The Dreams That Stuff is Made of

by Stephen Hawking

  The calculated separation between the levels and 2lt P3/2 is 0.365 cm−1 and corresponds to a wave-length of 2.74 cm. The great*Publication assisted by the Ernest Kempton Adams Fund for Physical Research of Columbia University, New York.

  **Work supported by the Signal Corps under contract number W 36–039 sc-32003.

  Reprinted with permission from the American Physical Society: Lamb & Rutherford, Physical Review, Volume 72, p. 241, 1947. ©1947, by the American Physical Society.

  wartime advances in microwave techniques in the vicinity of three centimeters wave-length make possible the use of new physical tools for a study of the n = 2 fine structure states of the hydrogen atom. A little consideration shows that it would be exceedingly difficult to detect the direct absorption of radiofrequency radiation by excited H atoms in a gas discharge because of their small population and the high background absorption due to electrons. Instead, we have found a method depending on a novel property of the level. According to the Dirac theory, this state exactly coincides in energy with the state which is the lower of the two P states. The S state in the absence of external electric fields is metastable. The radiative transition to the ground state is forbidden by the selection rule ΔL = ±1. Calculations of Breit and Tellerlv have shown that the most probable decay mechanism is double quantum emission with a lifetime of 1/7 second. This is to be contrasted with a lifetime of only 1.6 × 10−9 second for the nonmetastable 22P states. The metastability is very much reduced in the presence of external electric fieldslw owing to Stark effect mixing of the S and P levels with resultant rapid decay of the combined state. If for any reason, the level does not exactly coincide with the level, the vulnerability of the state to external fields will be reduced. Such a removal of the accidental degeneracy may arise from any defect in the theory or may be brought about by the Zeeman splitting of the levels in an external magnetic field.

  In brief, the experimental arrangement used is the following: Molecular hydrogen is thermally dissociated in a tungsten oven, and a jet of atoms emerges from a slit to be cross-bombarded by an electron stream. About one part in a hundred million of the atoms is thereby excited to the metastable state. The metastable atoms (with a small recoil deflection) move on out of the bombardment region and are detected by the process of electron ejection from a metal target. The electron current is measured with an FP-54 electrometer tube and a sensitive galvanometer.

  If the beam of metastable atoms is subjected to any perturbing fields which cause a transition to any of the 22P states, the atoms will decay while moving through a very small distance. As a result, the beam current will decrease, since the detector does not respond to atoms in the ground state. Such a transition may be induced by the application to the beam of a static electric field somewhere between source and detector. Transitions may also be induced by radiofrequency radiation for which hv corresponds to the energy difference between one of the Zeeman components of and any component of either or 22P3/2. Such measurements provide a precise method for the location of the state relative to the P states, as well as the distance between the latter states.

  We have observed an electrometer current of the order of 10−14 ampere which must be ascribed to metastable hydrogen atoms. The strong quenching effect of static electric fields has been observed, and the voltage gradient necessary for this has a reasonable dependence on magnetic field strength.

  We have also observed the decrease in the beam of metastable atoms caused by microwaves in the wave-length range 2.4 to 18.5 cm in various magnetic fields. In the measurements, the frequency of the r-f is fixed, and the change in the galvanometer current due to interruption of the r-f is determined as a function of magnetic field strength. A typical curve of quenching versus magnetic field is shown in Fig. 1. We have plotted in Fig. 2 the resonance magnetic fields for various frequencies in the vicinity of 10,000 Mc/sec. The theoretically calculated curves for the Zeeman effect are drawn as solid curves, while for comparison with the observed points, the calculated curves have been shifted downward by 1000 Mc/sec. (broken curves). The results indicate clearly that, contrary to theory but in essential agreement with Pasternack’s hypothesis,3 the state is higher than the by about 1000 Mc/sec. (0.033 cm−1 or about 9 percent of the spin relativity doublet separation. The lower frequency transitions = . have also been observed and agree well with such a shift of the level. With the present precision,

  FIG. 1 A typical plot of galvanometer deflection due to interruption of the microwave radiation as a function of magnetic field. The magnetic field was calibrated with a flip coil and may be subject to some error which can be largely eliminated in a more refined apparatus. The width of the curves is probably due to the following causes: (1) the radiative line width of about 100 Mc/sec. of the 2P states, (2) hyperfine splitting of the 2S state which amounts to about 88 Mc/sec., (3) the use of an excessive intensity of radiation which gives increased absorption in the wings of the lines, and (4) inhomogeneity of the magnetic field. No transitions from the state have been observed, but atoms in this state may be quenched by stray electric fields because of the more nearly exact degeneracy with the Zeeman pattern of the 2P states.

  we have not yet detected any discrepancy between the Dirac theory and the doublet separation of the P levels. (According to most of the imaginable theoretical explanations of the shift, the doublet separation would not be affected as much as the relative location of the S and P states.) With proposed refinements in sensitivity, magnetic field homogeneity, and calibration, it is hoped to locate the S level with respect to each P level to an accuracy of at least ten Mc/sec. By addition of these frequencies and assumption of the theoretical formula Δv = for the doublet separation, it should be possible to measure

  FIG. 2 Experimental values for resonance magnetic fields for various frequencies are shown by circles. The solid curves show three of the theoretically expected variations, and the broken curves are obtained by shifting these down by 1000 Mc/sec. This is done merely for the sake of comparison, and it is not implied that this would represent a “best fit.” The plot covers only a small range of the frequency and magnetic field scale covered by our data, but a complete plot would not show up clearly on a small scale, and the shift indicated by the remainder of the data is quite compatible with a shift of 1000 Mc.

  the square of the fine structure constant times the Rydberg frequency to an accuracy of 0.1 percent.

  By a slight extension of the method, it is hoped to determine the hyperfine structure of the state. All of these measurements will be repeated for deuterium and other hydrogen-like atoms.

  A paper giving a fuller account of the experimental and theoretical details of the method is being prepared, and this will contain later and more accurate data.

  The experiments described here were discussed at the Conference on the Foundations of Quantum Mechanics held at Shelter Island on June 1–3, 1947 which was sponsored by the National Academy of Sciences.

  THE ELECTROMAGNETIC SHIFT OF ENERGY LEVELS

  BY

  HANS. A. BETHE

  BY very beautiful experiments, Lamb and Retherfordlx have shown that the fine structure of the second quantum state of hydrogen does not agree with the prediction of the Dirac theory. The 2s level, which according to Dirac’s theory should coincide with the level, is actually higher than the latter by an amount of about 0.033 cm−1 or 1000 megacycles. This discrepancy had long been suspected from spectroscopic measurements.ly,lz However, so far no satisfactory theoretical explanation has been given. Kemble and Present, and Pasternackma have shown that the shift of the 2s level cannot be explained by a nuclear interaction of reasonable magnitude, and Uehlingmb has investigated the effect of the “polarization of the vacuum” in the Dirac hole theory, and has found that this effect also is much too small and has, in addition, the wrong sign.

  Schwinger and Weisskopf, and Oppenheimer have suggested that a possible explanation might be the shift of energy levels by the interaction of the electron with the radiation
field. This shift comes out infinite in all existing theories, and has therefore always been ignored. However, it is possible to identify the most strongly (linearly) divergent term in the level shift with an electromagnetic mass effect which must exist for a bound as well as for a free electron. This effect should properly be regarded as already included in the observed mass Reprinted with permission from the American Physical Society: Betje, Physical Review, Volume 72, p. 339, 1947.

  ©1947, by the American Physical Soceity.

  of the electron, and we must therefore subtract from the theoretical expression, the corresponding expression for a free electron of the same average kinetic energy. The result then diverges only logarithmically (instead of linearly) in non-relativistic theory: Accordingly, it may be expected that in the hole theory, in which the main term (self-energy of the electron) diverges only logarithmically, the result will be convergent after subtraction of the free electron expression.mc This would set an effective upper limit of the order of mc2 to the frequencies of light which effectively contribute to the shift of the level of a bound electron. I have not carried out the relativistic calculations, but I shall assume that such an effective relativistic limit exists.

  The ordinary radiation theory gives the following result for the self-energy of an electron in a quantum state m, due to its interaction with transverse electromagnetic waves:

  (1)

  where k = kw is the energy of the quantum and v is the velocity of the electron which, in non-relativistic theory, is given by

  (2)

  Relativistically, v should be replaced by cα where α is the Dirac operator. Retardation has been neglected and can actually be shown to make no substantial difference. The sum in (1) goes over all atomic states n, the integral over all quantum energies k up to some maximum K to be discussed later.

  For a free electron, v has only diagonal elements and (1) is replaced by

  (3)

  This expression represents the change of the kinetic energy of the electron for fixed momentum, due to the fact that electromagnetic mass is added to the mass of the electron. This electromagnetic mass is already contained in the experimental electron mass; the contribution (3) to the energy should therefore be disregarded. For a bound electron, v2 should be replaced by its expectation value, (v2)mm . But the matrix elements of v satisfy the sum rule

  (4)

  Therefore the relevant part of the self-energy becomes

  (5)

  This we shall consider as a true shift of the levels due to radiation interaction.

  It is convenient to integrate (5) first over k. Assuming K to be large compared with all energy differences En − Em in the atom,

  (6)

  (If En − Em is negative, it is easily seen that the principal value of the integral must be taken, as was done in (6).) Since we expect that relativity theory will provide a natural cut-off for the frequency k, we shall assume that in (6)

  (7)

  (This does not imply the same limit in Eqs. (2) and (3).) The argument in the logarithm in (6) is therefore very large; accordingly, it seems permissible to consider the logarithm as constant (independent of n) in first approximation.

  We therefore should calculate

  (8)

  This sum is well known; it is

  (9)

  for a nuclear charge Z. For any electron with angular momentum l ≠ 0, the wave function vanishes at the nucleus; therefore, the sum A = 0. For example, for the 2p level the negative contribution A1s2,p balances the positive contributions from all other transitions. For a state with l = 0, however,

  (10)

  where n is the principal quantum number and a is the Bohr radius.

  Inserting (10) and (9) into (6) and using relations between atomic constants, we get for an S state

  (11)

  where Ry is the ionization energy of the ground state of hydrogen. The shift for the 2p state is negligible; the logarithm in (11) is replaced by a value of about −0.04. The average excitation energy (En − Em)AV for the 2s state of hydrogen has been calculated numericallymd and found to be 17.8 Ry, an amazingly high value. Using this figure and K = mc2, the logarithm has the value 7.63, and we find

  (12)

  This is in excellent agreement with the observed value of 1000 megacycles.

  A relativistic calculation to establish the limit K is in progress. Even without exact knowledge of K, however, the agreement is sufficiently good to give confidence in the basic theory. This shows(1) that the level shift due to interaction with radiation is a real effect and is of finite magnitude,

  (2) that the effect of the infinite electromagnetic mass of a point electron can be eliminated by proper identification of terms in the Dirac radiation theory,

  (3) that an accurate experimental and theoretical investigation of the level shift may establish relativistic effects (e.g., Dirac hole theory). These effects will be of the order of unity in comparison with the logarithm in Eq. (11).

  If the present theory is correct, the level shift should increase roughly as Z4 but not quite so rapidly, because of the variation of (En − Em)AV in the logarithm. For example, for He+, the shift of the 2s level should be about 13 times its value for hydrogen, giving 0.43 cm−1, and that of the 3s level about 0.13 cm−1. For the x-ray levels LI and LII, this effect should be superposed upon the effect of screening which it partly compensates. An accurate theoretical calculation of the screening is being undertaken to establish this point.

  This paper grew out of extensive discussions at the Theoretical Physics Conference on Shelter Island, June 2 to 4, 1947. The author wishes to express his appreciation to the National Academy of Science which sponsored this stimulating conference.

  Chapter Eight

  As we have seen, the problem of infinite energy plagued the development of quantum electrodynamics for much of the 1930s and 40s. In 1947, Hans Bethe figured out a method for removing the infinity from the electron self-energy calculation. His calculation was non-relativistic, but set the stage for the relativistic completion of quantum electrodynamics done independently by Sin-Itiro Tomanaga, Julian Schwinger, and Richard Feynman. The method for removing infinite energy from calculations in quantum electrodynamics is now called renormalization. The basic idea of renormalization is to impose a cutoff in the energy and carefully take the limit as the cutoff approaches infinity. A cutoff in energy amounts to imposing a length scale below which variations in the field are ignored. A higher cutoff energy corresponds to a smaller length scale. With a finite length scale the mathematical terms that produced infinite energy become regularized and finite. Most importantly they cancel other terms that also produce infinite energies. With these troubling terms safely canceling each other, the limit that the length scale approaches zero can be taken in such a way that the calculation remains finite.

  Schwinger’s and Tomonaga’s approaches to quantum electrodynamics were similar and are now grouped together. Feynman’s, on the other hand, was completely different. Schwinger’s approach was more straightforwardly based on previous work, but involved mathematics so complicated that it obscured the basic physical processes. In fact Freeman Dyson once said that it was “something that needed such skills that nobody besides Schwinger could do it.”me Feynman’s approach was more intuitive. He constructed pictorial representations, known as Feynman diagrams, of the process in question and a set of rules describing how to interpret the diagrams. Using these rules it is possible to set up mathematical expressions that when evaluated give contributions to the probability for a particular transition to occur. In 1948 Dyson published a proof showing the equivalency of the Tomonaga, Schwinger and Feynman methods. Incidentally, because Feynman was slow to publish his work, Dyson’s proof appeared before Feynman had even formally published his method!

  In developing his method for solving quantum electrodynamics, Feynman, with the aid of John Wheeler, gave a new interpretation to the positron. They claimed that a positron is an electron moving backward through time. I
t had been known for some time that when matter and anti-matter interact the result is a total annihilation of both and the production of high-energy photons. In Feynman’s method, the annihilation of a positron-electron pair can be understood as a single electron interacting with an electromagnetic field in the form two photons and then reversing direction in time—ie. becoming the incident positron. Although an electron moving backward in time seems strange, it is in fact completely equivalent to a positron moving forward in time.

  The renormalization method is not without its detractors. It relies on mathematics that at the time seemed murky. Paul Dirac said about renormalization, “This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small—not neglecting it just because it is infinitely great and you do not want it!”mf Nevertheless, renormalized quantum electrodynamics is one of the most successful theories ever produced. Its predictions match observations to an astounding degree of accuracy.