Collected Papers of L. D. Landau

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D. Ter Haar (Editor)

Collected Papers of L.D. LANDAU

Collected Papers of L. D. LANDAU E D I T E D A N D W I T H AN I N T R O D U C T I O N

BY

D. TER HAAR

G O R D O N AND B R E A C H , S C IE N C E P U B L I S H E R S NEW YORK * LONDON

■ PARIS

Gordon and Breach, Science Publishers, Inc., 150 Fifth Avenue, New York 11, N Y . Gordon and Breach, Science Publishers Ltd., 171 Strand, London W.C. 2

Copyright (cy 1965 Pergamo.n Press Lid. and Gordon and Breach, Science Publishers, Inc.

First edition 1965

This book, is published jointly by Cordon and Breach, Science Publishers Inc. and Pcrgainon Press Ltd.

Library of Congress Catalog Card No. 64-17191

CONTENTS Frontispiece

Facing page

Proface

iii xi

Introduction

xiii

1. On the theory of the spectra of diatomio molecules (Z. P h y s 10, 621,1926)

1

2. The damping problem in wave mechanics (Z . Phys.t 45, 430, 1927)

8

3. Quantum electrodynamics in configuration space (Z. Phys., G2, 188, 1930; with R, Peierls)

19

4. Diamagnetism of metals (Z. Phys., 04, 629, 1930)

31

5. Koto on tho scattering of hard gamma-rays (Natunoiss., 18, 1112, 1930)

39

6. Extension of the uncertainty principle to relativistic quantum theory (Z. Phys., 69, 1931; with R. Peierls)

40

7.

theory of energy transfer on collisions {Phys. Z. Sowjei., 1, 88, 1932)

8. Gi the theory of stars

(Phys. Z. Sowjet., 1, 285, 1932)

9. 1, theory of energy transfer II (Phys. Z. Sowjet., 2, 46, 1032) 10.

Beotron motion in crystal lattices (Phys. 7j . Sowjei., 3, 664, 1933)

52 60 03 67

IL 0\\ the second law of thermodynamics and the universe (Phys. Z. Sowjct., 4, 114, 1933; with M. Bronstein)

69

12. A possible explanation of the field dependence of the susceptibility at low temperatures (Phys. Z. Sowjet., 4, 675, 1933)

73

13- Internal temperature of stare (Nature, 132, 567, 1933; with G. Ganiow)

77

14. Structure of the undisplaced scattering line (Phys. 7t. Sowjct., 6, 172, 1934; with G. Placzck)

79

15. On tho theory of the slowing down of fast electrons by radiation, (JETP*, 5, 255, 1935; Phys. Z. Sowjct., 5, 761, 1934)

80

16. On the production of electrons and positrons by a collision of two particles (Phys. rZ. Sowjct., 6, 244, 1934; with K Lifsbitz) "

84

17. OA the theory of specific heat anomalies (Phys. Z. Sowjei., 8, 113, 1935)

96

* JE T P — Journal of Experimental and Theoretical Physics of the U.S.S.R.

VI

CONTENTS

18. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies (Phys. Z. Sowjet., 8> 153, 1935; with E. Lifstatz)

1

q1

19. On the relativistic correction of the Schrodinger equation foT the many-body problem (Phys. Z. Soiojet., 8, 487, 1935) U5 20. On the theory of the accommodation coefficient {Phys. Z. Soiojet., 8, 489, 1935)

117

21. On the theory of the photoelcctromotivo force in semiconductors (Phys. Z. Sovjjet 9, 477, 1936; with E. Lifshitz)

126

22. On the theory of sound dispersion (Phys. Z. Sowjet., 10, 34, 1936; with E. Teller)

147

23. On the theory of uni-molecular reactions (Phys. Z. Soiojet., 10, 67, 1936)

Igg

24. The transport equation hi the case of Coulomb interactions (JETP, 7, 203, 1937; Phys. Z . SowjcL, 10, 154, 1936)

17, 548, 1940)

211

40. On tho nature of the nuclear forces (Phys. Rev., 58, 1006, 1940; O. R ♦ Acad. Sci. URSS, 29, 656, 1940; with I. Tamm)

272

41. On tho “ radius” of the elementary particles (J. Phys. U.S.S.R., 2, 485, 1940; Phys. Rev., 58, 1006, 1940;JE T P , 10, 718, 1940)

274

42. On tho scattering of mesotrons by “ nuclear forces” (7. Phys. U.S.8.R., 2, 483, 1940; JETP, 10, 721, 1940) '

278

43. The angular distribution of the shower particles (J. Phys. U.8.S.R., 3,237,1040; JE TP, 10, 1007, 1940)

280

:4» On tho theory of secondary showers (J . Phys. J7.xSf.jSU?., 4, 376, 1941; JE T P , 11, 32, 1941) '

289

46. On tho scattering of light by mesotrons (J. Phys. J7.jSf.x9-J?,, 4,455,1941; J E T P ,1 1 ,35, .1941; with J. Smorodinski)

292

£6. The theory of superfluidity of helium II (J. Phys. J7.j$f.$f£, 17, 42, 1942 ; with B. Levich)

355

49. On the theory of the intermediate state of superconductors (J. Phys. UJ3.S.R., 7, 99, 1943; JETP, IB, 377, 1943)

365

50. On the relation between the liquid and the gaseous states of metals (Acta Phys.-chim. URS8, 18, 194, 1943; JE T P , 14, 32, 1944; with J. Zeldovich)

380

51. A new exact solution of the Navier-Stokes equations (O. R. Acad. Sci. URSS, 48, 286, 1944; Dokl. Akad. Nauk 8SSR, 43, 299,1944)

383

52. On the problem of turbulence (O. R. Acad. jSfci. URSS, 44, 311, 1944; Dokl. Akad. Nauk 88SR, 44, 339, 1944)

387

53. On tho hydrodynamics of helium II (J. Phys. U.8.S.R., 8, 1, 1944; JE T P , 14, 112, 1944)

392

54. On the theory of slow combustion (Acta Phys.-chim. URSS, If), 77, 1944; JETP, 14, 240, 1944)

390

65. On the theory of scattering of protons by protons (J. Phys. U.S.8.R., 8, 154, 1944; JETP, 14, 269, 1944; with J. Smorodinski) 404 56. On the cnorgy loss of last particles by ionisation (J. Phys. U .8.8.R., 8, 201, 1944) 67.

417

On a study of the detonation of condensed explosives {0 . R . Acad. Sci. URSS, 46, 362, 1945; Dokl. Akad. Nauk S8SR, 40, 399, 1945; with K. I \ Staninkovich) 425

viii

CONTENTS

58* The determination of the flow velooity of the detonation products ofsdmo gaseous mixtures (0. R. Acad. 8ci. UR88y 47, 199, 1945; DoH. Akad. Nauh S88R, 47, 205, 1945; with It. V. Staniukovioh) 429 59. Determination of the flow velocity of tho detonation products of condensed explosives (C. 11. Acad. Sci. UR88. 47, 271, 1945; Dole. Akad. Nauh 88SR, 47, 273, 1945; with K. P. Sfcaniukovich)

432

60. On shock waves at large distances from tho place of their origin (J. Phy$. U.8.S.R.% 9, 496, 1945; Prikl. Mat. Mehli. 9, 286, 1945) 437 61. On tho vibrations of the electronic plasma (J. Pkys. JJ.8.8.R. 10, 25, 1946; JETP, 16, 574, 1946)

445

62. On the thermodynamics of photolumincsconcc [J. Pkys. U.8.8.R., 10, 603, 1946)

46:1

63. On the theory of superfluidity of helium II (J. Phys. U.8.8.R., 11, 91, 1947)

46$

64. On the motion of foreign particles in helium TI {Dohl. Akad. Nauk 8 8 8 R, 59, 669* 1948; with I. Pomeranehuk) 46$ 65. On tho angular momentum of a system of two photons {Dohl. Akad. Nauh 888R, 69, 207, 1948)

471

66.. On the theory of superfluidity (Pkys. Rev., 76, 884, 1949; Dohl. Ahad. Nauh 888R, 61, 253, 1948)

474

67. The effective mass of the polaron (JETP, 18, 419, 1948; with S. I. Pekar)

478:

68. On tho theory of energy transfer during collisions III (JETP, 18, 750, 1948; with E. Lifshitz)

484

69. The theory of tho viscosity of helium II: 1. Collisions of elementary excitations in helium II {JETP, 19, 637, 1949; with I. M. KJialatnikov) 494 70. Tho theory of tho viscosity of helium II. II. Calculation of the viscosity coefficient (JETP, 19, 709, 1949; with I. M. Khalatnikov)

511

71. On the electron-positron interaction (JETP, 19, 673, 1949; with V. B. Bovcstetskii)

532

72. The equilibrium form of orystals (A. P, Ioffe Festschrift, Moscow 1950, p. 44)

540

73. On the theory of superconductivity {JETP, 20, 1064, 1950; with V. L. Ginzburg)

546

74. On multiple production of particles during collisions of fast partioles {Izv. Akad. Nauh 888R , Ser. fiz., 17, 51, 1953) 569 75. The limits of applicability of the theory of Bremsstrahlung by electrons and of the oreation of pairs at large energies (Dohl. Akad. Nauh SSSR, 92, 535, 1953; with I. Pomeranehuk)

586

76. Electron-cascade processes at ultra-high energies (Dokl. Akad. Nauk SSSR, 92,735, 1953; with I. Pomeranehuk)

589

77. Emission of y-quanta during tho collision of fast ^-mesons with nuoleons {JETP, 24, 505, 1953; with I. Pomeranehuk)

594

INTRODUCTION

XV

types of elementary excitations. I t would be more correct to speak simply of the long wave and short wave excitations ”. In that second paper he had fitted the energy spectrum to agree with Peshkov’s data on second sound propaga­ tion, and found that the two branches of the spectrum should continuously merge into one another—a feature since then abundantly corroborated by microscopic theories. In the earlier paper, Landau calculates the specific heat of liquid helium following from the energy spectrum, and shows that superfluidity is also a consequence of the spectrum inasmuch as the fluid, if moving with suf­ ficiently low velocity, will be unable to slow down by the excitation of a single elementary excitation. He also evaluates the velocity of sound in liquid helium, and finds th at apart from the ordinary sound waves, temperature waves may be propagated with a velocity different from that of ordinary sound. These waves are the so-called second-sound waves. Finally, he started the discussion of hydrodynamics of helium II, a subject further developed by him in a later paper53,66. In a paper with Pomeranchuk 64 he shows that impurities in helium II move with the normal and not with the superfluid part of the liquid. Landau's other contribution to the theory of liquid helium has been in two papers with Elhalatnikov69,70 in which they gave an extensive discussion of viscosity phenomena in liquid helium, based upon the idea that transport phenomena can be described in terms of collisions between the elementary excitations. Together with Lifshitz85 he has considered the problem of rotation in liquid helium; although the picture given there is not in agreement with the now generally accepted Feynman-Onsager theory it may be correct under certain conditions. I t is nowadays generally accepted that the peculiar properties of liquid 4He are due to the fact that the 4He-atoms are bosons. In three—by now classical— papers90*n *95 Landau developed a theory of Fermi liquids, that is, of a system of interacting fermions, the most important of which are the conduction electrons in a metal and liquid 3He. As in the case of 4He, the first problem is to find the energy spectrum for the elementary excitations. Landau 90 found this spectrum by assuming that in switching on the interaction between the fermions, the classification of the levels remains invariant, “ dressed 55 fermions taking the place of the original “ bare'' fermions. We find then that the energy of the excitations is a functional of the distribution function of the particles— and is thus temperature dependent. In the second paper91 Landau discusses the propagation of waves in a Fermi liquid and finds a new kind of “ sound” : zero sound or “ high frequency” sound. While ordinary “ low-frequency” sound with its rarefactions and compressions corresponds to the oscillation of the radius of the Fermi surface which remains spherical about its centre, zero sound corresponds to a periodic oscillation of the shape of the Fermi surface. In the third paper95 Landau discusses the forward scattering of the “ dressed” fermions, which determines the general properties of a Fermi liquid. 2. S o l id S t a t e P h y s i o s

Among Landau’s contributions to solid state physics we should first of all mention his paper on the diamagnetism of metals4. Classically one knows that a system of charged particles will have a vanishing magnetic susceptibility.

XVI

INTRODUCTION

However, quantum mechanically a magnetic field introduces a quantisation of the levels—the so-called Landau levels—and this discreteness of the levels leads to a non-vanishing diamagnetism. This effect becomes expecially notice­ able at low temperatures, and the susceptibility shows a periodic variation with the magnetic field (de Haas-van Alphen effect) due to the changing rela­ tive position of the highest occupied level and the Termi level. This effect has been studied experimentally, notably by Shoenberg, and was theoretically treated by Landau38. Landau discussed12 the field-dependence of the low-temperature magnetic susceptibility of such anti-ferromagnetic substances as chromium chloride and with Lifshitz 18 gave a thermodynamic theory of domain structure as well as the basic equation of motion for the magnetisation in ferromagnetics and a first theory of ferromagnetic resonance. In two papers in 193729 Landau developed a theory of second-order phase transitions, pointing out the close connexion between such transitions and symmetry properties. In a paper with Khalatnikov 82 he showed that near a second-order phase transition point anomalous sound absorption occurs, while he has also considered32 the scattering of X-rays near such a transition point. In an earlier paper on a related topic17 Landau had considered the specific heat anomalies near a critical point and found below the critical point a {Tc — T)lj2 law. Zeldovich and Landau 50 discussed the relations between the liquid, dielectric, and metallic forms of a substance, while Landau also con­ sidered the scattering of X-rays by crystals with a variable lamellar structure 33 and the equilibrium shape of crystals72. Together with Rumer3Sthe absorption of short wavelength sound was investigated, and the absorption was considered to be due to sound-wave-lattiee-wave collisions. In 1933 Landau 10 suggested the possibility that an electron might dig its own hole in a crystal through its polarising action. This idea is essential for polaron theory, and in a later paper with Pekar 67 the effective mass of a polaroii was evaluated. With Lifshitz30 photoconductivity in semi-conductors was studied. We finally mention a paper with Pomeranchuk23 on the electrical conduc­ tivity and thermo-electric power of metals at low temperatures and a paper20 on the theory of the accommodation coefficient both at low'temperatures where quantum effects become important and at high temperatures where the classi­ cal theory holds. 3. P l a s m a P h y s i c s

In plasma physics Landau has made two important contributions. The first one24 was his derivation of a transport equation for a system of charged-par­ ticles, where the long range of the Coulomb forces makes it impossbile to use the normal Boltzmann equation. The second one61 was a paper -where he dis­ cusses in some detail plasma oscillations and shows th at these are always damped (Landau damping).

INTRODUCT!ON

XVH

4, H y d r o d y n a m i c s

Apart- from providing physics with one of the best existing text books on hydrodynamics, Landau has also from time to time contributed short notes on various problems in hydrodynamics and aerodynamics. With Levich *6*8 he studied the formation of a thin liquid layer upon the surface of a solid which is dragged through the liquid, and in a later note 51 he showed that the motion in an axially symmetric jet can be determined for arbitrary Reynolds numbers by solving the Navier-Stokes equations rigorously. Landau investigated 50 the behaviour of shock waves at large distances from their point of origin, pointing to the existence of two (rather than one) shock waves. With Liishitz 83 he used the Euler-Tricomi equations to study wreak discontinuities at the sonic line. The onset of turbulence as an eigenvalue problem was studied 52 in order to further the understanding of turbulence, while with Lifshitz93 hydrodynamic fluctuations were considered. We finally mention three papers with Staniukovich57, 58>59 on the hydro­ dynamics of detonating gaseous mixtures and condensed explosives. 5. A s t r o p h y s i c s

Landau has also made a few brief excursions into astrophysics, and again as one is bound to expect, with interesting results. In his first paper on this subject8 he discusses in general terms stellar equilibrium. He points out the importance of quantum statistics and relativity for the equation of state of matter at the centre of a star, and comes to the conclusion that for stars heavier than about 1*5 solar masses, quantum mechanics can no longer hold for parts of such stars. He discards the idea that “ some mysterious process of mutual annihilation of protons and electrons’" would supply stellar radiation, but suggests that rather the law- of conservation of energy should no longer be valid in the relativistic region6*. However, a year later he considers with Gamow13 the consequences of nuclear reactions for the temperature at the centre of a star and concludes that either lithium will not be present in any appreci­ able amount, or the central temperatures can not exceed a few million degrees. In his final paper on stellar energy27 Landau suggests that this energy is released by transforming ordinary matter into “ neutronie” matter, th at is, by pushing all electrons into the nuclei. He shows that this would be energeti­ cally favourable for systems with masses more than about one thousandth of a solar mass. 6. H rcLEAR P h y s i c s

and

C osmic R a y s

In 1937 Landau applied31 Bohr’s concept of a statistical theory of nuclei to find a relation between the level density and the neutron scattering width. He has also35 pointed out the importance of selection rules—especially the forbiddenness of transitions between states of different parity—for the

xvm

INTRODUCTION

stability of nuclei against ^-particle disintegration. With Tamm40he sketched a theory of nuclear forces. Landau has at various times studied scattering processes (see also the next section), for instance, scattering of mesons by nuclear forces42, scattering of light by mesons (with Smorodinski45), and proton-proton scattering (also with Smorodinski55). With Lifshitz 68 he has considered the energy transfer during neutron-deuteron and proton-deuteron scattering*, while with Pomeranchuk 77 he discussed the emission of gamma-quanta- when fast pions are scattered by nucleons. With Rumer, Landau has considered various processes leading to cosmic ray showers84, 36 especially the cascade theory of electronic showers. Follow­ ing these calculations Landau evaluated the angular distribution in a shower43 and also has developed a theory of secondary showers produced by mesons44. He has also studied the ionisation energy loss of fast particles58. One of his most widely quoted contributions to the theory of cosmic ray phenomena is the theory of multiple production of particles by fast incoming particles74. This theory is an extension of ideas, first proposed by Fermi. The essential fea­ tures of the production process are th at when two nucleons collide, a compound system is produced and within a small volume, Lorentz-contracted because of the high velocity of the centre-of-mass system, a large number of particles is created. The density in that small volume is so high and the mean free path so small that statistical equilibrium is established. After that the system ex­ pands, at first “ hydrodvnamically” , that is, in such a way that the mean free path of the constituents remains small, and finally in such a way that the indi­ vidual products of the collision-process can be distinguished. This theory was further developed by Belen’kii and Landau88. With Pomeranchuk76 Landau applied their considerations of Bremsstrahlung processes75 to electron-cascade processes at very high energies. 7. Q u a n t u m Me c h a n i c s

Among the many contributions by Landau to quantum mechanics we must first of all mention a fundamental paper with Peierls6 in which the applicabi­ lity of the uncertainty relations to relativistic quantum mechanics is studied. This attempt to establish in relativity restrictions on measuring processes whieh are not included in the uncertainty relations led to the well-known papers by Bohr and Rosenfeld on the measurement of electromagnetic fields. Landau also discussed10 the relativistic corrections to the many-body Schrodinger equation, the so-called Breit Hamiltonian. The problem of the angular momen­ tum of a system of two photons which is of importance in the theory of elec­ tron-positron annihilation processes was discussed by Landau in 194865. * It is interesting to note the title of this paper. Instead of a title such as L'Neutron-deuteron and proton-deuteron scattering, ” the authors gave the paper a title so that it looked like a continuation of refs. 7 and 9 in order to be able to publish this paper at a time when nuclear physics papers seldom obtained permission to be published.

INTRODUCTION

XIX

Apart from the scattering processes mentioned in the previous section, Lan­ dau has also made a general study of inelastic collisions7-0 and applied this theory, for instance, to excitation of nuclei, and excitation of vibrations during optical transitions. Bremsstrahlung emitted by fast electrons was evaluated by a method slightly simpler than Heitler’s original one15, and the limitations of the applicability of the theory were considered. With Pomeranchuk, Landau returned to this problem 75 in a paper in which pair-production was also taken into account, a topic also treated in a paper with Lifschitz16. The cross-section for photon-photon scattering was calculated for the case of very high energies25 with Akhieser and Pomeranchuk. In a short note in the Physical Review™ Landau pointed out that the failure to observe the theoretically prodicted polari­ sation of electrons through scattering was likely to be caused by the fact that most electrons underwent multiple rather than single scattering. Berestetskii and Landau 71 considered the second-quantised wave equation of the elec­ tron-positron system. Among Landau's earlier papers there are a few dealing with spectra. In the first paper he wrote alone1 Landau evaluated the band spectrum of diatomic molecules, the intensity distribution in the spectrum, and the Zeeman and Stark effect in the band spectrum. In a paper on damping in wave mechanics Landau introduced the density matrix to discuss coupled systems. In a short note in the Naturwissenschaften5Landau pointed out the importance of sum rules for estimating the intensity of spectral lines. Together with Placzek, Landau studied the fine structure of Rayleigh scattering lines, but a detailed paper on this topic never saw the light of day, and only the main conclusions of this study were published14. S. Q u a n t u m F i e l d T h e o r y

With Peierls, Landau derived5 the Heisenberg-Pauli quantum electrodyna­ mics in a slightly different manner while Landau41 has considered the limits beyond which electrodynamics loses its validity in quantum mechanics, and applied these considerations to electrons and mesons. With Abrikosov and Khalatnikov, Landau has considered the elimination of infinities in quantum electrodynamics75, the propagators or Green functions for photons80 and electrons79, and the electron mass81, while Landau and Pome­ ranchuk 86 discussed point interactions in quantum electrodynamics. A survey of the results of these papers were presented in the Nuovo Gimento supplement dealing with Russian physics89 and in the Bohr Festschrift84. The gauge trans­ formation of charged particle propagators and vertex operators were considered by Landau and Khalatnikov87. Landau94 also studied some spectral properties of the temperature-dependent single-particle Green functions. Abrikosov, Galanin, Gorkov, Landau, Pomeranchuk and Ter-Martirosyan96 have shown the impossibility of constructing a consistent strong-coupling fermion theory. The difficulty here is that as one increases the so-called cut-off limit, the physical interaction tends to zero independent of how large the bare coupling constant is.

XX

rNTBOBircTiojr

Recent developments in quantum field theory are mainly concerned with constructing theories without Hamiltonians and wave functions in which dis­ persion relations and diagram methods play the main role. One of the tasks to be accomplished is to find the singularities of the various quantities occurring in the theory, Landau98 {see also ref, 99 and a general discussion in ref, 100) has developed a general method to find such singularities. In connection with the developments following Lee and Yang’s theory of non-conservation of parity, Landau proposed92 the hypothesis of conservation of combined parity and also discussed in that connexion the properties of the neutrino. 9. M i s c e l l a n e o u s

In this last section we have collected all those papers which do not fall easily into the earlier categories. In an interesting paper with Bronstein 11 Landau discussed the connection with the second law of thermodynamics and the universe as a whole. Discussions with Vavilov led Landau to a study of the thermodynamics of photo-luminescence62 and the limitations imposed by thermodynamics upon its yield. The temperature dependence of the dis­ persion of sound in gases was studied together with Teller-2, Landau has also made some contributions to theoretical chemistry. He stu died the pressure dependence of the decomposition rate of large molecules25, the theory of slow combustion54, and the stability of sols and the coalescence of charged particles in electrolytic solutions under the action of van der Waals forces47. Finally, Landau has studied with Meiman and Khalatnikov 97 finite differ­ ence methods for solving differential equations.

1. ON TH E T H EO R Y OE THE SPECTRA OF DIATOMIC MOLECULES The model of the diatomic molecule as a rotator with internal momentum is treated by means of the new quantum mechanics. The familiar band theory is obtained for the frequencies. All intensities are calculated. The behaviour of the molecule in electric and magnetic fields (Stark and Zeeman effects in the bands) has been examined.

The Hamiltonian for a diatomic molecule isf H = U + -r— J > 2+ — - P I + -zrzr2m ^

2

1

2J£>

'

(1 )

where the small letters refer to electrons and the capital letters to the nuclei; V denotes the potential energy. In order to separate the translatory motion of the molecule as a whole, we make the following transformation of co-ordinates: r =r —

jl

"f" ilL R% -r

.

R ~ It-2 —Rl?

m £ r + Afxfi, + M 2 R z Y m + M1 + M2

( 2)

the new co-ordinates of the electrons are their vector distances from the centre of mass of the nuclei, R is the distance between the nuclei, and C is the radius vector of the centre of mass of the molecule. After this transformation the Hamiltonian becomes a

-

u +

-S

+

2 (Z m + M 1 + M t) T ‘

]. L. Landau, Zur Theorie der Spektren der zweiatomigen Molekule, Z. Phys. 40, 621 (1926). t AH co-ordinates and momenta which occur are treated as matrices. OPL

I

1

(4)

9

COLLECTED PAPEBS OP L , D , LAND ATT

If finally we use polar co-ordinates i?, 6,

2 + -2(M1 + m 2)

* 4 « r * '« W

+ J ( ~ m T + ' F ^ j **

< -4 * * A (* -W

I -

(5)

Jf* =» - Simp p 8 - — cote(cos

M v = cos

(10) I denoting these new quantum numbers. Since the solution already found for any quantity can be put in the form A

n Z_

An

we can write (ii) where $ is independent of the nature of the quantity A. If n = nlt then < - < .

A%\ = A l.

(12)

For any function of 0, ‘, - Liz. Jn l

~\nl

{Ml + M l + M l + D\. + Dp - DP)

-[■ 2 J since from (9) D^ “ = Dy>* = 0 . I*

AnX

(14)

4

COLLECTED PAPERS OF L . D. LANDAU

Bom, Heisenberg and Jordan have shown that < f = (Ml + M I + Ml),, t = %*j (j + 1):

(15)

let j be one of the two quantum numbers 1. From (14), (15) and (12) we find E(l)n,l = «»?'(?' + ! ) + & • , or •S'il,1 — -®(0)n,l + & (1)», I ~

+ an j (j + 1)>

(16)

where An and an depend only on n and not on I. By the j selection rule 1 the well-known formula (16) gives the following frequencies-: < 4 = , sin0 srn^, cos0), since1 (Mx +

= (Mx - i M $ l ~ x = W (? + ™)(? - m + 1),

( 22 )

( J f,)& = * » .

The relations sin 0 ei*’cos0 — cos0 sin 0 elr/' — 0 , sin 0 e ^ sin 0 e“J* + cos20 = 1 give finally 2 _ / o' + fc) (j - &) J V (2j + 1) (2j —1)

M %r 'hj~ 1

(23)

Formula (23) shows that we must hare

(24) If we recall that the f terms of x' and y' are zero (equation (9)), we can put sc| = z‘%sin# cos^,

y\ = z% sin # sin p ,

z\ = z'%cos

Since also for the nucleus co-ordinates X \ = sin# cos9 , etc., it follows th at similar relations hold for all possible f , 17, 'Q. This gives m _ Jkj

A

h

A*

fkj

/&? —1

Jk

_

A f k jM - 1

A 1

«/ &?

//

(i + h) (j - k)

«/ & *yj

(25)

ri ( i + i ) ’

with /f independent of j. In order to calculate the /*_!, we use the the commutation relation. z cos# —cos# 2 = 0 Then

terms of

f* * = fh ^ M + *) U - &+ !) J i - l i Jh-1 j(j+i) fh t /i-ii-i

= _ fh 1 U j + ^ U + J c - l ) Jh-x j V (2; + l) (2 j - 1) ’

(26)

fh i -1 _. fh _L / (■? ~ ^)Q ~ & + 1) J h - i j y f (2j + l ) ( 2 j - l )

y j - ij

Similar formnlae are obtained for the /j^ 1. Since ^ is approximately equal to v%, on account of the smallness of Ex> and therefore is independent of j and m (to a first approximation), we can use formulae (25), (26) also for time derivatives of £ r\, £ of any order. They there­ fore give the radiation amplitudes also. In the field-free case the system is degenerate with respect to m. I t is easily seen th at we then have for the intensity of {unpolarised) radiation j = * i (4.)*, ?»= - j

(27)

c

COLLECTED PAPERS OF L . D . LA2TDAXT

where A m is the amplitude of the radiation polarised in the direction of the z axis (in the non-degenerate system). S u b stitu tio n of A m from (18) gives A - (/?)2 / 0' + 1)( 2 ? + 1),

) (28)

^ , = ( / U 2 ?(2 j - l ) ( 2 ? + l).. or ^ rki

= 4(-r + _

7+ 1

*2>

n U + k)U - * )

~ 'i lc

. 3

“ ^1-1

»

^ ^ U+

(29)

U “ ^ + *)>

j------------» (j - k ) ( j - k + 1) A - l t 1 = J*-l jk-lj _

7k

j

jk-lj

_

jk

3-1

rk-lj-1 _

Tk

j

A magnetic field along the z-axis gives in a first approximation the following addition to the Hamiltonian | AH =

* \n \

2(*c

(30)

A-

Hence e \S \ d E = A _ l n COS0 2 fxc

*

eh ■ .

k2m

(31)

7 ( 7 + 1) ’

(A )n = {Dy)n ~ 0 from (9). This completes the investigation of the Zeeman effect for the bands: the splitting is obtained from (31), the intensities and pola­ risations from (18), and (25), (26). An “.-band system for which k = k' = 0 shows, according to (31), no splitting in the first approximation. For sufficiently strong fields we must carry the approximation to (7) one step further and also take into account the quadratic terms in the field perturbation. The calculation f Here and below we denote the electron mass by fi to avoid, confusion with the quantum number m.

THEORY OJF SPECTRA OP DIATOMIC MOLECULES

7

is somewhat involved; the result is W * « o = \ M\ y n m + \R\*Kn

3 (j + 1) 4- (m - 1) (m + I) (2j ~ l ) ( 2j + Z)

where nx depends only on the position of the centre and satisfies the condition «; = 1 .

(18)

Substitution of (17) in (15) gives Ex =

E,

cos