Tamm-Dancoff Method Research Articles

Yang-Feldmanformalismus und einzeitige Wellenfuktionen

The integral equations of the new Tamm-Dancoff method are derived with the aid of the Yang-Feldman formalism. The inhomogeneous terms of these equations contain the boundary conditions at the timet=±∞.

Pion-nucleon Scattering by Tamm-Dancoff Method

The pion-nucleon scattering for the isotopic spin I=3/2 state in Tamm-Dancolf approximation has ~ treated by· Fubinji), and the phase shifts thus obtained conSiderably agreed. with the experiment~). for the coupling .constant g~/47r=10.5 and the cut-o!f' momentum = O.92M, the cutting off. being assumed in order to take account of the relativistic effects sw:h as .recoil non-relativistically. However,. similar treatmentment for I = 1/2 state has not been available SO far~>, since the mass and. charge renormalization is necessary. We have done this keeping correspoiidence with the renormalizatlon . procedure· in covariant; fo1'l1lalism4), and cal~ted tbephase shifts for I=1/2 state in the same approximation as Fubini's, or taking into account all graphs with n;:;;; 1 nucleon pairs and m ;:;;; 2 mesons except for graphs with closed loops, the graphs involving meson nucleon interactions in the state with the nucleon pairs and the graphs in Fig. 1. We first eliminate the nucleon pair states from the equations of motion, and· discuss the problem comparing with the Dyson's and Foldy's transformation. The interaction Hamiltonian consists of three parts: scattering H.,. pair creation He and pair annihilation Ha of nucleons. The integral equation for the state vector r/J with the nucleon pair eliminated, turns out

TheS-wave in pion-nucleon scattering

A calculation is made of the twoS-phase shifts of pion-nucleon scattering corresponding to isotopic spin statesT=1/2 andT=3/2, using the Tamm-Dancoff method and an extended source model for the nucleon. The magnitude and energy variation of the α3 phase-shift agree reasonably well with experiment. The sign of the α1 phase-shift is correct, but its magnitude is completely wrong. It is likely, however, that α1 is theS-phase shift which will be seriously altered by taking account of renormalization effects in a more rigorous treatment of the problem.

Nuclear Surface Oscillations

The extreme shell model of the nucleus is extended by allowing the last odd nucleon to interact with collective modes of motion of the even-even nuclear core. The interaction between a nucleon and the surface oscillations of the core was given by Bohr and is employed here with a Fock space representation for the core excitation. Because of the interaction neither the nucleon nor the core conserve their angular momentum, but the total angular momentum is conserved. The problem is treated by the Tamm-Dancoff method. In terms of the Tamm-Dancoff wave function, formulas are derived for the magnetic moment and the quadrupole moment of an odd nucleus. The energy of excitation of the core is taken empirically from the first excited state of the corresponding even-even nucleus. With this procedure all the parameters of the theory are fixed. The magnetic moment deviations calculated in this manner agree with experiment for nuclei in which the total angular momentum is one half, but the agreement does not exist for higher angular momenta. It is also possible to obtain a qualitative fit for the large quadrupole moments of nuclei with odd numbers of nucleons between fifty and eighty-two. The problem of two equivalent nucleons outside a closed shell is treated in the same manner, and it is shown that the first and second excited states have spins 2 and 4 as usually observed. In addition there is a pairing energy binding effect in the ground state which is larger when the individual angular momenta of the two nucleons is larger. This is the pairing energy rule usually employed in shell model calculations.

Remarks on the validity of the tamm-dancoff method

An example of a field which has already been investigated by Wentzel and Blatt is discussed in detail. The exact solution of the problem corresponding to the scattering of a meson by a nucleon is compared with that obtained by the first approximation of the Tamm-Dancoff method. It is found that this method leads to results, which, though better than those obtained in perturbation theory are essentially qualitative. The disagreement is particularly strong, if the coupling constant is nearly sufficient to give rise to bound states.

The Wave Function of a Relativistic System

Three methods of handling problems in relativistic quantum mechanics are discussed. One is the Tamm-Dancoff method, one is the Bethe-Salpeter method, and the third is a new method intermediate between the other two. The connections between the three methods are investigated. It is shown that the new method provides a kind of bridge between the 3- and 4-dimensional points of view. The new method is illustrated with applications to simple examples.

Mass-Renormalization with the Tamm-Dancoff Method

Mass-Renormalization with the Tamm-Dancoff Method

A Covariant Meson-Nucleon Equation

A relativistic meson-nucleon two-body equation is derived in a form suitable for carrying out renormalization. Methods for determining the interaction kernel and classifying its terms are discussed. A reduction of the equation to three dimensions is carried out and the approximations involved in this procedure are examined. The resulting equation agrees with a corresponding one derived by Tamm-Dancoff methods.

Non-adiabatic treatment of nucleon pion scattering

The Tamm-Dancoff non adiabatic method is applied to the study of nucleon-pion scattering. An integral equation is obtained. It is resolved, making use of some approximations, getting numerical results in satisfactory agreement with experiment. The existence of an siobaric state of the nucleon with spin 3/2 and isotopic spin 3/2 is also deduced.

Non-Adiabatic Treatment of the Relativistic Two-Body Problem

The method of Tamm and Dancoff, for the non-adiabatic treatment of the relativistic interaction of two nucleons, is generalized in order to include nucleon pair creation and higher order effects in the exchange of mesons. This generalized form of the Tamm-Dancoff method is shown to give results which are equivalent to those obtained from the relativistic equation of Bethe and Salpeter. A detailed study is made of two limiting cases: (a) no pair of nucleons is created in the intermediate states, but an arbitrary number of mesons can be present at the same time; (b) the maximum number of mesons present at a given time is one, but the number of pairs is unrestricted.The two methods are applied to the calculation of the lowest order correction to the scalar meson interaction of two nucleons. It is shown that the exact correction, which is of the second order in the nucleon velocities, can only be obtained through the inclusion of the fourth- and sixth-order interaction processes involving, in the corresponding Feynman diagrams, the crossing of the meson lines.