INHOMOGENEOUS ELECTRON GAS IV CONCLUDING REMARKS inhomogeneous electron gas and may suggest further In the preceding sections we have developed a theory of the electronic ground state which is exact in two ACKNOWLEDG MENTS limiting cases: The case of a nearly constant density This work was begun and, to a considerable extent (n=no+n(n), n(r)/nu<<1) and the case of a slowly varying density. Actual electronic systems do not belong authors(P. Hohenberg) acknowledges with thanks a to either of these two categories. The most promising NAto Postdoctoral Fellowship; the other author(w formulation of the theory at present appears to be that Kohn)a Guggenheim Fellowship. Both authors wish obtained by partial summation of the gradient expan- to thank the faculties of the Ecole Normale Super sion (Sec. III. 4). It has, however, not yet been tested Paris, and the Service de Physique des Solides, Orsay in actual physical problems. But regardless of the out- for their hospitality, and Professor A. Blandin, P come of this test, it is hoped that the considerations of J. Friedel, Dr. R. Balian, and Dr. C. De D this paper shed some new light on the problem of the for valuable discussions PHYSICAL REVIEW VOLUME 136. NUMBER 3B 9 NOVEMBER 1964 Scattering of a High-Intensity, Low-Frequency Electromagnetic Wave by an Unbound Electron ZOLTAN FRIEDt U.S. Noval ordnance Lo and Universily of California, Santa Barbara, california JoSEPH H. EBERLY U. S. Naval Ordnance Laboralory, Silver Spring, Maryland Received 15 June 1964) Thomsonscattering of a high-intensity, low-frequency, circularly-polarized electromagnetic wave by a free electron is considered. We find that by neglecting radiative corrections and pair effects, the Feynman Dyson perturbation expansion is summable, and the sum can be analytically continued in the form of a sum of continued fractions. By imposing the boundary conditions that at he photons and target electron opagate as free particles, we obtain results which differ from those reported by Brown and Kibble and by ldman. In particular our results differ in two aspects. The first difference is in the kinematics; nd Kibble and of Goldman using a mixed set of classical and quantum boundary valr I INTRODUCTION second area of concentration is the question of proper lE advent of masers and lasers has stimulated a description of the electromagnetic radiation emanating great deal of interest in the interaction of intense from a laser; i.e. questions of coherence and correla electromagnetic fields with matter. This activity has tion. And finally, the problem of interaction of laser been focused on three different aspects of the subject. light with matter has attracted considerable interest First, a great deal of attention has been devoted to the It is this latter question to which we are devoting our dynamics of production of high-intensity light. 1 A selves in this paper The particular problem of immediate interest is the A preliminary version of this work was presented at the effect of the presence of the high-intensity field on the Compton(Thomson) scattering amplitude. Recall that ological Instit Massachusetts; on leave from the U. S. Nava rell, the Thomson amplitude describes the scattering of a Council R Glauber, Phys. Rev. 130, 2529(1963); E C G, Sudarshan, 10,277(196 hysics, Varenna, 19(3(unpublished) 8(1963I NHOMOGENEOUS ELECTRON GAS IV. CONCLUDING REMARKS In the preceding sections we have developed a theory of the electronic ground state which is exact in two limiting cases: The case of a nearly constant density (Is=np+rI(r), rI(r)/ep((1) and the case of a slowly varying density. Actual electronic systems do not belong to either of these two categories. The most promising formulation of the theory at present appears to be that obtained by partial summation of the gradient expan￾sion (Sec. III.4). It has, however, not yet been tested in actual physical problems. But regardless of the out￾come of this test, it is hoped that the considerations of this paper shed some new light on the problem of the inhomogeneous electron gas and may suggest further developments. ACKNOWLEDGMENTS This work was begun and, to a considerable extent, carried out at the University of Paris. One of the authors (P. Hohenberg) acknowledges with thanks a NATO Postdoctoral Fellowship; the other author (W. Kohn) a Guggenheim Fellowship. Both authors wish to thank the faculties of the Ecole Normale Superieure, Paris, and the Service de Physique des Solides, Orsay, for their hospitality, and Professor A. Blandin, Professor J. Friedel, Dr. R. Balian, and Dr. C. De Dominicis for valuable discussions. PHYSICAL REVIEW VOLUME 136, NUM BER 3 B 9 iVOVEM B ER 1964 Scattering of a High-Intensity, Low-Frequency Electromagnetic Wave by an Unbound Electron* ZOLTAN FRIED) U. S. iVaval Ordnance Laboratory, Silver Spring, Maryland und Univ' sity of California, Santa Barbara, California AND IosEPII H. EszRLvf. U. S. Naval Ordnance Laboratory, Silver Spring, Maryland (Received 15 June 1964) "Thomson" scattering of a high-intensity, low-frequency, circularly-polarized electromagnetic wave by a free electron is considered. We find that by neglecting radiative corrections and pair e6ects, the Feynman￾Dyson perturbation expansion is summable, and the sum can be analytically continued in the form of a sum of continued fractions. By imposing the boundary conditions that at t=&~ the photons and target electron propagate as free particles, we obtain results which differ from those reported by Brown and Kibble and by Goldman. In particular our results dier in two aspects. The 6rst difference is in the kinematics; namely, we find no intensity-dependent frequency shift in the scattered photon. The second difference is in the dynamics; that is, we obtain a different expression for the scattering amplitude. Both of these changes originate in the choice of boundary conditions. Instead of treating the asymptotic radiation 6eld classically, we choose our states as linear combinations of occupation-number states. Finally, contact is made with the results of Brown and Kibble and of Goldman using a mixed set of classical and quantum boundary values. I. INTRODUCTION 'HE advent of masers and lasers has stimulated a great deal of interest in the interaction of intense electromagnetic 6elds with matter. This activity has been focused on three different aspects of the subject. First, a great deal of attention has been devoted to the dynamics of production of high-intensity light. A *A preliminary version of this work was presented at the Pasadena Meeting of the American Physical Society, Bull. Am. Phys. Soc. 8, 615 (1963). f Present address: Lowell Technological Institute, Lowell, Massachusetts; on leave from the U. S. Naval Ordnance Laboratory. $ National Academy of Sciences—National Research Council Postdoctoral Research Associate, 1962-64. ' J. R. Singer, %users (John Wiley R Sons, Inc., New York, 1900); F. Schwabl and W. Thirring (to be published); W. E. Lamb, Jr., Lecture Notes, Enrico Fermi International School of Physics, Varenna, 1963 (unpublished). second area of concentration is the question of proper description of the electromagnetic radiation emanating from a laser; i.e., questions of coherence and correla￾tion. ' And finally, the problem of interaction of laser light with matter has attracted considerable interest. ' It is this latter question to which we are devoting our￾selves in this paper. The particular problem of immediate interest is the effect of the presence of the high-intensity field on the Compton (Thomson) scattering amplitude. Recall that the Thomson amplitude describes the scattering of a ' R. Glauber, Phys. Rev. 130, 2529 (1963);E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963); E. Wolf, Proc. Phys. Soc. (London) 80, 1269 (1962). ~ J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962); Z. Fried s.nd W. M. Frank, Nuovo Cimento 27, 218 (1963)