Volume 4, Issue 3, March 1963
Index of content:

Scattering, Weak Decays, and Final‐State Interactions in Model Theories in a Dressed‐Particle Picture
View Description Hide DescriptionElastic and inelastic collisions in the Lee model are treated in a dressed‐particle picture. The procedure by which the renormalization constant Z disappears from the integral equations for the transition matrix is studied in detail. A new procedure is given for obtaining the exact transition matrix for N − θ scattering. A hypothetical fermion B is introduced and the decays B → N + θ, B → V + θ, as well as the decay in the Goldberger‐Treiman model are studied in the dressed‐particle picture. The iterative expansion of the B → V + θ decay amplitude is obtained and is shown to agree with the renormalized power series for this decay, though such is not the case for B → N + θ decay, or the decay in the Goldberger‐Treiman model. It is shown that in the final‐state interaction of B → V + θ decay, there are cancellation effects between one‐ and two‐meson contributions, which are crucial in avoiding divergences.

Mechanical Model for Quantum Field Theory
View Description Hide DescriptionWe consider two harmonic oscillators, coupled during a finite time. Initial and final states can be defined unambiguously, and if the duration of the coupling is sufficiently short, the S matrix can be computed explicitly. The coupling gx ^{2} y is investigated in detail, for complex values of g. It is found that, along the real axis, the S matrix behaves smoothly as a function of g and tends to the unit matrix for g → 0, as it should. However, the S matrix has a line of essential singularities along the imaginary g axis, including the origin, so that it cannot be expanded into powers of g. If such an expansion is sought by means of a perturbation procedure, it is found that each term of the series is finite (no need of renormalization), but the series as a whole diverges.

Approximate Solution of a Nonlinear Field Equation
View Description Hide DescriptionVariational principles for obtaining the eigensolutions ψ_{ i } of the equation are developed. Variational solutions to the first two spherically symmetric eigenstates are obtained. Variational solutions of odd parity are also obtained.

Generalized Momentum Operators in Quantum Mechanics
View Description Hide DescriptionThe usual form P _{0} for the quantum‐mechanical operator P conjugate to a generalized coordinate q _{1} is, in atomic units,,where g is the Jacobian of the transformation from Cartesian coordinates to the generalized coordinates. However, in some cases, this plusible form for P is not Hermitian with respect to physically acceptable bound‐state wavefunctions, as it must be if it is to represent a real observable quantity. In this paper, the general formis justified. Here A = h _{1}q̂_{1}, where h _{1} is the metric scale factor corresponding to q _{1}, and q̂_{1} is the unit vector in the direction of increasing q _{1}. This general form reduces to P _{0} if the usual formula for the divergence is applied. In the cases where P _{0} is not Hermitian, it transpires that q̂_{1} is ill‐defined at one or both of the end points α and β of the range of q _{1}, and the divergence formula is thus invalid at such points. It is shown that, in order to obtain a Hermitian form for P, certain terms involving delta functions similar to Dirac's must be added to the usual formula for div A. These terms can be regarded as implicit in div A. If q̂_{1} is ill‐defined at the lower limit q _{1} = α, then the resulting new Hermitian form for P, which we propose as the correct one, is.If q̂_{1} is, in addition, ill‐defined at the upper limit q _{1} = β, then the extra term +½i δ_(β − q _{1}) must be added. Corresponding new forms are obtained for the Laplacian operator. In addition, the new formulas for P are applied to hypervirial relations. In the Appendix, Charles Goebel obtains a similar expression for the momentum operators by replacing the metric scale function by θg where θ is a step function, unity inside and zero outside the range of definition of the generalized coordinates. The differentiation of θ then produces the delta functions.

On the Nonorthogonality of Generalized Momentum Eigenfunctions in Quantum Mechanics
View Description Hide DescriptionThe orthogonality requirement usually specified for the eigenfunctions φ of a quantum‐mechanical operator P which has a continuous spectrum of eigenvalues λ, is The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound‐state wavefunction ψ in the form The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground‐state hydrogen atom.
When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single‐valued eigenfunctions.

Electrostatic Interactions in Complex Electron Configurations
View Description Hide DescriptionSimplified expressions for the matrix elements of electrostatic interactions both within and between several types of complex electron configurations have been obtained by the application of angular‐momenta recoupling techniques. The use of these recoupling techniques avoids the usual extensive calculation of the sums of products of the matrix elements of tensors of the types V ^{κk } and U^{k} . The derived expressions involve the sums of products of coefficients of fractional parentage and n − j symbols, and as such, are amenable to machine computation.

Partial Wave Analysis of the Scattering of Charged Spinless Particles
View Description Hide DescriptionThe partial wave series for a relativistic charged spinless particle is not uniformly convergent and is difficult to evaluate numerically in the forward direction. The singularity in the scattering amplitude at the forward direction which leads to this nonuniform convergence is separated and given explicitly in a closed form so that the partial wave series may be accurately evaluated numerically.

Regge Poles and Branch Cuts for Potential Scattering
View Description Hide DescriptionThe analytic properties of partial wave amplitudes are studied for complex energy and angular momentum. The properties of the wavefunctions are first obtained by standard methods in the theory of differential equations for general classes of potentials, and the effects of the dominant singular term in the potential near the origin are investigated. These include the appearance of branch cuts in the angular‐momentum variable for potentials which are singular like z ^{−2}, and the location of Regge poles for more singular potentials. The trajectories of Regge poles are also studied with particular reference to their behavior in the angular‐momentum plane as the energy tends to infinity. An example is given of a singular potential in which the trajectories move to infinity in a complex direction, contrary to the normal behavior for which they tend to negative integers. The real sections of Regge surfaces are also briefly discussed.

Potential Scattering
View Description Hide DescriptionThe potential‐scattering model is discussed from the point of view of analyticity properties of the scattering amplitude. Dynamical schemes based on the Mandelstam representation and the use of Regge poles are reviewed. Consequences for strong interaction physics are also briefly reviewed.

New Approach to Low‐Energy Potential Scattering
View Description Hide DescriptionThe problem of low‐energy potential scattering is reformulated in a manner suggested by the ``invariant imbedding'' techniques of transport theory. The differential equations thus obtained have several conceptual and computational advantages over the Schrödinger equation. Some new bounds and approximations are derived and a rigorous investigation of the Born approximation is carried out.

Relativistic Coulomb Scattering
View Description Hide DescriptionContour integration is employed to evaluate analytically the angular dependence of the relativistic correction terms to the Rutherford cross section for the scattering of electrons by nuclei at high energies. The phase shifts obtained by Mott from the solution of the Dirac equation are expanded in powers of the fine structure constant, and the resulting infinite sums are converted into integrals in the complex angular‐momentum plane. In turn, these integrals are evaluated by distorting the path of integration and by the use of integral representations. In this manner, the angular dependence of the cross section is obtained in closed form up to terms of the fifth (fourth) order in the cross section (wavefunction). The form of the correction term corresponding to an arbitrary power of the fine structure constant is found in terms of two‐dimensional integrals involving elementary transcendental functions. A related problem, the nonrelativistic scattering for an attractive 1/r ^{2} potential is also discussed.

Binary Kernel Formulation of a Heisenberg Model of Ferromagnetism
View Description Hide DescriptionAn ideal Heisenberg model of a ferromagnet for spin is studied by considering the model in terms of a spin‐deviation lattice gas. Utilizing the general methods of Yang and Lee, a binary kernel function is obtained in terms of which the thermodynamic properties of the lattice gas can be completely expressed. As an example, Dyson's results are rigorously obtained.

On a New Method in the Theory of Irreversible Processes
View Description Hide DescriptionA new method is presented for obtaining irreversible equations describing the approach to equilibrium in systems of many particles. The basic idea is the removal of secular terms arising in a perturbation expansion by the technique used in nonlinear mechanics. The irreversible equations then appear as consistency conditions for the existence of a well behaved expansion. The method relies heavily on the existence of the natural fine‐scale mixing occurring in the dynamics.

Some Properties of the Reduced Density Matrix
View Description Hide DescriptionTwo conjectures made in a previous paper are proved. A further remark is made concerning the largest eigenvalue of the reduced density matrices.

Equivalence and Antiequivalence of Irreducible Sets of Operators. I. Finite Dimensional Spaces
View Description Hide DescriptionA fundamental problem which arises in determining whether two quantum mechanical systems are essentially identical is whether a unitary or antiunitary transformation exists which maps one set of dynamical variables into another. Since an elementary dynamical system is specified by giving an irreducible set of dynamical variables, we are led to investigate the following problem: Given two irreducible sets of operators with a one‐to‐one correspondence between them, find the algebraic properties of the two sets which make it possible to infer the existence of a unitary or antiunitary operator relating them. A series of theorems is obtained from such considerations for finite dimensional spaces. It is shown that if the second set of operators contains some of the algebraic properties of the first set, the two sets are related by a similarity transformation. By altering the requirements, this transformation is a unitary transformation. Indications are also given to show how the theorems can be extended to Hilbert spaces. The rigorous statements of the theorems and the proofs will be given in a second paper. Finally, in the Appendix there is given a definition of invariance of elementary quantum‐mechanical systems based on the above theorems, giving the same results as Wigner's definition in terms of transition probabilities.

On Strain Energy Functions for Isotropic Elastic Materials
View Description Hide DescriptionThis article deals with isotropic elasticmaterials which possess a strain energy function. For such materials the strain energy of a material point is given, of course, by a symmetric function σ̄ of the principal stretches v _{1}, v _{2}, v _{3} at that point. It is known that a necessary condition to have an isotropic material compatible with the axioms for thermostatics proposed by Coleman and Noll is that the function σ̄ be jointly and strictly convex in its three variables v _{1}, v _{2}, v _{3}. Here we show that this condition is not sufficient for compatibility with the thermostatic axioms.

Note on the Riccati Equation
View Description Hide DescriptionA simple modification of a method introduced by R. Bellman is proposed, which under certain circumstances produces both upper and lower bounds for the solution of the Riccati equation. An application to scattering theory is suggested.

Symmetry Functions of the Cube
View Description Hide DescriptionA simple method is presented for reducing the 2J + 1 state vectors of good total angular momentumJ to cubic symmetry types (basic vectors of the irreducible representations of the cubic rotation group). The cubic symmetry functions are written explicitly in terms of the eigenfunctions of J_{z}, J_{x} , and J_{y} , and formulas and tables which facilitate their use in obtaining eigenfunctions and energy splittings of ions in fields of cubic symmetry are presented and applied.

On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions
View Description Hide DescriptionIt is intended in the present series of papers to discuss explicit constructive determinations of the representations of the semisimple Lie groupsSU_{n} by an extension of the Racah‐Wigner techniques developed for the two‐dimensional unimodular unitary group (SU _{2}). The present paper defines, and explicitly determines, a symmetric vector‐coupling coefficient for the group SU_{n} . These coefficients are utilized to construct a series of canonical invariants for SU_{n} , of which the first I _{2} is the familiar Casimir invariant, and it is proved (by construction) that these invariants form a complete system of independent invariants suitable for uniquely labeling the irreducible inequivalent representations of SU_{n} .

Moments of the Neutron Time‐Energy Distribution and the Use of Random Functionals
View Description Hide DescriptionThe time moments of the neutron time‐energy distribution are derived under the following conditions: infinite homogeneous moderator with no absorption,elasticscattering, and distribution of scattering angle independent of energy in center‐of‐mass frame. However, no assumption is made on the variation of the scattering cross section with energy. This is in contrast to previous papers in which a scattering cross section proportional to v ^{δ} (δ any real number) is assumed. The moments are derived in two ways: first, by means of Laplace transforms, and then through the use of random functionals. Random functionals have not been previously employed in neutron moderation theory and they offer certain advantages which are discussed.