Volume 13, Issue 5, May 1972
Index of content:
13(1972); http://dx.doi.org/10.1063/1.1666018View Description Hide Description
An explicit algebraic expression [containing the minimal number (n − 1) sums] for the general reduced Wigner coefficients associated with the multiplicity free Kronecker product [h 1 … hn ] × [p0 … 0] of irreducible representations of U(n), is determined. The calculation employs a combined use of recursive relations derived for the Wigner coefficients, and matrix elements (with respect to Gel'fand basis states) of a generator of U(n) raised to an arbitrary power. We also give an alternative procedure using the techniques of the ``pattern calculus.'' The method is illustrated first for the case of U(3) and then generalized to arbitrary U(n). It is found that the results can be expressed succinctly in terms of a new algebraic function Snm ‐ thereby elucidating the structural details of the underlying algebra.
13(1972); http://dx.doi.org/10.1063/1.1666019View Description Hide Description
13(1972); http://dx.doi.org/10.1063/1.1666021View Description Hide Description
The Cauchy problem for the coupled Maxwell‐Diracequations is solved within an arbitrary bounded region of space‐time. An integral part of the proof is that the Cauchy problem for a cut‐off version of these equations has a global solution. The analysis requires that the size of the Cauchy data or the coupling constant be suitably restricted.
13(1972); http://dx.doi.org/10.1063/1.1666022View Description Hide Description
Two functional equations of the form ψ2(s) − E(s)φ2(s) = V(s), where s is a complex variable and E(s) and V(s) are given even polynomials, are solved for the even entire functions ψ and φ which are required to behave like cosh[αs + o(s)] for large |Rls|. Two cases are considered: (i) V of degree zero and E of degree two and (ii) V of degree eight and E of degree six. In the second case the polynomials must satisfy a condition in order for ψ and φ to have the right asymptotic behavior. These functional equations arise in solving the Percus‐Yevick equation for a mixture of hard spheres with nonadditive diameters.
13(1972); http://dx.doi.org/10.1063/1.1666023View Description Hide Description
For a system which may be partitioned into M subsystems A 1, A 2, …, AM , such that configurations of subsystems Ai, Aj correspond to realizations of discrete, random variables si, sj , not necessarily isomorphic, and such that the probability for a configuration of the total system is p(s 1, s 2, …, sM ) = f 1(s 1) … fM (sM )g 1(s 1, s 2) × g 2(s 2, s 3) … g M−1(s M−1, sM ), we prove that p(s 1, s 2, …, sM ) = p 1(s 1 | s 2)p 2(s 2 | s 3) … p M−1(s M−1 | sM )pM (sM ), where pi (si | s i+1) is the conditional probability for si given s i+1, and pi (si ) is the reduced probability for si . This result yields a decomposition of the total entropy into ``single‐subsystem entropies'' and ``nearest‐neighbor'' cumulant‐like terms only. This Markovian decomposition applies to systems with short‐range interactions for which transfer‐matrix methods are introduced.
Nonrelativistic Time‐Dependent Scattering Theory and von Neumann Algebras. I. Single Channel Scattering13(1972); http://dx.doi.org/10.1063/1.1666024View Description Hide Description
For single‐channel nonrelativistic time‐dependent scattering we analyze the von Neumann algebras generated by the wave operators Ω± and the algebra generated by the spectral projections of both the total and free Hamiltonians. Each of the algebras is the direct sum of an Abelian algebra and a factor of type I∞, but both summands need not occur. For a large class of central potentials is the direct sum of a countable number of factors of type I∞ and its commutant the direct sum of a countable number of finite factors of type I. In the latter case the type of each direct summand is uniquely determined by the angular momentum of the corresponding partial wave. Similar decompositions are obtained for algebras related to and in a natural way that have certain connections with the absolutely continuous and point spectra of the total Hamiltonian. The structure of some algebras related to is connected with certain properties of the bound states.
13(1972); http://dx.doi.org/10.1063/1.1666025View Description Hide Description
It is shown that the bound‐state solutions of regular and singular Sturm‐Liouville systems have the same asymptotic behavior, in the limit of infinitely deep potential wells. This conclusion is illustrated explicity by showing the profound similarity that exists between two seemingly unconnected problems: Mathieu's equation in a finite domain and Schrödinger's equation with Morse's potential.
13(1972); http://dx.doi.org/10.1063/1.1666026View Description Hide Description
By a purely infinitesimal method we derive the eigenvalues of the Biedenharn's invariant operators from I 2 to I 6 for the SU(n) group. We show that the general formula, quoted from the Racah's work by Baird and Biedenharn, to obtain the eigenvalues of the invariant operators, is valid only for the two first I 2 and I 3 operators and not for the higher order invariants. We give the correct values for the first invariants till I 6.
13(1972); http://dx.doi.org/10.1063/1.1666027View Description Hide Description
A state vector (vector‐valued distribution) of an unstable particle can be a 3‐momentum eigenvector in at most one inertial frame as a consequence of the necessary width of the energy spectrum. We investigate this frame dependence of momentum eigenvectors of unstable particles and demonstrate that:
(i) It is compatible with the principle of relativity.
(ii) It leads to a distinction between two ways of defining the spin of the particle which are equivalent for stable particles.
(iii) One definition, called kinematical spin, yields a precise value, and is determined by considerations of detailed balancing and other means of counting the degeneracy of the momentum eigenvectors.
(iv) The second definition, called dynamical spin, need not yield a sharp value, is determined by the dynamics of the decay process, and is measured by observing angular distributions of decay products.
It is shown that an unstable particle with vanishing kinematical spin (pion) may have a small admixture of non‐isotropic angular distribution of unpolarized decay products in the rest frame of the decay products. The order of magnitude of the effect is estimated and the coupling between the mass and spin spectrum is studied in the case of local interactions without derivative coupling.
13(1972); http://dx.doi.org/10.1063/1.1666028View Description Hide Description
13(1972); http://dx.doi.org/10.1063/1.1666029View Description Hide Description
In this paper, the multipole radiation solutions are derived for the integer and half‐integer spin massive particle wave equations of Weaver, Hammer, and Good. The rest system and massless limits of the solutions are found, the latter limit being shown to coincide with previous work by the present authors. Finally, as an application, the massive particle description is second quantized in terms of the multipole solutions and compared with some other formulations.
13(1972); http://dx.doi.org/10.1063/1.1666030View Description Hide Description
The conventional WKB method for phase shift calculations is known to fail for singular repulsive potentials at small incident energies. We show in this paper that the modified WKB method due to Miller and Good can be used instead and that it gives better agreement with the exact results of the scattering phase shifts at small incident energies and in the small coupling constant limit, as more terms in ℏ2 are included.
13(1972); http://dx.doi.org/10.1063/1.1666031View Description Hide Description
It is shown that by constructing explicit realizations of the Clebsch‐Gordan decomposition for tensor products of irreducible representations of a group G, one can derive a wide variety of special function identities with physical interest. In this paper, the representation theory of the harmonic oscillator group is used to give elegant derivations of identities involving Hermite, Laguerre, Bessel, and hypergeometric functions.
13(1972); http://dx.doi.org/10.1063/1.1666032View Description Hide Description
13(1972); http://dx.doi.org/10.1063/1.1666033View Description Hide Description
This series of papers presents an extension of Keller's diffraction ray method to problems involving two or more parallel plates by introducing a modified diffraction coefficient which takes care of coupling along a shadow boundary automatically. In Part I, a canonical problem was solved and the expression for the modified diffraction coefficient derived. In this part, we give a recipe for how to use this set of rays and illustrate it through several examples including (i) open‐end parallel‐plate waveguide, (ii) bifurcated waveguide, and (iii) an infinite array of parallel plates. The above three examples represent the only three types of problems in the edge diffraction theory that can be solved exactly by analytical techniques based on complex variables. In this paper it is demonstrated that all three exact solutions can be recovered by the present ray method. Moreover, in some problems where the analytical techniques cannot be conveniently applied, the ray method can often provide a useful approximate solution.
13(1972); http://dx.doi.org/10.1063/1.1666034View Description Hide Description
In this paper an attempt is made to define the class of equivalent reference frames in special relativity in terms of a purely kinematical characterization of the notion of free material point and using homogeneity of space and time, isotropy of space, and unidirectional time flow. The problem of finding the possible forms of the relativity group and the significance of a ``space‐time homogeneity axiom'' which was used in previous papers are discussed.
13(1972); http://dx.doi.org/10.1063/1.1666035View Description Hide Description
The generalized covariant Dirac equation for a certain five‐dimensional universe is studied. If a torsion invariant is included in the free Lagrangian, it is shown that particlelike stable solutions exist having definite positive rest energy, spin, and corresponding antiparticles. The treatment is throughout classical.
13(1972); http://dx.doi.org/10.1063/1.1666036View Description Hide Description
Let be the class of functions which are bounded by Cr −1+ε and Cr −3−ε, the class of potentials V(r) such that V(v), rV′(r), and r 2 V″(r) belong to . is dense in the class of potentials with finite norm in which almost all the results of potential scattering are derived. In this paper a complete solution of the inverse scattering problem at fixed energy is given in a class of potentials which contains . This means that given any set of phase shifts bounded by Cl −1−ε, we construct all the potentials of which fit this set of phase shifts. They depend on an arbitrary function. The fundamental tool in the solution is the ``scattering structure function.'' The method is derived in such a way that an approximation theory and numerical computations are feasible. These, together with various studies of the solutions, are the object of forthcoming papers.
On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior13(1972); http://dx.doi.org/10.1063/1.1666037View Description Hide Description
A computer has been used to determine the stability character of periodic orbits for the Hamiltonian oscillatorsystem. Using procedures developed by Greene [J. Math. Phys. 9, 760 (1968)], empirical evidence has been obtained indicating that this system has a dense or near dense set of unstable periodic orbits throughout its stochastic (unstable) regions of phase space. The extent to which such stochastic regions exhibit C‐system behavior, i.e., ergodicity and mixing, is discussed. Finally, the above Hamiltonian system is shown to be intimately related to the Fermi‐Pasta‐Ulam system as well as to the Toda lattice.