Abraham Klein

Fermions and bosons interacting with arbitrarily strong external fields

Abstract The question, “What happens to the electron orbitals as the charge of the nucleus is increased without bounds?” has inspired much of the interest in the description of particles bound strongly by external fields. Interest in this problem and in the related Klein paradox extends back nearly to the beginnings of relativistic quantum mechanics. However, the correct interpretation of the theory for overcritical potentials, where the parts of the complete set of single particle solutions associated with particles and antiparticles are no longer distinct, was given only recently. The understanding of the spectrum of the Dirac and Klein-Gordon equations is essential in order to obtain an appropriate physical description with quantum field theory. The strong binding by more than twice the rest mass of the particles in overcritical external potentials leads to qualitatively new effects. In the case of fermions we find spontaneous positron emission accompanied by creation of a charged lowest energy state, i.e. a charged vacuum. The number of positrons produced spontaneously is limited by the Pauli exclusion principle. For bosons we find that depending on the character of the external potential, either neutral or charged Bose condensates develop. While the questions associated with the meson fields seem academic at the moment, the effects attributed to the fermion field stand a good chance of being tested in an experiment in the near future. It is expected that in heavy ion collisions such as uranium on uranium near the Coulomb barrier overcritical electromagnetic fields will be created.

Collective states of positive parity in 16O

Abstract A restricted shell model calculation of the so-called excited rotational bands of 16O is outlined. The main advance over previous efforts is the serious calculation of a two-particle - two hole deformed state of minimum energy separated by an energy gap from excited states of a similar character

New method for studying the microscopic foundations of the interacting boson model

Abstract We describe (i) A mapping, using a multishell seniority basis, from a prescribed subspace of a shell model space to an associated boson space. (ii) A new dynamical procedure for selecting the collective variables within the boson space, based on the invariance of the trace. (iii) A comparison with exact calculations for a multi-level pairing model, to demonstrate that the method works.

Study of boson expansion methods in an exactly soluble two-level shell model☆

Abstract A two-level numerically soluble shell model previously utilized by Lipkin, Meshkov, and Glick to investigate the accuracy of the random phase and related approximations is applied to the study of boson expansion methods for the description of a vibrational spectrum. Since the model is expressed completely in terms of quasi-spin operators, the required correspondence to boson operators is given by well-known results from the theory of ferromagnetism. The expansions required are also obtained independently by the methods current in nuclear physics. The spectra computed from several harmonic and anharmonic oscillator approximations to the exact Hamiltonian are compared with the results of an exact diagonalization, and possible analogies with the case of vibrational nuclei are drawn.

Invariant Operators of the Unitary Unimodular Group in n Dimensions

An elementary derivation is given of Biedenharns construction of a complete set of independent invariants for the group SU(n). The basic tool is the mapping of the adjoint representation onto the linear space of generators in the defining representation. The trace of any algebraic function of the matrix thus associated is seen to constitute an invariant of the adjoint representation and yields by substitution an invariant operator. The independent invariants are recognized by their isomorphy to the invariant forms under the permutation group.

Mandelstam Representation for Potential Scattering

A proof of the Mandelstam representation for the nonrelativistic scattering amplitude is given when the potential is of the Yukawa form or (by obvious extension) a suitable linear combination of such forms. The analytic properties of the scattering amplitude as a function of momentum transfer are established by using only a finite sequence of equivalent definitions of the scattering amplitude. By studying the Born series for individual partial waves, it is shown in addition that there cannot be an essential singularity at infinity. Together, these results imply both dispersion relations for individual partial waves and the Mandelstam representation.

Reaction paths and generalized valley approximation

The generalized valley approximation has been developed as a method of approximately decoupling one or a few low‐frequency nonlinear modes from the remaining higher frequency modes of a multiparticle system. This decoupling will be best when the difference in frequencies is large; this is the case of adiabatic motion. We describe the application of this method to chemical reactions, relying in some measure on our earlier work, and contrast it with reaction‐path theories. We give an algorithm for the incorporation of our method in a chemical calculation of the Born–Oppenheimer type. Detailed calculations are reported for several standard models that couple a double well to a harmonic oscillator. The decoupling procedure leads to an effective or renormalized one‐dimensional double‐well problem. The energy splitting of the lowest doublet in this well is contrasted with the exact splitting obtained by numerical integration of the two‐dimensional Schrodinger equation. Results are good when the adiabatic condit...

Boson mappings for schematic nuclear models with the symmetry of SO(5)

Abstract In the past, several schematic nuclear models have been proposed with the symmetry of the Lie algebra SO(5) . We derive a set of mappings from the shell model spaces in which these models are defined to suitable boson spaces. The derivations are carried out by two distinct algebraic methods, which are customarily associated with the names Holstein-Primakoff and Dyson, respectively. The relative utility of the two is seen to depend on the particular mapping. The mappings are useful for the study of the collective properties of the various models and of generalizations of each of them. This is illustrated with one of the models. Some algebraic details, some ideas concerning schematic model building, and an extension of the results of this paper are presented in appendices.

PHONONS IN LIQUID HELIUM

Abstract By writing a dispersion relation for the density propagation function we obtain its most general form consistent with known sum rules, for any macroscopic system of bosons or fermions. The Fourier transform of the pair correlation Sk appears explicitly in the dispersion relation. An inequality for Sk is derived for any macroscopic system. We specialize the results to liquid He4 by making the single assumption that for k → 0, Sk “saturates” the inequality. This assumption is consistent with experiments and is later proved theoretically. The nature of the low-lying excited states of He4 can then be deduced. The result is that for given small momentum k, there is a group of states having an energy distribution peaked about ck, with a Lorentzian shape. The constant c is the velocity of sound at absolute zero, defined in terms of the macroscopic compressibility. The wave functions of these states are in some average sense Feynmans phonon wave function. We prove the assumption mentioned above by assuming Bose-Einstein condensation and by making essential use of the gauge invariance associated with the conservation particles. The mathematical technique is simple and does not require perturbation expansions, or summation of diagrams.

THE DESCRIPTION OF ROTATING NUCLEI

A generalization of the Hartree-Fock method used in studies of vibrational states of spherical nuclei and shapes of non-spherical nuclei is used to obtain dynamical properties of nuclear rotational states. (L.N.N.)