Quantummechanical solutions of the laser masterequation. II

A Fokker-Planck equation for a distribution function over the macroscopic observables of the laser essentially equivalent to that recently obtained byRisken,Schmid andWeidlich is derived from the fundamental quantummechanical laser masterequation. The general method used is the expansion of the statistical operator in a complete set of projection operators of the atoms and the lightfield. The assumptions leading from the microscopic equation of motion to the macroscopic semiclassical Fokker-Planck equation are explicitly introduced and discussed.     

Quantum mechanical solutions of the laser masterequation

We treat a laser consisting of one mode described by a running wave and a set of atoms with two optically active levels which are homogeneously broadened. We start from the laser density matrix equations ofWeidlich andHaake and define a distribution functionf for lightfield and atomic variables, where we use for the lightfield the coherent state representation and for the atomic system a modified version of the distribution function used bySchmid andRisken in a previous paper. We derive a partial differential equation forf which is completely exact and is of the type of a generalized Fokker-Planck equation, i.e. it contains higher derivatives. Using a recently stated theorem ofHaken andWeidlich we show that this distribution function allows to calculate single-time as well as multitime quantum mechanical correlation functions. If the leading terms of the generalized Fokker-Planck equation are retained we find the semiclassical Fokker-Planck equation ofRisken,Schmid andWeidlich. Our treatment can be extended to several modes connected with standing waves and multilevel atoms.     

On transport theory of Bose systems. I

The concept of thermodynamical correlation functions for non-equilibrium states used byKadanoff andBaym in transport theory is generalized by means ofBogoliubov's quasi-averages in order to describe transport phenomena in systems with Bose-Einstein condensation. The one-particle Green's function is decomposed in one part describing the condensed particles and a second one describing the elementary excitations. The coupled equations of motion for these functions are derived and some consequences for the case of slowly varying disturbances are discussed.     

A covariant polarization operator for s=1 particles

An operator corresponding to the 4-vector operator byMichel andWightman is given for the Duffin Kemmer Petiau equation.     

Dispersion of the stress optic coefficient of the alkali halides

The dispersion of the stress optic coefficient C=n³/2 (q₁₁?q₁₂) of the alkali halides, NaCl, KCl, KBr and KI have been measured from the visible to the ultraviolet region. In general the value of “C” decreases with wavelength for all crystals. While the dispersion is only a few per cent in the visible region of wavelengths, it is enormous in the ultraviolet. NaCl shows a dispersion of about 100% from 5800 to 2400 Å; KCl about 200% from 5000 to 2400 Å; KBr about 300% from 5000 to 2400 Å; and KI about 400% from 5000 to 2800 Å. Also the potassium halides exhibit a change in sign of their “C” values in the ultraviolet. In KCl the sign reversal occurs at about 2550 Å; in KBr at 2760 Å and in KI at 3380 Å. Below these wavelengths, the potassium halides belong to the same class inMueller's classification as sodium chloride. The theory ofRamaseshan andSivaramakrishnan based on the assumption that a stress causes a change in the frequencies and oscillator strengths of atoms is unable to explain the observed behaviour of the alkali halides. On the other hand, the mere variation of the ionic refractivities with wavelength is also unable to explain the observed dispersion onMueller's theory. One is forced to assume that the strain polarisability constantK inMueller's theory varies with wavelength. When “K” is calculated from the experimentally observed values of “C”, it is found to increase with decreasing wavelength for all alkali halides. The variation with wavelength of “K” for all the alkali halides can be fitted up well by a formula of the type given byRamaseshan andSivaramakrishnan. Hence it appears that the total dispersion ofC can be explained only when we take into account the variation with wavelength of 1. theLorentz andCoulomb contributions fromMueller's theory and 2. the strain polarisability constant fromRamaseshan andSivaramakrishnan's theory.     

The Adler-Weisberger sum rule

Starting from a weak form of the axial current algebra and usingFubini's covariant dispersion method we give a fairly rigorous derivation of the Adler-Weisberger sum rule. Special attention is given to the necessary continuation in the “pion mass”. Combining the sum rule with the forwardπ ^± p dispersion relation we obtain a sum rule containing essentially low energy data. A numerical evaluation is given for two sets of existing s-waveπN scattering lengths. The one reproducesAdler's andWeisberger's results, the other improves the agreement with experiment giving r_A=1.20.     

Relativistic Slater's integrals for theK andL shells

TheZ-dependent form of theSlater's integrals for electrons in theK andL shells is studied by expanding the relativistic hydrogenic radial wave functions in the manner explained byLayzer andBahcall. When screening is not taken into consideration, theseSlater's integrals can be put in the form AZ(1+a Z²) wherea is positive for all integrals considered except G¹(1s, 2p) and G²(2¯p, 2p). Values ofA anda are given.     

Neuberechnung der Bindungsenergie des Wasserstoffmolekülions

The procedure of quantum mechanical treatment of a three-body problem described in the preceding paper byDiehl, Flügge, Schröder, Völkel andWeiguny has been specialized to the ground state in which the eigenfunction only depends upon the three distances between the particles. The solution of the Schrödinger equation has been approximated by variational methods, using the electronic functions ofFinkelstein andHorowitz, and ofGuillemin andZener, but including the nuclear vibration in the trial function.     

Ein neuer Variationsansatz für das Polaron

A new trial function for the self-energy of the polaron is presented which is a combination of the weak and strong coupling methods ofLee,Low andPines resp.Pekar. The result for the ground-state energy is a smooth interpolation between both methods which is better than either one but not as good asFeynman's result.     

Deduction of generalised Kirchhoff's Laws from the basic principles of electromagnetism

The deduction of the so well-renowned and established Laws of Kirchhoff relating to the currents flowing in a network, which are considered to be almost axiomatic in electrical engineering sciences, seems to be preposterous and non-sensical at first sight. However, on a closer examination it will appear that these laws are based on two principles, viz, the steady-state condition under which accumulation of free electrical charge is precluded, and, the experimentalOhm's Law, which propounds a linear relation between the difference of potential across a conductor and the total current flowing through the same. But, there are two things, which are normally overlooked. The conductors inKirchhoff's Laws are “wires”, which are one-dimensional lines, and theOhm's Law is the macroscopicOhm's Law for total currents applied to these lumped resistors. Strictly speaking, the “Laws” are to be deduced from the more fundamental electromagnetic equations for continuous media and the microscopicOhm's Law. It is to be noted that theKirchhoff Nodal Law is but a consequence of the steady state condition derivable from these basic equations, viz, the current-density is divergence-free. For a continuous medium, what the form of theKirchhoff's Laws will be, is difficult to guess unless deduced from the basic equations. Once these are established from the basic equations of electromagnetism for a continuous medium, the usual form ofKirchhoff's Laws will follow as corollary of the general case, as has been shown here.     

Note on the significance of the Aharonov-Bohm effect

The fringe shift first predicted byEhrenberg andSiday and subsequently more fully discussed byAharonov andBohm has been consistently explained in terms of a classical interaction energy arising from overlap of the electron's and solenoid's magnetic fields.Kasper criticizes this theory on the ground that “skin effect” would prevent penetration of the electron's magnetic field into the interior of the solenoid. This criticism is shown to be invalid.     

Genäherte 3-Punkt-Funktionen in der Heisenbergschen Theorie

Heisenberg's approximation for the wave function of a fermion is modified by a more extended account for the corresponding function of three field operators. This may also contain a certain type of form factor, although the calculations already finished are done without any dependence on the internal coordinates. The result depends remarkably on the ambiguous sign of the non-linear term inHeisenberg's 3 field equation.     

On the combinatorics of Feynman graphs. I

As a preliminary step towards a rigorous proof ofDyson's conjectures on the combinatorics of Feynman graphs, the following lemma is proved rigorously in an explicit and elementary way: The generating functional of the time-orderedGreen's functions in perturbation theory is an exponential, the exponent of which is a sum of contributions from connected graphs only, with no contributions from vacuum graphs. The model used to demonstrate this is a scalar hermitean fieldA(x) with anA ⁴ selfcoupling.     

Gruppentheoretische Behandlung der Aufspaltung von Kristalltermen vermöge der Wechselwirkung äquivalenter Gitterteilchen

First an analysis of the Hamiltonians related toBethe's crystal levels and to the energy levels of the whole crystal is given. Then the splitting ofBethe's levels by the interaction between the electrons of equivalent lattice particles is described in the group ring (factor group splitting,Davydov splitting). By thisWinston's rule derived for this splitting is deduced from a new group theoretical point of view. FinallyBethe's levels degenerated according to time reversal symmetry are considered and spin is regarded.     

Beugung elektromagnetischer Wellen in Braunbekscher Näherung. II

The extension, given in a previous paper, ofBraunbek's approximation method to the diffraction of electromagnetic waves, is applied to the near field of the circular aperture and the circular disk. The field distributions along the central axis and in the plane of the screen are calculated and compared with the results of both the rigorous solution andKirchhoff's approximation. The relation ofBraunbek's method to other approximation methods for the optical region is discussed.     

Comportement asymptotique de la fonction de distribution radiale dans les liquides

We study the influence of pair potential on the asymptotic behaviour of the radial distribution function in liquids. We use a perturbation expansion from a model of rigid spherical core inKirkwood's integral equation. It is pointed out that in metals as well as in insulators the asymptotic behavior is connected to that of the pair potential. However, the results are numerically different from those obtained from thePercus-Yevick approximation.     

On the combinatories of Feynman graphs. II

Some ofDyson's conjectures on the combinatories of Feynman graphs especially the following theorem is proved rigorously: the truncated time-orderedGreen's functions will be correctly calculated if, in their series expansions, all the reducible graphs are excluded and appropriate insertions in the irreducible ones are made. Along with the lemma presented in I, it leads to a similar theorem for the scattering operator. The model used to demonstrate these properties is a scalar hermitean field withA ⁴ self-coupling.     

Note on calculating energy values of rotationally hindered linear molecules

The variational principle based on an integral equation equivalent to the Schrödinger equation is employed to calculate energy levels ofDevonshire's model.     

Zur Deutung der Struktur der Bremsstrahl-Isochromate des Wolframs an der kurzwelligen Grenze

Ohlin's maximum of the tungsten isochromat may well be explained on the basis of the energy bands of tungsten calculated byManning andChodorow. The photons near the threshold voltage are then assumed to arise from transitions of the primary electrons to the unoccupied levels of the 5d and 6s bands. Using this explanation one has to drop, however, a selection rule formulated byNijboer, admitting only transitions to final s states. The validity of this ?s selection rule“ is critically examined. For a strict treatment one has to calculate the matrix elements connected with the transitions, using the exact wave functions of the tungsten crystal. These, however, are not known sufficiently. ThereforeNijboer appliedNedelsky's results for free-free transitions at the isolated atom to the transitions to bound states of the solid. Yet according to calculations made recently byGuggenberer the s selection rule is put into question even for free-free transitions. — A better approximation is obtained considering electron transitions from the continuum to the 5d and 6s states of the isolated atom. A calculation of the corresponding matrix elements is carried out. It shows that both kinds of transitions are of about the same probability at energies around 1 keV which are of special concern to us. Therefore it is allowed to use the 5d band for explaining the tungsten isochromat.     

A new approach to potential scattering

Potential scattering calculations based onCalogero's equation and given in two former papers are now improved to include unitarity of theS-matrix. It is shown that the method gives results better than the second Born approximation and even reasonable in the two limits of very strong coupling and very low energies. As test examples, the square-well potential and the Yukawa potential are discussed.