Tamm-Dancoff Method Research Articles

Interacting-boson-model and open-shell-Tamm-Dancoff-approximation interpretations of quadrupole collectivity

The collective $S\ensuremath{-}D$ states used by Otsuka, Arima, and Iachello in their microscopic theory of the interacting boson model are reinterpreted in terms of conventional concepts of collective motion. It is shown that a suitable seniority projected one-body excitation operator acting on a state composed of $S$ and $D$ pairs excites a new $D$ pair. The excitation operator can be calculated using an open-shell Tamm-Dancoff method. The $S\ensuremath{-}D$ states are then pictured as arising from many-quantum excitations of a single collective mode.NUCLEAR STRUCTURE IBM shell model subspace constructed; open-shell TDA excitation operator used. Seniority scheme, pairing, and quadrupole degrees of freedom for describing low energy collective motion.

Relativistic description of nuclear bound states

The Tamm-Dancoff method is used to give a relativistic description of nuclear bound states; in particular we discuss exchange effects and deformation of the nucleon meson clouds stressing the importance of the self-energy effects. Application to hadronic atoms is also discussed.NUCLEAR STRUCTURE Relativistic treatment of nucleon interaction, Tamm-Dancoff, self-energy corrections to nucleon binding energy.

On the Mathematical Structure of the New-Tamm-Dancoff Procedure

Abstract The New Tamm-Dancoff method is a procedure for the approximate determination of differences of eigenvalues in quantum field theory. This procedure can be formulated mathematically in the framework of the theory of C*-algebras, especially in our case by using von Neumann's infinite tensor products. Calculational rules are presented for operators which obey the canonical anticommutation relations. The concept of the CAR tensor product is introduced for the joint treatment of a system algebra and the associated functional algebra, A conjugation is discussed which will be needed for a proof of equivalence in II.

A New Tamm-Dancoff Method Based on the SO (2N+1) Regular Representation of Fermion Many-Body Systems

A new Tamm-Dancoff method is developed on the basis of the SO(2N + 1) regular representation formalism of Fermion many-body systems. The Schrodinger equation in the SO(2N + 1) generator coordinate representation is decomposed into symmetry subspaces by means of a Peierls-Yoccoz, projection procedure. The decomposed Schrödinger equation is transformed to a quasi-particle frame by a transformation of the generator coordinate and expanded by the symmetry projected Tamm-Dancoff bases. The Tamm-Danco[f equation and wave function thus obtained satisfy all symmetry requirements at every order of the expansion. The first order Tamm-Dancoff wave function is neither an independent particle nor quasi-particle approximation but represents Bose condensed Fermion pairs (and an unpaired particle) performing a coherent motion. The higher order terms represent fluctuations of Fermion pairs (and the unpaired particle) from the Bose condensate.

Anharmonic-oscillator energies with operator recursion mechanics

The procedure of the new Tamm-Dancoff method of Heisenberg et al. is considered in the context of the nonlinear oscillator. Recurrence equations which result from the operator algebra and equation of motion of the oscillator's displacement operator are converted into a meaningful characteristic energy eigenvalue problem via an infinite-dimensional basis transformation of the form $\mathrm{SAT}$, where $A$ is a matrix representing the recursion equation and where $S$ is not the inverse of $T$. This procedure, appropriate to the situation where the recursive matrix $A$ is not orthogonal or Hermitian, is numerically seen to lead to convergent approximation sequences for energy eigenvalues for several nonlinear oscillators. The matrix elements of $\mathrm{SAT}$ are defined by summable but nonconvergent infinite series. In each order of approximation eigenvalues exist which are locally independent of the single parameter upon which $S$ and $T$ depend, a fact which implies that this recursive method belongs to an as yet not defined variational principle. As the approximation order is increased the eigenvalues are numerically seen to converge to the proper limits for several different oscillators, the convergence being independent of the parameter.

Wiek-Regeln mit Zwischenzuständen und die Neue Tamm-Dancoff-Methode mit Zwischenzuständen / On Wick Rules with Intermediate States and the New Tamm-Dancoff Method with Intermediate States

Abstract Instead of emphasizing the ground state as is done in Green's function method, we take a finite-dimensional subspace of the Hilbert space: the space of the "intermediate states". A systematic introduction of intermediate states is effected by an extension of the method of generating functionals: we combine the generating functionals of the n-point Green's functions to a "matrix functional" T, and form new matrix functionals, which are matrix functions of T. The aim of this paper is to develop the functional calculus in such a way that the transition from scalar functionals to matrix functionals is straightforward, and the method of obtaining further results becomes clear. Following the lines of Dürr and Wagner we get u η-and ζ-rules with intermediate states". Using them we define a truncation procedure for the equations of motion of the n-point Green's functions, the "New Tamm-Dancoff method with intermediate states". This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In ad-dition to well-known approximations, such as the Hartree-Fock and the Hartree-Bogoliubov theory, the RPA and the quasiparticle RPA, we obtain a series of new approximations. Among these are the "Hartree-Fock theory with intermediate states" and the "random-phase approximation with intermediate states", which we already applied with great success to some exactly soluble models.

Die Hartree-Fock-Theorie mit Zwischenzust�nden: Kl�rung ihrer Struktur bei Zeitumkehrinvarianz und Anwendung auf das Kleinsche Rotations-Vibrationsmodell

In an earlier paper we have extended the new Tamm-Dancoff method to the “New Tamm-Dancoff method with intermediate states”. This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In addition to well-known approximations, such as the Hartree-Fock theory and the Hartree-Bogoliubov theory, we obtain a series of new approximations. The “Hartree-Fock theory with intermediate states”, which is the subject of the present investigation, is one of these. By using time reversal invariance we have succeeded in clarifying its structure, and we give the solution procedure. The exchange terms in theN-particle intermediate states can be represented by an additional potentialY, which (as is the case for the generalized density matrixρ) has to be determined selfconsistently. In this way we have overcome the difficulties, that Kerman and Klein met in their “generalized Hartree-Fock approximation”, which has some close similarities with our Hartree-Fock theory with intermediate states. We demonstrate our method for the exactly soluble rotational-vibrational model of Klein et al. Hereby we show how to treat conservation laws and the degeneracy of levels. The Hartree-Fock equations with three intermediate states turn out to give analytical expressions for the energies and the matrix elements. These agree excellently with the exact values in the rotational part of the spectrum.

Erweiterung der Neuen Tamm-Dancoff-Methode auf Phasen- übergänge in Vielteilchensystemen / Extension of the New Tamm-Dancoff Method to Phase Transitions in Many-Body Systems

Abstract We extend the New Tamm-Dancoff method by introducing intermediate states. In this way we are able to treat with the Green's function method the effect of nearby levels in many-body systems. We formulate the η-and the ζ-function method with intermediate states. Aldready in first order, the ζ-function method yields a whole series of new approximations in addition to known theories such as the Hartree-Fock theory and the Hartree-Bogoliubov theory. As an example we study intensively "the Hartree-Fock theory with intermediate states". The ζ-function method which is based on the η-function method yields in first order in addition to RPA, quasi-particle RPA and other approximations "RPA with intermediate states". We apply the Hartree-Fock theory with intermediate states and RPA with intermediate states to the exactly solvable Lipkin model.

Magnons in a model with antiferromagnetic properties. - II

The analysis of the model studied in I is continued with the intention to clarify two problems: the compatibility of the results concerning the magnons (angular momentum zero around the axis) with the requirements of the Goldstone theorem, and the existence of « strange » electrons of spin zero. With regard to the first problem the results of a conventional perturbation theory are compared with the results of the Tamm-Dancoff method; the comparison leads to a better understanding of the dynamics of the magnon vibrations and of their role in the Goldstone theorem. Strange electrons of spin zero are a natural consequence of the model and may be compared with,e.g., the singlet states in the optical spectrum of Ca. The role of the corresponding states in elementary-particle physics is discussed, especially in connection with an approximateSU 3 behaviour in the spectrum.

Spin polarization effects in odd-mass deformed nuclei

Within the framework of the Tamm-Dancoff method the spin polarization effects are interpreted as caused by the “scattering” of the odd nucleon on the 1 +( K=0 orl) excitations of the even core. On this assumption the values of the effective factors g s z and g s + are connected with the “scattering” process on the 1 + 0 and 1 + 1 excitations respectively. It is pointed out that it is necessary to take the latter process into account in calculating the decoupling parameter for the K= 1 2 spin up states. The numerical calculation is performed for the 1 2 − [510] Nilsson state.

One-neutron transfer reactions on tin nuclei and higher quasi-particle Tamm-Dancoff microscopic theories

The spectroscopic factors for stripping and pick-up reactions on the tin isotopes are calculated with the nuclear wave functions of microscopic theories of the one- and three-quasi-particle Tamm-Dancoff (QTD 13) and of the quasi-particle second Tamm-Dancoff (QSTD) methods, which describe the states of the odd-mass and even-mass isotopes, respectively. The residual two-body nuclear force is the Gaussian Wigner force of Kuoet al. The results are compared both with experiment and with the previous results of Kuoet al., where the nuclear wave functions of the even isotopes are approximated by the quasi-particle Tamm-Dancoff (QTD) eigenvectors.

Tamm-Dancoff Method in the 2V Sector of the Lee Model

The 2V sector of the Lee model with boson sources at zero separation is considered from the Tamm- Dancoff point of view. This involves three singular integral equations which are solved by appealing to results previously gained in the Lehmann-Symanzik-Zimmermann method of solution. Two scattering amplitudes and a production amplitude are rederived via the state vectors and an equation for the determination of the two-meson exchange potential is discussed.

Spectroscopic Factors for Two-Neutron Transfer Reactions in Vibrational Nuclei

The spectroscopic factors of two-nucleon transfer reaction amplitudes are calculated with the microscopic theories known as the quasiparticle Tamm-Dancoff (QTD) and "second" Tamm-Dancoff (QSTD) methods. The latter theory describes vibrational states in terms of zero-, two, and four-quasiparticle (qp) excitations. The theory is applied to study ($p,t$) and ($t,p$) reactions in even tin isotopes. All the important lowest-lying states of these isotopes of both even and odd parity are included in the analysis. The QTD and QSTD eigenvectors correspond to several different residual interaction potentials. One of these potentials is the realistic potential of Tabakin renormalized for core polarization. The four-qp correlations cause a quite considerable reduction of the spectroscopic factors relative to those of the QTD theory. These factors are also appreciably sensitive to the effective nuclear force assumed.

Two-nucleon transfer reactions with vibrational nuclei and two- and four-quasiparticle Tamm-Dancoff microscopic theories of vibrational states

The spectroscopic factors of the two-nucleon transfer reaction amplitudes are calculated with the microscopic theories of the quasi-particle Tamm-Dancoff (QTD) and “second” Tamm-Dancoff (QSTD) methods. The latter theory describes vibrational states in terms of zero-, two- and four-quasi-particle excitations. The theory is applied to study (p, t) and (t, p) reactions in even tin isotopes. The QTD and QSTD eigenvectors correspond to two different residual interaction potentials. The four-quasi-particle correlations cause quite a considerable reduction of the spectroscopic factors relative to those of the QTD theory.

Positive mass distribution in the non linear spinor theory

Defining generalised functions products by analytic continuation, a two-point function whose mass distribution is positive may be used in the non linear spinor theory. As an application, we show that in an approximation where only strong interactions are considered — so that no imaginary parts have to be neglected, correct values for the π and η meson masses (and eventually, using enough parameters, for the meson nucleon coupling constants) may be obtained, in the first order of the new Tamm-Dancoff method, if the mass distribution is positive ; if not, it seems that the contribution of the negative part must be negligible.

The 28Si nucleus in the Hartree-Fock model

An estimate of the energies of all the low-lying states of 28Si has been made using the Hartree-Fock and Tamm-Dancoff methods. A one-one correspondence with the correct ordering has been obtained for all observed levels below 8 MeV, however the calculated spectrum is too dense by about a factor of two. Possible reasons for this compression are discussed. Relative E2 transition rates are in fair agreement with experiment. Extensive comparisons are made with a parallel investigation using the SU 3 model, and the failure of the latter model in this case is discussed. An interpretation is made of some of the low-lying states in terms of the Bohr-Mottelson model of a rotating-vibrating nucleus.

A study of some approximation schemes in quantum mechanics

By means of numerical examples in one dimensional Hamiltonian problems, we study the behavior of the approximation scheme known as the New Tamm-Dancoff method. It is concluded that for good convergence of this, and other related methods, one should arrange to start with a symmetric set of equations.

Zur Konvergenz der einzeitigen Tamm-Dancoff-Methode beim anharmonischen Oszillator

As in the usual quantum field theory, the states, and therefore also the eigenvalue spectrum of an anharmonic oscillator can be characterized by means of the so-called τ-functions, that is the matrix element of the type 〈0 | qn | j〉. For the calculation of these matrix elements, the equation of motion of the anharmonic oscillator can be used to obtain an infinite set of equations, which define an eigenvalue problem. To solve it a new set of functions, the so-called φ-functions, are introduced by means of a transformation, whose matrix corresponds formally to the WICK rule. An analysis of this infinite system of φ-equations shows that a convergent secular polynomial can be obtained, which exists as a limiting value of the polynomials for the truncated N φ -equationsystems in the limit N → ∞ . It is therefore permissible to calculate the eigenvalues of the infinite system in an approximate way from the truncated systems. Such an approximation procedure is the essential content of the so-called TAMM-DANCOFF method. The above mentioned convergence of the determinants therefore provides its justification. The convergence of the eigenvalues of the truncated systems to the exact oscillator values is numerically examined up to N= 20. The results are satisfactory.

Sur certaines difficultés des théories spinorielles non-linéaires

Two problems arising in non-linear spinor field theories are discussed. One of these problems concerns general non-linear field equations and is connected with the behaviour of the anticommutator at the origin of the light cone. We show that the resolution of the nonlinear equation by means of the new Tamm-Dancoff method is not self-consistent, if we assume anticommutativity of the fields at the same point. A consistent resolution of the equation requires, as a consequence of microscopic causality, an anticommutator with at least δ-type singularities on the light cone. The other problem concerns Heisenberg's theory of elementary particles; we show that, in this theory, the nucleon-nucleon scattering amplitude vanishes. In order to get some insight on the validity of the Tamm-Dancoff approximation we compute, in an appendix, first and second order contributions to the Möller scattering amplitude in quantum electrodynamics by means of the Tamm-Dancoff method and compare with the results of perturbation theory. The calculations indicate that the first-order contribution is in accordance with perturbation theory ( e 2 terms) while this is no longer the case for the second-order contribution.

Hermiticity of Hamiltonians and existence of eigenstates in soluble field theories

The model in which a charged, elastically bound point electron interacts with the dipole part of the radiation field is studied within the framework of the new Tamm-Dancoff method. It is shown that assumption of the existence of eigenstates forces the conclusion that the Hamiltonian is not hermitian. Since this behaviour is shared by the Lee model when the coupling constant exceeds the critical value, the difficulties of the two models may well have a common origin. It is shown that, if imaginary values for energy differences are disregarded, the N.T.D. method gives the same energy shifts and phase shifts as the method of transformation to normal coordinates.