Werner had an elder brother, Erwin, who became a chemist. In 1910 August Heisenberg was appointed professor of medieval and modern Greek at the University of Munich, the only such chair in Germany, and the family moved there.

At the Maximiliansgymnasium Werner stood out in mathematics. He taught himself calculus and read about number theory while still at school. He was also an accomplished pianist. The war years brought hunger to Munich, and in 1919, during the fighting that ended the short-lived Bavarian Soviet Republic, he served as a teenage auxiliary with volunteer troops. He was active in the German youth movement, with its hikes, camps and long conversations about the future of the nation, and kept these friendships for life.

He passed the Abitur in 1920 and won a place in the Maximilianeum, a Bavarian foundation for gifted students. He first hoped to study pure mathematics with Ferdinand von Lindemann, who turned him away after a brief interview. On his father’s advice he went to the theoretical physicist Arnold Sommerfeld instead.

Sommerfeld’s student, 1920–1923

Sommerfeld admitted the 18-year-old to his advanced seminar at once. There Heisenberg met Wolfgang Pauli, a year older and already the author of an encyclopedia article on relativity. Pauli became his closest critic for the next four decades. Sommerfeld’s method was to hand students unsolved problems from atomic spectroscopy, and within months Heisenberg proposed a model of the anomalous Zeeman effect that used half-integer quantum numbers. It fitted the data and violated the rules of the old quantum theory. It was published in 1922.

In June 1922 Sommerfeld took him to Göttingen to hear Niels Bohr’s week of lectures, later called the Bohr Festival. Heisenberg raised an objection in the discussion, and Bohr invited him for a walk on the nearby hills. Heisenberg later dated the real beginning of his scientific career to that afternoon. He spent the winter of 1922–23 in Göttingen with Max Born while Sommerfeld was in the United States.

The doctorate

Sommerfeld wanted his student to have a classical thesis as well, and set him the stability of laminar flow and the onset of turbulence. The dissertation, Über Stabilität und Turbulenz von Flüssigkeitsströmen, gave an asymptotic estimate of the critical Reynolds number for plane Poiseuille flow. At the oral examination in July 1923 the experimentalist Wilhelm Wien questioned him on the resolving power of optical instruments, which he had not studied, and he nearly failed. He passed with a modest grade and left for Göttingen the next day.

Göttingen and Copenhagen, 1923–1927

From October 1923 Heisenberg was Born’s assistant in Göttingen. Born was trying to replace the orbit pictures of the old quantum theory with a systematic “quantum mechanics”, a phrase he introduced in 1924, in which differential quotients would be replaced by difference quotients. Heisenberg received his habilitation in Göttingen in July 1924, at 22, with work on the anomalous Zeeman effect.

From September 1924 to May 1925, with a fellowship from the Rockefeller-funded International Education Board, he worked at Bohr’s institute in Copenhagen. With Hendrik Kramers he extended Kramers’s dispersion theory, in which an atom scatters light as if it were a set of virtual oscillators with the frequencies of its spectral lines. The Kramers–Heisenberg formula of 1925 already worked only with transition frequencies and amplitudes, never with orbits.

Back in Göttingen, Heisenberg tried to apply the same idea to the intensities of hydrogen lines and got stuck. In early June 1925 a severe attack of hay fever drove him to the island of Helgoland in the North Sea. There, in about ten days, he worked out the scheme of Lesson 04. He returned with a manuscript, sent a copy to Pauli, and gave it to Born, who sent it to the Zeitschrift für Physik. In 1926 he became lecturer at Copenhagen, where in February 1927 he found the uncertainty relations. That autumn, at 25, he became professor of theoretical physics at Leipzig.

Helgoland: the Umdeutung

For the general reader. Nobody has seen an electron orbit. What one sees are the colours of light an atom gives off, and how bright each one is. Heisenberg decided to build a theory using only such observable quantities. Each colour belongs to a jump between two states, so the natural bookkeeping is a table with two indices. Multiplying two such tables gave a rule in which the order of multiplication matters, and that rule turned out to be the heart of quantum mechanics.

The paper Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (“On the quantum-theoretical reinterpretation of kinematic and mechanical relations”) appeared in the Zeitschrift für Physik in 1925.

From Fourier series to transition arrays

In the old theory, a periodic orbit in the stationary state \(n\) is a Fourier series \(x_n(t)=\sum_{\alpha}X_\alpha(n)\,e^{i\alpha\omega(n)t}\), and the radiation it emits has the harmonic frequencies \(\alpha\omega(n)\). Observed spectral lines do not have this form. By the Ritz combination principle they are differences of terms, \(\omega(n,m)=(E_n-E_m)/\hbar\), satisfying

Heisenberg replaced each Fourier component \(X_\alpha(n)e^{i\alpha\omega(n)t}\) by a quantity \(X(n,m)\,e^{i\omega(n,m)t}\) belonging to the transition \(n\to m\). Reality of \(x\) becomes \(X(m,n)=X(n,m)^*\).

The multiplication rule

What represents \(x^2\)? Its components must again oscillate with frequencies \(\omega(n,k)\). The only combination of products of components with this property, given the Ritz rule, is

Heisenberg noticed the non-commutativity and was troubled by it. Born recognized the rule within days as matrix multiplication.

The quantum condition

The old rule \(\oint p\,dq=nh\) refers to an orbit. Heisenberg differentiated it with respect to \(n\) and translated the result, guided by the correspondence principle and the Thomas–Kuhn sum rule for dispersion. In modern notation, with \(\omega_{kn}=(E_k-E_n)/\hbar\), it reads

With the equation of motion \(\ddot x+\omega_0^2x=0\), applied component by component, the oscillator gives \(\big(\omega_0^2-\omega(n,k)^2\big)X(n,k)=0\): only transitions with \(\omega(n,k)=\pm\omega_0\) survive. Labelling states so that these connect \(n\) to \(n\pm1\), the sum rule becomes a recursion whose solution, with a lowest state, is \(|X(n,n-1)|^2=n\hbar/(2m\omega_0)\). Energy conservation then gives \(E_n=\hbar\omega_0\big(n+\tfrac12\big)\). Heisenberg did this on Helgoland, and also treated the anharmonic oscillator by perturbation theory. He checked, late at night, that energy was conserved in his scheme.

Matrix mechanics

For the general reader. Within two months of Heisenberg’s paper, Max Born and Pascual Jordan turned his tables into the algebra of infinite matrices and wrote the rule that underlies everything since: position times momentum minus momentum times position equals a fixed imaginary number. With Heisenberg they then built a complete mechanics on it, the first full quantum theory.

Born and Jordan’s Zur Quantenmechanik was received by the Zeitschrift für Physik about two months after Heisenberg’s paper. The “three-man paper”, Zur Quantenmechanik II by Born, Heisenberg and Jordan, followed in November 1925. Independently, Paul Dirac reached the same structure in Cambridge, replacing Poisson brackets by commutators.

Diagonal elements of the commutation relation are the sum rule of Lesson 04. The equations of motion give \(\dot q=\partial H/\partial p\) and \(\dot p=-\partial H/\partial q\) for polynomial \(H\), so Hamilton’s equations survive as matrix equations. Energy is conserved because \([H,H]=0\).

The oscillator algebraically

Let \(H=\frac{p^2}{2m}+\tfrac12m\omega^2q^2\) and \(a=\sqrt{\tfrac{m\omega}{2\hbar}}\big(q+\tfrac{ip}{m\omega}\big)\). Then \([a,a^\dagger]=1\) and \(H=\hbar\omega(a^\dagger a+\tfrac12)\). From \([H,a]=-\hbar\omega a\), if \(H\psi=E\psi\) then \(H(a\psi)=(E-\hbar\omega)a\psi\). Since \(\langle a^\dagger a\rangle=\|a\psi\|^2\ge0\), the ladder must stop at a state with \(a\psi_0=0\), whose energy is \(\tfrac12\hbar\omega\).

Angular momentum

The proof uses the ladder operators \(J_\pm=J_x\pm iJ_y\), with \([J_z,J_\pm]=\pm\hbar J_\pm\) and \(J_\mp J_\pm=J^2-J_z^2\mp\hbar J_z\). This is how half-integer values, which Heisenberg had introduced by hand in 1922, became legitimate. In early 1926, independently of Schrödinger, Pauli used matrix mechanics and the conserved Runge–Lenz vector to derive the hydrogen spectrum.

The uncertainty relations

For the general reader. To see where an electron is, you must hit it with light. Short waves locate it precisely, but each short-wave photon gives it a hard, unpredictable kick. Heisenberg showed in 1927 that this trade-off is built into quantum mechanics itself: the better a state fixes position, the more it spreads momentum. The product of the two spreads can never fall below a fixed amount set by Planck’s constant.

Heisenberg described the idea in a letter to Pauli on 23 February 1927. The paper Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik was received by the Zeitschrift für Physik on 23 March 1927. Bohr, who was skiing in Norway when it was written, criticized the microscope analysis on his return: Heisenberg had overlooked the finite aperture of the lens. Heisenberg added a postscript acknowledging this.

The microscope

Light of wavelength \(\lambda\) entering a lens of half-angle \(\varepsilon\) resolves positions to \(\Delta x\approx\lambda/\sin\varepsilon\). A scattered photon may enter anywhere within the aperture, so the recoil of the electron is uncertain by \(\Delta p_x\approx(h/\lambda)\sin\varepsilon\). The product \(\Delta x\,\Delta p_x\approx h\) is independent of the instrument.

The theorems

Proof. Replace \(A\) by \(A-\langle A\rangle\) and \(B\) likewise; this changes neither the deviations nor the commutator. By Cauchy–Schwarz, \(\sigma_A^2\sigma_B^2=\|A\psi\|^2\|B\psi\|^2\ge|\langle A\psi|B\psi\rangle|^2\). Write \(\langle A\psi|B\psi\rangle=\langle AB\rangle=\tfrac12\langle\{A,B\}\rangle+\tfrac12\langle[A,B]\rangle\). The first term is real and the second imaginary, so \(|\langle AB\rangle|^2\ge\tfrac14|\langle[A,B]\rangle|^2\). Equality in Cauchy–Schwarz needs \(B\psi=c\,A\psi\); for \(x,p\) this is a first-order equation whose solutions are Gaussians.

Heisenberg also stated an energy–time relation \(\Delta E\,\Delta t\gtrsim h\). Time is not an operator in quantum mechanics, and the rigorous forms, such as the Mandelstam–Tamm inequality (1945) \(\sigma_E\,\tau_A\ge\hbar/2\) with \(\tau_A=\sigma_A/|d\langle A\rangle/dt|\), refer to how fast a state changes. A separate question, how much a measurement of \(x\) disturbs \(p\), was made precise only much later (Ozawa 2003; Busch, Lahti and Werner 2013).

Exchange: helium, hydrogen, magnets

For the general reader. Two electrons are perfectly identical, so swapping them cannot change anything observable. Heisenberg found in 1926 that this simple fact produces a powerful effective force, the exchange interaction, with no counterpart in classical physics. It explained why helium has two separate families of spectral lines, why hydrogen gas comes in two forms, and, in 1928, why iron is magnetic.

Helium (1926)

In Mehrkörperproblem und Resonanz in der Quantenmechanik (1926), Heisenberg treated two electrons in orbitals \(a\) and \(b\). Spatial wave functions must be symmetric or antisymmetric, \(\psi_\pm=\frac{1}{\sqrt2}\big[a(1)b(2)\pm b(1)a(2)\big]\), and the total wave function including spin must be antisymmetric. To first order in the repulsion \(e^2/(4\pi\varepsilon_0r_{12})\),

\(\psi_+\) pairs with the spin singlet (parahelium), \(\psi_-\) with the spin triplet (orthohelium). Since \(K>0\) for helium, the triplet lies lower. Transitions between the two families are forbidden because the dipole operator does not act on spin, which is why helium was long thought to be two different gases spectroscopically.

Two kinds of hydrogen

In 1927 Heisenberg applied the same reasoning to the two protons of the hydrogen molecule. Protons are fermions with spin ½, so nuclear-spin singlets (para) must have even rotational quantum number \(J\) and triplets (ortho) odd \(J\). At room temperature their ratio is 3 to 1; at low temperature conversion is slow, which explained the anomalous specific heat of hydrogen. The Nobel Prize of 1932 was awarded for the creation of quantum mechanics, whose application had, among other things, led to the discovery of these allotropic forms of hydrogen. Karl Friedrich Bonhoeffer and Paul Harteck separated them in 1929.

Ferromagnetism (1928)

For spins \(\tfrac12\), \(\mathbf S_1\cdot\mathbf S_2\) equals \(-\tfrac34\hbar^2\) on the singlet and \(\tfrac14\hbar^2\) on the triplet. So the singlet–triplet splitting \(2K\) can be rewritten as a spin–spin coupling. Heisenberg proposed that neighbouring atoms in a crystal interact this way.

The exchange constant is electrostatic in origin, about a thousand times stronger than the magnetic dipole interaction between spins, which resolved the old puzzle of the enormous “molecular field” in Pierre Weiss’s theory. The Heisenberg model remains a central object of statistical mechanics; the Mermin–Wagner theorem (1966) shows it has no spontaneous magnetization at positive temperature in one or two dimensions.

Neutrons, protons and isospin

For the general reader. James Chadwick discovered the neutron in February 1932. Within months Heisenberg proposed that nuclei consist of protons and neutrons only, held together by a force that lets them trade identity. He described the neutron and proton as two states of one particle, the start of the idea of isospin, an internal symmetry that later became a model for the quark symmetries of particle physics.

The three papers Über den Bau der Atomkerne appeared in the Zeitschrift für Physik in 1932 and 1933. Dmitri Ivanenko had independently proposed that nuclei contain no electrons, and Ettore Majorana soon modified Heisenberg’s exchange force.

The nitrogen problem

In the proton–electron model, nitrogen-14 would contain 14 protons and 7 electrons: 21 spin-½ fermions, giving half-integer spin and Fermi statistics. Band spectra of N\(_2\), measured by Franco Rasetti in 1929, show that \({}^{14}\)N has integer spin and obeys Bose statistics. With 7 protons and 7 neutrons it has 14 fermions: integer spin and Bose statistics, as observed.

Isospin

Heisenberg assigned each heavy particle a two-valued coordinate, with \(\rho^\zeta=+1\) for a neutron and \(-1\) for a proton, and wrote the nuclear Hamiltonian with Pauli matrices acting on it. In today’s notation the nucleon is a doublet \(N=(p,n)\) under an \(\mathrm{SU}(2)\) with generators \(\mathbf T=\tfrac12\boldsymbol\tau\). The name isotopic spin and the hypothesis that nuclear forces are exactly symmetric under this \(\mathrm{SU}(2)\) came later, with Eugene Wigner (1937) and others, after proton–proton scattering showed charge independence.

Heisenberg’s neutron–proton force exchanged charge, modelled on the binding of the hydrogen molecular ion. In 1935 Hideki Yukawa reinterpreted such exchange forces as due to a massive meson, with range \(\hbar/(m_\pi c)\).

Fields, vacuum and the S-matrix

For the general reader. After quantum mechanics, the next task was a quantum theory of fields such as the electromagnetic field. Heisenberg and Pauli wrote its general framework in 1929. In 1936 Heisenberg and his student Hans Euler showed that, according to Dirac’s theory, empty space behaves like a weakly nonlinear medium, so that light can scatter off light. When infinities blocked further progress, Heisenberg proposed in 1943 to describe particles only by what goes into and comes out of a collision: the scattering matrix.

Quantization of wave fields

In Zur Quantendynamik der Wellenfelder (1929–30), Heisenberg and Pauli took a classical Lagrangian density \(\mathcal L(\phi,\partial\phi)\), defined the conjugate field \(\pi=\partial\mathcal L/\partial\dot\phi\), and imposed \([\phi(\mathbf x,t),\pi(\mathbf y,t)]=i\hbar\,\delta^3(\mathbf x-\mathbf y)\), with anticommutators for fermions. They applied the scheme to electrodynamics and confronted the gauge constraint and the self-energy divergences that would dominate the next twenty years.

Nonlinear vacuum (1936)

The correction depends only on the two Lorentz invariants \(\mathbf E^2-\mathbf B^2\) and \(\mathbf E\cdot\mathbf B\), as gauge and Lorentz invariance require. It predicts light-by-light scattering and vacuum birefringence in strong magnetic fields. Euler and Heisenberg also gave the full non-perturbative expression, whose imaginary part in an electric field, made explicit by Julian Schwinger in 1951, describes spontaneous pair creation. Light-by-light scattering was observed by the ATLAS collaboration at the LHC in 2017.

The S-matrix (1943)

In Die „beobachtbaren Größen“ in der Theorie der Elementarteilchen, submitted in September 1942 and published in 1943–44, Heisenberg proposed that a fundamental theory, with a smallest length, need only predict the matrix linking incoming to outgoing free states. In modern terms \(S=\lim_{t\to\infty}U_I(t,-t)\), and \(\langle f|S|i\rangle\) is the amplitude for \(i\to f\).

The programme of building physics from unitarity and analyticity alone was revived in the 1960s by Geoffrey Chew and others for the strong interactions. It led to the Veneziano amplitude and so to string theory; the modern amplitudes programme is a descendant.

Fission and the uranium machine

For the general reader. In December 1938 Otto Hahn and Fritz Strassmann in Berlin found barium among the products of uranium bombarded with neutrons. Lise Meitner and Otto Frisch explained this as fission. Each fission releases about 200 MeV and two or three neutrons, so a chain reaction is possible. Heisenberg’s main wartime work was the theory of a “uranium machine”, a reactor. This lesson gives the physics he and others used, as background for the history in Lesson 11.

Multiplication

Let \(k\) be the average number of fission neutrons produced per neutron absorbed, counting all losses. The neutron population grows like \(e^{(k-1)t/\ell}\), with \(\ell\) the mean generation time. A system is critical when \(k=1\). Natural uranium contains 0.7% \({}^{235}\)U, which fissions readily with slow neutrons. Fast fission neutrons must therefore be slowed by a moderator of light nuclei without being absorbed. For an infinite lattice of natural uranium and moderator,

where \(\eta\) is the number of fast neutrons per thermal neutron absorbed in uranium (about 1.3 for natural uranium), \(\varepsilon\) the fast-fission factor, \(p\) the probability of escaping resonance capture in \({}^{238}\)U while slowing down, and \(f\) the fraction of thermal neutrons absorbed in uranium rather than in the moderator. With \(\eta\approx1.3\), the margin is small, and the moderator must absorb very little.

Critical size

Heisenberg’s reports to Army Ordnance at the end of 1939 and the start of 1940 concluded that a natural-uranium machine was possible with heavy water, or with very pure carbon, as moderator. In early 1941 Walther Bothe at Heidelberg measured the absorption of neutrons in graphite and found a value too high for carbon to work. Most accounts attribute the result to impurities, probably boron, in the sample. The project therefore depended on heavy water, produced mainly at the Norsk Hydro plant at Vemork in occupied Norway, which Allied and Norwegian sabotage and bombing put out of action by 1943. The United States, using purified graphite, reached criticality in Chicago on 2 December 1942.

Experiments

In 1942 the Leipzig experiment L-IV of Robert Döpel and Heisenberg, concentric spherical shells of uranium metal powder and heavy water immersed in a water tank, showed a net increase of neutrons. In June 1942 it was destroyed in a chemical explosion and fire. The last German experiment, B-VIII, was assembled in a cave beneath the castle church at Haigerloch in early 1945: 664 uranium cubes on chains in a tank of heavy water, surrounded by graphite. It showed strong neutron multiplication but did not become critical. A 2019 Monte Carlo study suggests that roughly 50% more uranium would have been needed; Kurt Diebner’s rival group held about 400 more cubes of the same kind.

Leipzig, the Nazi state and the war, 1927–1945

At Leipzig, where Peter Debye directed experimental physics, Heisenberg built a leading school of theoretical physics. His doctoral students and assistants included Felix Bloch, Rudolf Peierls, Edward Teller, Hans Euler and Carl Friedrich von Weizsäcker, among many visitors. In November 1933 he was awarded the Nobel Prize in Physics for 1932, which had been reserved the previous year, “for the creation of quantum mechanics”.

Under National Socialism

After January 1933 many of Heisenberg’s colleagues and students were dismissed or emigrated. He stayed. He helped some colleagues privately and joined Planck’s efforts to limit the damage, but he did not resign or protest publicly. Supporters of a racial “German physics” led by Philipp Lenard and Johannes Stark attacked quantum theory and relativity as Jewish. In 1937 the SS weekly Das Schwarze Korps called Heisenberg a “white Jew”. A long SS investigation followed. In July 1938 Heinrich Himmler closed it, clearing Heisenberg while asking him to separate scientific results from the persons of their authors in his lectures. The same dispute blocked his appointment to Sommerfeld’s chair in Munich. In April 1937 he married Elisabeth Schumacher; they had seven children.

The uranium project

When war began on 1 September 1939, Army Ordnance (Heereswaffenamt) took control of German fission research, with Kurt Diebner as administrator, and Heisenberg was drafted to its theoretical work. The Kaiser Wilhelm Institute for Physics in Berlin was placed under the army. The groups working on uranium, sometimes called the Uranverein, were scattered across Berlin, Leipzig, Heidelberg, Hamburg and elsewhere, and competed for scarce uranium and heavy water.

Early in 1942 the army concluded that the project could not affect the war and passed it to the civilian Reich Research Council. On 4 June 1942, at the Harnack House in Berlin, Heisenberg briefed armaments minister Albert Speer and military leaders. He said a bomb was possible in principle but could not be built within the time and means available, and the meeting decided to continue at modest scale, aimed at a reactor for power. From 1942 he was director at the Kaiser Wilhelm Institute for Physics, and from 1943 also professor at Berlin. As bombing intensified, the institute moved much of its work to Hechingen and Haigerloch in Württemberg. The German project spent roughly 8 million Reichsmarks, against about 2 billion dollars for the Manhattan Project.

Copenhagen, 1941

In September 1941, with German armies advancing in Russia, Heisenberg travelled with von Weizsäcker to a German cultural institute conference in occupied Copenhagen and met Bohr privately. No contemporary record of the conversation exists. Bohr was deeply upset. Heisenberg later said, notably in a letter quoted in Robert Jungk’s 1956 book, that he had sought to discuss the moral problem of atomic weapons. Bohr wrote several replies that he never sent; his family released them in February 2002. They record Bohr’s memory that Heisenberg spoke as if Germany would win the war and as if work on weapons was under way. Michael Frayn’s 1998 play Copenhagen dramatized the uncertainty.

Capture and Farm Hall

In April 1945 the Allied Alsos mission reached Haigerloch and Hechingen, dismantled B-VIII and recovered its uranium. Heisenberg had cycled to his family at Urfeld in Bavaria, where Colonel Boris Pash took him into custody on 3 May. He and nine other German scientists, including Otto Hahn, Max von Laue, Walther Gerlach, Paul Harteck, Kurt Diebner and von Weizsäcker, were interned from 3 July 1945 to 3 January 1946 at Farm Hall, a house near Cambridge whose rooms were secretly bugged. When they heard of Hiroshima on 6 August 1945, Heisenberg at first doubted it. The transcripts, released in 1992, show that he cited a critical mass of a ton or more of uranium-235, far too high, and that on 14 August he lectured to the group on a largely correct theory of the bomb.

Interpretations

After the war Heisenberg and von Weizsäcker presented the project as one that never aimed at a bomb and was steered toward a reactor. Thomas Powers (Heisenberg’s War, 1993) argued that Heisenberg quietly discouraged a bomb. Paul Lawrence Rose (1998) argued the opposite: that he misunderstood the physics and would have built one if he could. Mark Walker and most historians conclude that the German leadership judged a bomb infeasible in wartime on technical and economic grounds, so the project remained at laboratory scale and the moral question was never forced. The record shows a Germany that was never close to a bomb.

Postwar years and legacy, 1946–1976

Released in January 1946, Heisenberg settled in Göttingen in the British zone, where the Kaiser Wilhelm Institute for Physics was refounded as the Max Planck Institute for Physics, with him as director. He became a leading spokesman for West German science. He signed the CERN convention on behalf of the Federal Republic in 1953, and from December 1953 he was president of the Alexander von Humboldt Foundation, which funds foreign researchers in Germany. On 12 April 1957 he was one of eighteen physicists who signed the Göttingen Manifesto, refusing to take part in making, testing or deploying nuclear weapons and opposing nuclear arms for the Bundeswehr. In 1958 the institute moved to Munich, as the Max Planck Institute for Physics and Astrophysics; he directed it until 1970.

His later research sought a unified theory of all elementary particles based on a nonlinear spinor field equation. A 1958 announcement, briefly with Pauli as co-author, drew wide press attention; Pauli soon withdrew, and the theory did not succeed. He wrote widely on philosophy, notably the Gifford Lectures at St Andrews in 1955–56, published as Physics and Philosophy, and the memoir Der Teil und das Ganze (1969; English Physics and Beyond, 1971), which reconstructs conversations from memory and is a source to use with care. He died of cancer at home in Munich on 1 February 1976.

Legacy

• Quantum mechanics: matrix mechanics, the Heisenberg picture and the commutation relations are the algebraic core of the theory; the uncertainty relations are its best-known consequence.
• Many-body and nuclear physics: exchange, the Heisenberg model of magnetism, the neutron–proton nucleus and isospin.
• Quantum field theory: canonical field quantization with Pauli, the Euler–Heisenberg Lagrangian and the S-matrix.
• Honours: Nobel Prize in Physics (1932); Max Planck Medal (1933); Matteucci Medal (1929); Foreign Member of the Royal Society (1955); Pour le Mérite for Sciences and Arts; Niels Bohr International Gold Medal (1970).
• Record: his choice to remain in Nazi Germany and lead its uranium research remains part of any account of his life, and the documents of Lesson 11 are the basis for judging it.

Sources and further reading

Biography and history

1. J. J. O’Connor and E. F. Robertson, Werner Heisenberg, MacTutor History of Mathematics.
2. Werner Heisenberg, Wikipedia (with extensive references).
3. American Institute of Physics, The Bohr–Heisenberg meeting, online exhibit by David Cassidy.
4. D. C. Cassidy, Beyond Uncertainty: Heisenberg, Quantum Physics, and the Bomb, Bellevue Literary Press, 2009.
5. M. Walker, German National Socialism and the Quest for Nuclear Power, 1939–1949, Cambridge University Press, 1989.
6. German nuclear program during World War II, Wikipedia.
7. Operation Epsilon (Farm Hall), Wikipedia.
8. J. Bernstein, Hitler’s Uranium Club: The Secret Recordings at Farm Hall, 2nd ed., Copernicus, 2001.
9. K. Gottstein, Werner Heisenberg and the German uranium project 1939–1945: myths and facts, arXiv:1609.02775.
10. T. Koeth and M. Hiebert, Tracking the journey of a uranium cube, Physics Today 72(5) (2019).
11. T. Powers, Heisenberg’s War, Knopf, 1993; P. L. Rose, Heisenberg and the Nazi Atomic Bomb Project, University of California Press, 1998.
12. Niels Bohr Archive, Release of documents relating to the 1941 Bohr–Heisenberg meeting, February 2002.

Physics

1. W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. Phys. 33 (1925) 879–893.
2. M. Born, P. Jordan, Zur Quantenmechanik, Z. Phys. 34 (1925) 858–888; M. Born, W. Heisenberg, P. Jordan, Zur Quantenmechanik II, Z. Phys. 35 (1926) 557–615.
3. B. L. van der Waerden (ed.), Sources of Quantum Mechanics, North-Holland, 1967 (English translations of the 1925 papers).
4. W. Heisenberg, Mehrkörperproblem und Resonanz in der Quantenmechanik, Z. Phys. 38 (1926) 411–426.
5. W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43 (1927) 172–198.
6. E. H. Kennard, Zur Quantenmechanik einfacher Bewegungstypen, Z. Phys. 44 (1927) 326–352; H. P. Robertson, The uncertainty principle, Phys. Rev. 34 (1929) 163–164.
7. W. Heisenberg, Zur Theorie des Ferromagnetismus, Z. Phys. 49 (1928) 619–636.
8. W. Heisenberg, W. Pauli, Zur Quantendynamik der Wellenfelder, Z. Phys. 56 (1929) 1–61; 59 (1930) 168–190.
9. W. Heisenberg, Über den Bau der Atomkerne I–III, Z. Phys. 77 (1932) 1–11; 78 (1932) 156–164; 80 (1933) 587–596.
10. W. Heisenberg, H. Euler, Folgerungen aus der Diracschen Theorie des Positrons, Z. Phys. 98 (1936) 714–732.
11. W. Heisenberg, Die „beobachtbaren Größen“ in der Theorie der Elementarteilchen, Z. Phys. 120 (1943) 513–538 and 673–702.
12. W. Heisenberg, The Physical Principles of the Quantum Theory, University of Chicago Press, 1930.