How to use this course
Most people know Oppenheimer as the director of Los Alamos and the defendant in the 1954 security hearing. Physicists also know him as the man who brought the new quantum mechanics from Europe to California and trained a generation of American theorists. His own papers are scattered across molecular physics, quantum electrodynamics, nuclear reactions, cosmic rays and relativistic astrophysics. Two of them, written in 1939 with his students, laid the foundations of neutron-star and black-hole physics.
- Quantum mechanics at the level of the hydrogen atom, perturbation theory and tunnelling.
- For Lesson 05, the Dirac equation in outline.
- For Lessons 08 and 09, the Schwarzschild metric and the idea of a perfect fluid in general relativity.
- For Lesson 10, basic nuclear physics: binding energy, neutrons, isotopes.
Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lesson 10 treats the physics of the Manhattan Project at the level of a nuclear physics textbook. It explains fission, chain reactions and isotope separation as general principles and contains no weapon design information.
Timeline
| 1904 | Born 22 April in New York City to Julius Seligmann Oppenheimer, a textile importer, and Ella Friedman, a painter. |
| 1911–21 | Pupil at the Ethical Culture Society School, New York. |
| 1922–25 | Harvard College. A.B. in chemistry, summa cum laude, after three years. |
| 1925–26 | Christ’s College, Cambridge, working in the Cavendish Laboratory. |
| 1927 | PhD at Göttingen under Max Born. Zur Quantentheorie der Molekeln with Born. |
| 1927–29 | National Research Council fellow at Harvard and Caltech; visits Paul Ehrenfest in Leiden and Wolfgang Pauli in Zürich. |
| 1929 | Assistant professor at both Berkeley and Caltech. |
| 1930 | Shows that the holes in Dirac’s theory cannot be protons. |
| 1935 | Oppenheimer–Phillips process, with Melba Phillips. |
| 1936 | Full professor at Berkeley and Caltech. |
| 1937 | Cascade theory of cosmic-ray showers, with J. Franklin Carlson. |
| 1939 | “On Massive Neutron Cores” with George Volkoff; “On Continued Gravitational Contraction” with Hartland Snyder. |
| 1940 | 1 November: marries Katherine (Kitty) Puening. |
| 1941 | Elected to the National Academy of Sciences. October: joins the uranium project through Ernest Lawrence. |
| 1942 | Summer theoretical study group at Berkeley. October: General Leslie Groves chooses him to lead a new weapons laboratory. |
| 1943 | Los Alamos Laboratory opens in New Mexico with Oppenheimer as director. |
| 1945 | 16 July: Trinity test. August: Hiroshima and Nagasaki. October: hands Los Alamos to Norris Bradbury. |
| 1946 | Medal for Merit. Shapes the Acheson–Lilienthal report on international control. |
| 1947 | Director of the Institute for Advanced Study, Princeton; chairman of the AEC General Advisory Committee. |
| 1949 | October: the GAC advises against a crash programme for the hydrogen bomb. |
| 1952 | Leaves the GAC at the end of his term. |
| 1953 | BBC Reith Lectures, Science and the Common Understanding. December: clearance suspended. |
| 1954 | 12 April – 6 May: AEC personnel security hearing. 29 June: the Commission denies reinstatement, 4–1. |
| 1962 | Foreign Member of the Royal Society. |
| 1963 | December: Enrico Fermi Award, presented by President Lyndon Johnson. |
| 1966 | Retires as director of the Institute. |
| 1967 | Dies 18 February at home in Princeton, of throat cancer, aged 62. |
| 2022 | 16 December: Energy Secretary Jennifer Granholm vacates the 1954 decision. |
New York and Harvard, 1904–1925
Julius Robert Oppenheimer was born in New York City on 22 April 1904. His father, Julius, had emigrated from Germany as a young man and prospered in the textile trade. His mother, Ella Friedman, was a painter. The family was Jewish but not observant, and lived in a large apartment on Riverside Drive with an art collection that included several Van Gogh paintings. A brother, Frank, was born in 1912 and also became a physicist.
From 1911 Robert attended the Ethical Culture Society School, founded on Felix Adler’s humanist ethics. He was a precocious, lonely child who collected minerals and lectured to the New York Mineralogical Club before he was a teenager. He graduated in 1921, but an illness contracted on a trip to Europe delayed college by a year.
He entered Harvard in 1922 intending to become a chemist. He read widely in languages, literature and philosophy and took far more than the usual course load. The physics of Percy Bridgman drew him away from chemistry, and he graduated in 1925 after three years, summa cum laude.
Cambridge and Göttingen, 1925–1927
In 1925 Oppenheimer went to Christ’s College, Cambridge, hoping to work under Ernest Rutherford at the Cavendish Laboratory. Rutherford was unimpressed, and J. J. Thomson accepted him on condition that he complete a basic laboratory course. Experimental work went badly. His tutor was Patrick Blackett, and the year was marked by depression and erratic behaviour. A story that he left a poisoned apple on Blackett’s desk appears in later biographies but has no documentary support.
Theory rescued him. In Cambridge he met Niels Bohr and Paul Dirac and began calculating with the new matrix and wave mechanics. In 1926 Max Born invited him to Göttingen, then the centre of the quantum revolution. There he worked alongside Pascual Jordan, Werner Heisenberg’s circle and many of the visitors who would lead physics for the next forty years.
He received his doctorate in the spring of 1927, aged 23, with a thesis on the quantum theory of continuous spectra. The same year he and Born published Zur Quantentheorie der Molekeln, which separated the motion of electrons from that of nuclei. It remains his most cited paper and is the subject of Lesson 04.
Wanderjahre and the Berkeley school, 1927–1941
A National Research Council fellowship took Oppenheimer to Harvard and then to Caltech in 1927–28. In 1928–29 an International Education Board fellowship took him to Paul Ehrenfest in Leiden and to Wolfgang Pauli at the ETH in Zürich. Pauli became an important influence on his taste for the deep problems of quantum field theory.
In 1929 he accepted posts as assistant professor at both the University of California, Berkeley, and Caltech. For thirteen years he spent autumn and winter in Berkeley and spring in Pasadena, and many students followed him back and forth. He was promoted to full professor at both in 1936. His early lectures were notoriously fast and obscure. Within a few years they had become the best course in theoretical physics in the United States.
At Berkeley he worked closely with Ernest Lawrence’s Radiation Laboratory, where the cyclotron produced a flood of new nuclear data. His doctoral students included Melba Phillips, Willis Lamb, Robert Christy, George Volkoff, Hartland Snyder, Philip Morrison, Sidney Dancoff, David Bohm and Bernard Peters. Robert Serber was his most constant collaborator. Hans Bethe later judged that Oppenheimer’s greatest achievement in these years was the school itself.
In the mid-1930s Oppenheimer became politically engaged. He supported Spanish Republican causes, unionisation of teaching staff and refugee physicists, and he had friends and relatives in or close to the Communist Party, including Jean Tatlock, his partner from 1936 to 1939, and his brother Frank. These associations would be the main subject of the 1954 hearing. In 1940 he married Katherine (Kitty) Puening, who had studied biology at Pennsylvania. Their son Peter was born in 1941 and their daughter Katherine (Toni) in 1944.
The Born–Oppenheimer approximation
For the general reader. A molecule contains light electrons and heavy nuclei. The electrons move so quickly that, at each instant, they settle into the state suited to wherever the nuclei happen to be. The nuclei then move slowly in an energy landscape created by the electrons. This separation is why chemists can speak of molecular shapes, bond lengths and vibrations at all.
- Derive the electronic and nuclear equations from the full molecular Hamiltonian.
- Identify the neglected non-adiabatic terms and when they matter.
- Recover the Born–Oppenheimer hierarchy of electronic, vibrational and rotational energies.
The separation
Let \(r\) denote electron coordinates and \(R\) nuclear coordinates, with nuclear masses \(M\) and electron mass \(m\). The molecular Hamiltonian is
\[ H = T_n + T_e + V(r,R), \qquad T_n=-\sum_A\frac{\hbar^2}{2M_A}\nabla_{R_A}^2,\quad T_e=-\sum_i\frac{\hbar^2}{2m}\nabla_{r_i}^2 . \]For each fixed nuclear configuration, solve the clamped-nuclei problem
\[ \big(T_e+V(r,R)\big)\,\phi_k(r;R)=E_k(R)\,\phi_k(r;R). \]The functions \(E_k(R)\) are the potential energy surfaces. Expanding the full wave function as \(\Psi(r,R)=\sum_k \chi_k(R)\,\phi_k(r;R)\) and projecting onto \(\phi_j\) gives the exact coupled equations
\[ \big(T_n+E_j(R)\big)\chi_j+\sum_k \Lambda_{jk}\,\chi_k = E\,\chi_j,\qquad \Lambda_{jk}=-\sum_A\frac{\hbar^2}{2M_A}\Big(2\langle\phi_j|\nabla_A\phi_k\rangle\cdot\nabla_A+\langle\phi_j|\nabla_A^2\phi_k\rangle\Big). \]If the surface \(E_j(R)\) is well separated from all others in the region the nuclei explore, the couplings \(\Lambda_{jk}\) may be dropped to leading order, and the nuclei obey \(\big(T_n+E_j(R)\big)\chi=E\chi\). Born and Oppenheimer organised the expansion in powers of \(\kappa=(m/M)^{1/4}\): electronic energies are of order \(\kappa^0\), vibrational energies of order \(\kappa^2\), and rotational energies of order \(\kappa^4\), all in units of the electronic energy scale.
Why \(\kappa=(m/M)^{1/4}\)
Near a minimum \(R_0\), \(E(R)\approx E(R_0)+\tfrac12 k(R-R_0)^2\), where the curvature \(k\sim \varepsilon_e/a^2\) is set by the electronic energy \(\varepsilon_e\sim\hbar^2/(ma^2)\) and the bond length \(a\). Then \(\hbar\omega=\hbar\sqrt{k/M}\sim\hbar^2/(a^2\sqrt{mM})=\varepsilon_e\sqrt{m/M}=\kappa^2\varepsilon_e\). The zero-point amplitude is \(\delta\sim\sqrt{\hbar/(M\omega)}\sim\kappa a\), so nuclei are localised to a fraction \(\kappa\) of the bond length. Rotational energies are \(\hbar^2/(Ma^2)=\kappa^4\varepsilon_e\).
When it fails
The neglected couplings contain \(\langle\phi_j|\nabla_A\phi_k\rangle=\langle\phi_j|\nabla_A V|\phi_k\rangle/(E_k-E_j)\) for \(j\ne k\). They become large where two surfaces approach. At conical intersections they control photochemistry, radiationless decay and vision. Max Born and Kun Huang gave the systematic treatment in 1954.
Exercise 4.1For H₂, take \(\varepsilon_e\approx 10\) eV and \(m/M=1/1836\). Estimate the vibrational and rotational energy scales and compare with the measured vibrational quantum (about 0.52 eV) and rotational constant (about 7.5 meV).
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Exercise 4.2Within the approximation, H₂ and D₂ have the same potential energy surface. Predict the ratio of their vibrational frequencies and compare with the observed 4401 cm⁻¹ and 3116 cm⁻¹.
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Exercise 4.3Show that for real electronic wave functions the diagonal first-derivative coupling \(\langle\phi_k|\nabla_A\phi_k\rangle\) vanishes.
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Holes, positrons and the electron–proton problem
For the general reader. Paul Dirac’s 1928 equation for the electron also allowed states of negative energy. Dirac proposed that these states are normally all filled, and that a missing electron, a hole, would look like a positive particle. He first hoped the hole was the proton. Oppenheimer showed in 1930 that this could not be so: holes must have the electron’s mass, and if they were protons, every hydrogen atom would destroy itself almost instantly. Carl Anderson found the positron in 1932.
- See why the Dirac equation has a negative-energy continuum.
- Follow the argument that a hole has the electron’s mass.
- Estimate the lifetime of hydrogen if protons were holes.
The negative-energy sea
The free Dirac equation \(i\hbar\,\partial_t\psi=\big(c\,\boldsymbol\alpha\cdot\mathbf p+\beta mc^2\big)\psi\) has plane-wave solutions with
\[ E=\pm\sqrt{p^2c^2+m^2c^4}. \]In a field, an electron can make radiative transitions into negative-energy states, so the one-particle theory is unstable. In A theory of electrons and protons (1930), Dirac proposed that all negative-energy states are occupied, so that the Pauli principle blocks such transitions. An unoccupied state of energy \(-|E|\) and momentum \(-\mathbf p\) then behaves like a particle of energy \(+|E|\), momentum \(\mathbf p\) and charge \(+e\).
If \(\psi\) solves the Dirac equation for charge \(-e\) in the potential \(A_\mu\), then \(\psi_c=C\bar\psi^{\,T}\) (with \(C=i\gamma^2\gamma^0\) in the Dirac representation) solves it for charge \(+e\) in the same potential, with the same mass \(m\). Hence a hole in the sea of electrons of mass \(m\) behaves as a particle of mass \(m\) and opposite charge.
Oppenheimer’s objections (1930)
In On the theory of electrons and protons (Phys. Rev. 35, 1930), Oppenheimer made two points. First, the electromagnetic properties of a hole are fixed by the Dirac equation itself, and the hole’s inertia equals the electron’s, so it cannot be the proton, 1836 times heavier. Hermann Weyl reached the same conclusion from symmetry. Second, if holes were protons, an electron could fall into the hole that a proton represents, and the two would annihilate into radiation. Ordinary matter would be unstable. Oppenheimer suggested instead that the sea is completely filled and that positive electrons, if they exist, are a separate species. Dirac accepted the argument in 1931 and predicted the anti-electron.
From holes to fields
In 1934 Wendell Furry and Oppenheimer reformulated hole theory as a quantum field theory in which electrons and positrons enter symmetrically, removing the infinite sea from the formalism. They noted that the vacuum itself becomes polarisable, an effect that contributes a small part of the Lamb shift. In 1930 Oppenheimer had also shown that the electron’s interaction with its own field gives an infinite energy shift in spectral lines. The problem of these infinities occupied him for twenty years, and its solution by renormalisation came from his former student Willis Lamb’s experiment and from the theorists at the 1947 Shelter Island conference, where he led the discussions.
Exercise 5.1Why can a free electron and positron at rest not annihilate into a single photon? What are the energies of the two photons?
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Exercise 5.2At low relative velocity \(v\), Dirac’s annihilation cross-section is \(\sigma\approx\pi r_e^2c/v\), with \(r_e=e^2/(4\pi\varepsilon_0 mc^2)\approx2.82\times10^{-15}\) m. Suppose the proton were a hole of electron mass. Estimate the lifetime of a hydrogen atom, using the electron density at the nucleus \(|\psi(0)|^2=1/(\pi a_0^3)\), \(a_0=5.29\times10^{-11}\) m.
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Exercise 5.3What is the minimum photon energy for pair creation \(\gamma\to e^+e^-\) near a heavy nucleus? Why is a nucleus needed?
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The Oppenheimer–Phillips process
For the general reader. To react with a nucleus, a positively charged projectile must climb over the electric repulsion of the nuclear charge. In the 1930s Lawrence’s cyclotron showed that deuterons, the nuclei of heavy hydrogen, transmuted heavy elements at energies where the repulsion should have kept them out. Oppenheimer and Melba Phillips explained why. The deuteron is a loosely bound proton and neutron. The electric field pushes the proton away while the neutron, which feels no repulsion, slips into the nucleus. The proton flies off alone.
- Compute Coulomb barriers and tunnelling exponents.
- Understand the polarisation of the deuteron in a nuclear field.
- Compute the Q-value of a (d,p) stripping reaction.
The Coulomb barrier
A nucleus of charge \(Ze\) and radius \(R\) presents a barrier of height \(V_C=Z z e^2/(4\pi\varepsilon_0R)\) to a projectile of charge \(ze\). With \(e^2/(4\pi\varepsilon_0)=1.44\) MeV fm, a deuteron meets a barrier above 10 MeV at a heavy nucleus. Below the barrier, the tunnelling probability is roughly \(e^{-2\pi\eta}\), where the Sommerfeld parameter is
\[ \eta=\frac{Zze^2}{4\pi\varepsilon_0\hbar v}. \]The mechanism
The deuteron is large and weakly bound: its binding energy is \(B_d=2.224\) MeV and its root-mean-square radius about 2 fm. In the Coulomb field of a heavy nucleus the deuteron is polarised. The proton end is pushed back while the neutron end leads. If the neutron comes within range of the nuclear force, it can be captured while the proton is still outside the barrier. The proton is then repelled. Oppenheimer and Phillips computed the transmutation probability in this picture in their 1935 Note on the transmutation function for deuterons (Phys. Rev. 48), and showed it falls much more slowly with decreasing energy than ordinary compound-nucleus capture.
For the reaction \({}^{A}X(d,p)\,{}^{A+1}X\), the energy released is
\[ Q=S_n\big({}^{A+1}X\big)-B_d , \]where \(S_n\) is the neutron separation energy of the final nucleus. The outgoing proton has kinetic energy at most \(E_d+Q\) (less recoil), with the maximum reached when the final nucleus is left in its ground state.
Since \(S_n\) is typically 5–8 MeV and \(B_d\) only 2.2 MeV, \(Q\) is usually positive. The ejected proton can then leave with more energy than the deuteron brought in, which was one of the puzzling observations the theory explained. After the war, Stuart Butler’s theory of stripping angular distributions (1950) turned the (d,p) reaction into a precise tool of nuclear spectroscopy.
Exercise 6.1Given \(S_n({}^{13}\mathrm C)=4.946\) MeV, find the Q-value of \({}^{12}\mathrm C(d,p){}^{13}\mathrm C\).
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Exercise 6.2Compare the Gamow exponent \(2\pi\eta\) for a proton and a deuteron of the same kinetic energy on the same target. Why did this make the high deuteron yields surprising?
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Exercise 6.3Estimate the Coulomb barrier for a deuteron on gold (\(Z=79\), \(A=197\)), taking the touching distance \(R=1.2\,(A^{1/3}+2^{1/3})\) fm.
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Cosmic-ray showers and the mesotron
For the general reader. A single very energetic cosmic ray striking the upper atmosphere can produce millions of particles that reach the ground together. In 1937 Oppenheimer and his student J. Franklin Carlson, and independently Homi Bhabha and Walter Heitler, explained these showers as a cascade: electrons radiate photons, photons make electron–positron pairs, and the number of particles doubles until the energy per particle runs out.
- Build the simple branching model of an electromagnetic cascade.
- Derive the logarithmic depth of shower maximum.
- See why penetrating particles had to be something new.
The cascade
Above a critical energy \(E_c\), electrons lose energy mainly by bremsstrahlung and photons mainly by pair production. Both processes are governed by the radiation length \(X_0\) (about 37 g cm⁻² in air). In the simplest model, due to Heitler, each particle splits in two after a depth \(d=X_0\ln2\), sharing its energy equally.
After \(n\) splitting lengths the shower contains \(N=2^n\) particles of energy \(E_0/2^n\). Multiplication stops when \(E_0/2^n=E_c\). Hence
\[ N_{\max}=\frac{E_0}{E_c},\qquad X_{\max}=X_0\ln\frac{E_0}{E_c}. \]Carlson and Oppenheimer solved the diffusion equations of the cascade with realistic cross-sections, including the fluctuations in the number of particles. The model captures the key facts: the size of the shower is proportional to the primary energy, and the depth of maximum grows only logarithmically. Both remain the basis of air-shower astrophysics.
The mesotron
The cascade theory also showed that the penetrating component of cosmic rays could not be electrons, which would radiate away their energy far too quickly. Carl Anderson and Seth Neddermeyer identified a particle of intermediate mass in 1937. Oppenheimer and Robert Serber suggested that same year that it was the quantum of Hideki Yukawa’s nuclear force. It was later shown to be the muon, which does not feel the strong force; Yukawa’s particle, the pion, was found in 1947.
Exercise 7.1Take \(E_c\approx85\) MeV and \(X_0=37\) g cm⁻² for air. For a primary photon of \(10^{15}\) eV, estimate \(N_{\max}\) and \(X_{\max}\). The whole atmosphere is about 1030 g cm⁻² thick. Does the shower peak above the ground?
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Exercise 7.2By how much does \(X_{\max}\) increase per factor of ten in primary energy?
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Neutron stars and the TOV limit
For the general reader. When a star exhausts its fuel, gravity crushes it until some pressure resists. Subrahmanyan Chandrasekhar showed that electron pressure cannot support a white dwarf above about 1.4 solar masses. Lev Landau, Walter Baade and Fritz Zwicky imagined still denser stars made of neutrons. In 1939 Oppenheimer and his student George Volkoff asked how heavy such a neutron star could be, using Einstein’s gravity. Their answer, about 0.7 solar masses for an ideal neutron gas, was the first calculation of a maximum mass for neutron stars.
- Derive the Tolman–Oppenheimer–Volkoff equation and its Newtonian limit.
- Understand why general relativity imposes a maximum mass.
- Compare the 1939 result with modern observations.
Hydrostatic equilibrium in general relativity
For a static, spherically symmetric perfect fluid with mass density \(\rho(r)\) (total energy density \(\rho c^2\)) and pressure \(P(r)\), write the metric as \(ds^2=-e^{2\Phi}c^2dt^2+\big(1-2Gm(r)/(rc^2)\big)^{-1}dr^2+r^2d\Omega^2\). Einstein’s equations reduce to the following system, derived in 1939 by Richard Tolman and by Oppenheimer and Volkoff in back-to-back papers.
With an equation of state \(P=P(\rho)\) and a central density \(\rho_c\), integration outward until \(P=0\) gives the radius \(R\) and mass \(M=m(R)\). Outside, the metric is Schwarzschild.
Each of the three relativistic corrections increases the inward pull: pressure has inertia (\(P/c^2\) added to \(\rho\)), pressure gravitates (\(4\pi r^3P/c^2\) added to \(m\)), and gravity is stronger close to the Schwarzschild radius. Raising the central pressure to resist gravity therefore also raises gravity. Beyond a critical central density, adding pressure makes the star lighter, and the configurations are unstable.
The 1939 result
Oppenheimer and Volkoff took the equation of state of a cold, ideal, degenerate Fermi gas of neutrons. With Fermi momentum \(p_F=\hbar(3\pi^2n)^{1/3}\) and \(x=p_F/(m_nc)\), it reads parametrically
\[ \rho c^2=\frac{m_n^4c^5}{8\pi^2\hbar^3}\Big[(2x^3+x)\sqrt{1+x^2}-\sinh^{-1}x\Big],\qquad P=\frac{m_n^4c^5}{24\pi^2\hbar^3}\Big[(2x^3-3x)\sqrt{1+x^2}+3\sinh^{-1}x\Big]. \]Integrating numerically, they found a maximum mass of about \(0.7\,M_\odot\) at a radius near 10 km. The hero figure of this page shows the curve. Since a white dwarf can reach \(1.4\,M_\odot\), the result suggested that the remnants of massive stars could find no stable resting place. Oppenheimer and Volkoff noted that nuclear forces, then unknown at these densities, could change the number.
Today
Repulsive nuclear forces stiffen the equation of state and raise the limit. Radio pulsars with masses near \(2\,M_\odot\) are now well measured, and PSR J0952−0607 is estimated at \(2.35\pm0.17\,M_\odot\). Analyses of the neutron-star merger GW170817 suggest a maximum mass of roughly 2.0–2.2 \(M_\odot\) for non-rotating stars. The limit still bears the names of Tolman, Oppenheimer and Volkoff.
Exercise 8.1Show that the TOV equation reduces to Newtonian hydrostatic equilibrium when \(P\ll\rho c^2\) and \(Gm/(rc^2)\ll1\).
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Exercise 8.2The maximum-mass ideal neutron star has \(M\approx0.71\,M_\odot\) and \(R\approx9.5\) km. Compute its Schwarzschild radius and compactness \(2GM/(Rc^2)\). (\(2GM_\odot/c^2=2.95\) km.)
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Exercise 8.3Landau’s 1932 argument balances the gravitational energy of the outermost neutron against its relativistic Fermi energy and gives a maximum mass of order \(M\sim(\hbar c/G)^{3/2}/m_n^2\). Evaluate this.
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Continued gravitational contraction
For the general reader. If no star can support itself above some mass, what happens to a heavier one? In 1939 Oppenheimer and Hartland Snyder followed the collapse of an idealised star in Einstein’s theory. The star falls inward through its own Schwarzschild radius in a finite time as measured by a clock riding on it. A distant observer instead sees the collapse slow and the light redden and fade, until the star is cut off from the outside. This was the first description of what John Wheeler would in 1967 call a black hole.
- Set up the collapse of a homogeneous pressureless ball.
- Derive the proper time of collapse from the cycloid solution.
- Explain the frozen appearance seen from outside.
The model
Oppenheimer and Snyder assumed a spherical, non-rotating ball of uniform density and zero pressure, released from rest at radius \(R_0\), with no radiation or mass loss.
Every spherically symmetric vacuum solution of Einstein’s equations is, locally, a piece of the Schwarzschild solution. So the exterior of any spherical collapsing star is Schwarzschild with fixed mass \(M\), whatever the interior does.
Inside, homogeneity forces the geometry to be that of a closed dust universe of the Friedmann type, \(ds^2=-c^2d\tau^2+a(\tau)^2\big(d\chi^2+\sin^2\!\chi\,d\Omega^2\big)\), with \(\tau\) the proper time of comoving dust. The surface is at fixed \(\chi=\chi_0\), so its areal radius is \(R(\tau)=a(\tau)\sin\chi_0\). The field equations reduce to an energy equation of Newtonian form
\[ \Big(\frac{dR}{d\tau}\Big)^2=\frac{2GM}{R}-\frac{2GM}{R_0}. \]The solution is the cycloid
\[ R=\frac{R_0}{2}(1+\cos\eta),\qquad \tau=\sqrt{\frac{R_0^3}{8GM}}\,(\eta+\sin\eta),\qquad 0\le\eta\le\pi, \]so the surface reaches \(R=0\) after the finite proper time \(\tau_c=\dfrac{\pi}{2}\sqrt{\dfrac{R_0^3}{2GM}}\). It crosses \(R=2GM/c^2\) with nothing special happening locally.
Seen from far away, the Schwarzschild time \(t\) of the surface diverges as \(R\to2GM/c^2\). Light emitted near the horizon is redshifted without limit. At late times the redshift grows exponentially, \(1+z\propto e^{c^3t/(4GM)}\), so the star dims within a few multiples of \(GM/c^3\). In the paper’s words, the star tends to close itself off from distant observers.
Reception
The paper appeared in Physical Review on 1 September 1939, the day Germany invaded Poland, and drew little attention. Wheeler resisted the conclusion until the late 1950s. David Finkelstein’s 1958 coordinates showed that the horizon is a regular one-way surface. Numerical collapse calculations in the 1960s, by Stirling Colgate’s group at Livermore and Yakov Zeldovich’s in Moscow, found that realistic stars behave much like the idealised model. Roger Penrose’s singularity theorem of 1965 showed that the outcome does not depend on the symmetry. Freeman Dyson considered the 1939 paper Oppenheimer’s greatest contribution to physics. Oppenheimer himself never returned to the subject.
Exercise 9.1Verify that the cycloid satisfies the energy equation and that \(\tau(\pi)=\tfrac{\pi}{2}\sqrt{R_0^3/(2GM)}\).
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Exercise 9.2A core of \(10\,M_\odot\) begins to collapse from rest at \(R_0=7000\) km. Find the proper collapse time and the e-folding time \(4GM/c^3\) of the redshift seen from far away. (\(GM_\odot=1.327\times10^{20}\ \mathrm{m^3\,s^{-2}}\), \(GM_\odot/c^3=4.93\ \mu\)s.)
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The physics of the Manhattan Project
For the general reader. In December 1938 Otto Hahn and Fritz Strassmann found barium among the products of uranium bombarded with neutrons. Lise Meitner and Otto Frisch explained that the nucleus had split, releasing enormous energy and, it soon turned out, further neutrons. A self-sustaining chain reaction became conceivable. This lesson covers the physical principles that made the project possible, at the level of a nuclear physics textbook. It contains no weapon design information.
- Estimate the energy of fission from the binding-energy curve.
- Understand the multiplication factor and the idea of criticality.
- See why separating uranium isotopes was so hard.
Energy from binding
The binding energy per nucleon peaks near iron at about 8.8 MeV and falls to about 7.6 MeV for uranium, because Coulomb repulsion grows like \(Z^2\). Splitting a heavy nucleus into two medium-mass fragments therefore releases about 200 MeV, mostly as kinetic energy of the fragments, roughly fifty million times the energy of a typical chemical reaction per atom. In 1939 Niels Bohr and John Wheeler’s liquid-drop theory showed that the rare isotope uranium-235 fissions with slow neutrons, while the abundant uranium-238 does not.
Chain reactions and criticality
Each fission releases on average two to three neutrons. Define the multiplication factor \(k\) as the mean number of neutrons from one generation that cause fission in the next. If \(\tau\) is the mean generation time,
\[ \frac{dN}{dt}=\frac{k-1}{\tau}\,N\quad\Longrightarrow\quad N(t)=N_0\,e^{(k-1)t/\tau}. \]In an infinite medium neutrons multiply by \(k_\infty\) per generation. In a finite body some neutrons leak out through the surface before causing fission. In one-group diffusion theory, a bare sphere of radius \(R\) has effective multiplication
\[ k_{\mathrm{eff}}=\frac{k_\infty}{1+L^2B^2},\qquad B=\frac{\pi}{R}, \]where \(L\) is the neutron diffusion length. The system is critical when \(k_{\mathrm{eff}}=1\). Production scales with volume and leakage with surface, so for \(k_\infty>1\) there is always a critical size.
Enrico Fermi’s graphite and natural-uranium pile in Chicago reached criticality on 2 December 1942. It was controllable because a small fraction of fission neutrons are emitted with delays of seconds, which lengthens the effective generation time enormously. Reactors at Hanford then produced plutonium-239 from uranium-238, while the plants at Oak Ridge separated uranium-235.
Isotope separation
Natural uranium contains only 0.72% uranium-235, and the two isotopes are chemically identical. The project used electromagnetic separation in Lawrence’s calutrons, thermal diffusion, and gaseous diffusion of uranium hexafluoride through porous barriers. In diffusion the separation exploits Graham’s law: lighter molecules effuse faster in proportion to \(1/\sqrt{\text{mass}}\).
Los Alamos and Trinity
Oppenheimer’s laboratory turned these materials into working devices. Its scientific problems included neutron cross-sections, the hydrodynamics of very fast processes and the theory of explosions. Robert Serber’s introductory lectures of April 1943, published in 1992 as The Los Alamos Primer, set out the programme. Before the first test, Emil Konopinski, Cloyd Marvin and Edward Teller checked whether a nuclear explosion could ignite the atmosphere and concluded that it could not. The Trinity test took place in the New Mexico desert on 16 July 1945.
Exercise 10.1Using a binding energy per nucleon of 7.6 MeV for a nucleus of mass number 236 and 8.5 MeV for two fragments of mass number about 118, estimate the energy released in one fission.
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Exercise 10.2Compare the energy from fissioning all nuclei in 1 kg of uranium-235 (200 MeV each) with burning 1 kg of coal (about \(3\times10^7\) J).
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Exercise 10.3For gaseous diffusion of UF₆ (molecular masses 349 and 352), find the ideal single-stage separation factor. How many ideal stages raise the abundance of uranium-235 from 0.72% to 3%, the level used in power reactors?
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Exercise 10.4A reactor with an effective generation time of 0.1 s (including delayed neutrons) runs at \(k=1.001\). How long does its power take to grow by a factor \(e\)? What would happen with prompt neutrons only, \(\tau\approx10^{-4}\) s?
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Los Alamos and Washington, 1942–1953
In September 1941 the Australian physicist Mark Oliphant briefed American colleagues on British estimates that a uranium bomb was feasible. In October Ernest Lawrence brought Oppenheimer into the American uranium project. In the summer of 1942 Oppenheimer led a study group at Berkeley, including Hans Bethe, Edward Teller and Serber, that surveyed the theory of a fission weapon. In October 1942 General Leslie Groves chose him to direct a new central laboratory, over objections that he was a theorist with leftist associations and no administrative experience. Oppenheimer knew northern New Mexico from summers at a ranch in the Pecos valley and proposed the Los Alamos Ranch School as its site.
The laboratory opened in spring 1943 with a few hundred people and grew to several thousand. Oppenheimer recruited Bethe to head the theoretical division and drew in Enrico Fermi, John von Neumann, Niels Bohr and the British mission. Colleagues described his grasp of every group’s problems as the decisive factor in the laboratory’s success. After Trinity he served on the scientific panel that advised on the use of the bomb; it did not see a workable alternative to military use. Hiroshima and Nagasaki were bombed on 6 and 9 August 1945.
Oppenheimer resigned as director in October 1945. That month he told President Harry Truman that he felt he had blood on his hands, and Truman was offended. He received the Medal for Merit in 1946. He was the chief scientific author of the Acheson–Lilienthal report on international control of atomic energy, which became the basis of the American Baruch Plan at the United Nations.
In 1947 he became director of the Institute for Advanced Study in Princeton and chairman of the General Advisory Committee of the new Atomic Energy Commission (AEC). He was the discussion leader at the 1947 Shelter Island conference that launched postwar quantum electrodynamics. In October 1949, after the first Soviet test, the GAC unanimously advised against a crash programme to build the hydrogen bomb, on technical and moral grounds. Truman decided to proceed on 31 January 1950. The opposition earned Oppenheimer powerful enemies, notably Lewis Strauss, later AEC chairman, and parts of the Air Force. Truman did not reappoint him, and he left the GAC in August 1952.
DiscussionThe GAC report of 1949 combined technical doubts with an ethical judgement. Should scientific advisers to a government confine themselves to technical questions? Use the Acheson–Lilienthal report and the hydrogen-bomb debate as evidence.
The hearing, Princeton and legacy, 1953–1967
The 1954 security hearing
On 7 November 1953 William Liscum Borden, former executive director of the congressional Joint Committee on Atomic Energy, wrote to FBI director J. Edgar Hoover that Oppenheimer was more probably than not a Soviet agent. On 3 December President Dwight Eisenhower ordered that Oppenheimer be cut off completely from secret information. AEC general manager Kenneth Nichols drafted a letter of 24 charges. Twenty-three concerned Communist associations between 1938 and 1946; the last concerned his opposition to the hydrogen bomb. On 21 December Strauss told Oppenheimer that his clearance was suspended. Oppenheimer chose a hearing rather than resignation.
The Personnel Security Board sat from 12 April to 6 May 1954. Its members were Gordon Gray (chair), Thomas Morgan and the chemist Ward Evans. Roger Robb acted as counsel for the board and conducted a prosecutorial cross-examination using classified files that Oppenheimer’s lawyers, led by Lloyd Garrison, were not cleared to see. The FBI had wiretapped Oppenheimer and his lawyers. The most damaging episode was the Chevalier affair: in 1943 Oppenheimer had given security officers a false account of an approach by his friend Haakon Chevalier, and under cross-examination he admitted it. Fermi, Isidor Rabi, Bethe, James Conant and Vannevar Bush testified for him. Teller testified that he believed Oppenheimer loyal but would feel more secure if the country’s vital interests were in other hands.
On 27 May the board voted 2–1 against restoring the clearance, with Evans dissenting. It found Oppenheimer a loyal citizen but questioned his associations and his conduct over the hydrogen bomb. On 29 June 1954 the Commission upheld the decision 4–1, with the physicist Henry DeWolf Smyth dissenting. Strauss’s majority opinion rested on what it called fundamental defects of character, not on disloyalty. The clearance would have expired the next day in any case. The AEC published an edited transcript, In the Matter of J. Robert Oppenheimer, in June 1954; the Department of Energy released the full transcript in 2014. On 16 December 2022 Energy Secretary Jennifer Granholm vacated the 1954 decision as the product of a flawed process, without ruling on the charges.
Princeton and the last years
Oppenheimer remained director of the Institute for Advanced Study until 1966 and built up its physics faculty. He gave the BBC Reith Lectures in 1953 and wrote and lectured widely on science and culture. He was made an officer of the Legion of Honour in 1957 and a Foreign Member of the Royal Society in 1962. In 1963 President John Kennedy announced the Enrico Fermi Award for him; Lyndon Johnson presented it in December 1963, a gesture widely read as public rehabilitation. He was nominated for the Nobel Prize in physics several times but never received it.
A lifelong heavy smoker, he was diagnosed with throat cancer in late 1965. He died at home in Princeton on 18 February 1967, aged 62. His ashes were scattered off the island of St. John in the US Virgin Islands, where the family had a beach house.
Legacy
- Physics: the Born–Oppenheimer approximation underlies all of quantum chemistry. The TOV equation and the Oppenheimer–Snyder collapse founded the physics of neutron stars and black holes.
- Teaching: the Berkeley school produced many leaders of postwar American physics and made the United States a centre of theoretical physics.
- Science and the state: as director of Los Alamos and as adviser he defined the role of the scientist in national security, and the hearing became the defining case of its limits.
- Memory: Los Alamos National Laboratory, the asteroid 67085 Oppenheimer and a lunar crater carry his name, and his life has been the subject of plays, operas and films.
Final projectChoose one paper from Part II. Write a two-page account for a strong undergraduate that states its main result, reproduces one calculation, and explains what later work confirmed, corrected or extended.
Connections
People linked to J. Robert Oppenheimer across this collection. A name without a link has no page here yet. See J. Robert Oppenheimer in the network of all 337 lives →
Teachers and advisors
- Max Born 1926-1927
Students
- David Bohm
- George Volkoff
- Hartland Snyder
- Melba Phillips 1933
- Philip Morrison
- Robert Christy
- Willis Lamb 1938
Influenced by
Influenced
- John Archibald Wheeler
- Roger Penrose
Colleagues and collaborators
- Albert Einstein 1947-1955
- Arthur Compton 1942-1945
- Enrico Fermi
- Ernest Rutherford 1925-1926
- Hans Bethe 1942-1967
- James Chadwick 1944-1945
- John von Neumann
- Niels Bohr
- Paul Ehrenfest 1928
- Ernest Lawrence 1929-1954
- Richard Tolman 1929-1948
- Robert Serber 1938-1945
- Wendell Furry 1934
Rivals
- Edward Teller
Friends
Family
- Frank Oppenheimer
Also mentioned in this life
- Charles-Augustin de Coulomb
- Benjamin Franklin
- George Gamow
- Otto Hahn
- Werner Heisenberg
- J. J. Thomson
- Lise Meitner
- Arnold Sommerfeld
- Hermann Weyl
Fields
- theoretical physics 11 lives
- quantum mechanics 6 lives
- nuclear physics 13 lives
- astrophysics 6 lives
On CoursesHub
Sources and further reading
Biography
- J. Robert Oppenheimer, Wikipedia (with extensive references).
- H. A. Bethe, J. Robert Oppenheimer 1904–1967, Biographical Memoirs of the National Academy of Sciences.
- Christ’s College Cambridge, J. Robert Oppenheimer (1904–1967).
- US National Park Service, The life of J. Robert Oppenheimer after the Manhattan Project.
- K. Bird and M. J. Sherwin, American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer, Knopf, 2005.
- R. Rhodes, The Making of the Atomic Bomb, Simon & Schuster, 1986.
- R. Monk, Inside the Centre: The Life of J. Robert Oppenheimer, Jonathan Cape, 2012.
The 1954 hearing
- Oppenheimer security hearing, Wikipedia.
- US Atomic Energy Commission, In the Matter of J. Robert Oppenheimer: Transcript of Hearing before Personnel Security Board, Government Printing Office, 1954; full transcript released by the Department of Energy, 2014.
- US Department of Energy, Order of the Secretary vacating the 1954 decision, 16 December 2022.
- C. Carson and D. A. Hollinger (eds.), Reappraising Oppenheimer: Centennial Studies and Reflections, Office for History of Science and Technology, UC Berkeley, 2005.
Physics
- M. Born and J. R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys. 389 (1927) 457–484.
- P. A. M. Dirac, A theory of electrons and protons, Proc. R. Soc. A 126 (1930).
- J. R. Oppenheimer, On the theory of electrons and protons, Phys. Rev. 35 (1930).
- W. H. Furry and J. R. Oppenheimer, On the theory of the electron and positive, Phys. Rev. 45 (1934) 245–262.
- J. R. Oppenheimer and M. Phillips, Note on the transmutation function for deuterons, Phys. Rev. 48 (1935) 500–502.
- J. F. Carlson and J. R. Oppenheimer, On multiplicative showers, Phys. Rev. 51 (1937).
- H. J. Bhabha and W. Heitler, The passage of fast electrons and the theory of cosmic showers, Proc. R. Soc. A 159 (1937).
- R. C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev. 55 (1939) 364–373.
- J. R. Oppenheimer and G. M. Volkoff, On massive neutron cores, Phys. Rev. 55 (1939) 374–381.
- J. R. Oppenheimer and H. Snyder, On continued gravitational contraction, Phys. Rev. 56 (1939) 455–459.
- Born–Oppenheimer approximation, Tolman–Oppenheimer–Volkoff limit, Oppenheimer–Snyder model, Wikipedia.
- R. Serber, The Los Alamos Primer, University of California Press, 1992.
- K. S. Thorne, Black Holes and Time Warps, Norton, 1994, chapters 5 and 6.
- M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford, 1954.