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The One-Body and Two-Body Density Matrices of Finite Nuclei and Center-of-Mass Correlations

Alexandr Victorovich Shebeko, P. Papakonstantinou, E. Mavrommatis

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

A method is presented for the calculation of the one-body and two-body density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance, in the case of a nucleus (a finite self-bound system). We restore translational invariance by using the so-called fixed center-of-mass approximation for constructing an intrinsic nuclear ground state wavefunction by starting from a non-translationally invariant wavefunction and applying a projection prescription. We discuss results for the one-body and two-body momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator orbitals, as the initial non-translationally invariant wavefunction. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.

Related papers

The One-Body and Two-Body Density Matrices of Finite Nuclei and Center-of-Mass Correlations

A. Shebeko

A method is presented for the calculation of the one‐body (1DM) and two‐body (2DM) density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance (TI), in the case of a nucleus (a finite self‐bound system). We restore TI by using the so‐called fixed center‐of‐mass (CM) approximation for constructing an intrinsic nuclear ground state wavefunction (WF) by starting from a non‐translationally invariant (nTI) WF and applying a projection prescription. We discuss results for the one‐body (OBMD) and two‐body (TBMD) momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator (HO) orbitals, as the initial nTI WF. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.

arXiv:nucl-th/0512054•AIP conference proceedings•Jan 1, 2006•0 citations

The one-body and two-body density matrices of finite nuclei with an appropriate treatment of the center-of-mass motion

Alexandr Victorovich Shebeko, Eirene Mavrommatis, P. Papakonstantinou

The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wavefunctions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators a^+ and a for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set {a^+} times other exponential of the set {a}. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of "Tassie-Barker" factors, and point out other model-independent results. The intrinsic wavefunction of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the 4He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wavefunction. We find that the TI introduces important effects in the MDs.

•Jan 1, 2006•7 citations

On the Center-of-Mass Free Density Matrix

Kazuhiro Yabana

Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.

Progress of Theoretical Physics•Aug 1, 1986•4 citations

Some universal properties of density matrices for finite nuclei (bound systems)

Alexandr Victorovich Shebeko

The properties of one-body and two-body density matrices of a nucleus as a nonrelativistic system are studied. Unlike the usual procedure applied in the theory of infinite systems, for a nucleus of A nucleons these quantities are determined as the expectation values of A-body multiplicative operators that depend on the relative coordinates and momenta (Jacobi variables) and affect the intrinsic wave functions. The translational invariance of these operators is thus ensured. An algebraic technique based on the Cartesian representation is developed for handling such operators. Within this technique, the coordinate and momentum operators are linear combinations of production and annihilation operators $$\hat \vec a^ + $$ and $$\hat \vec a$$ with commutation relations for bosons (oscillators). Each of the multiplicative operators reduces to the normally ordered product of two exponents, one of which depends on set $$\left\{ {\hat \vec a^ + } \right\}$$ only and is located to the left of the second, which depends on $$\left\{ {\hat \vec a} \right\}$$ only. This procedure offers a new view of the origin of the so-called Tassie–Barker factors and other model-independent results. In addition, the proposed method yields a fair description of the available experimental data and can easily be extended to investigate the role of short-range nucleon-nucleon correlations within these translationally invariant calculations of nucleon density and momentum distributions in light nuclei.

Bulletin of the Russian Academy of Sciences Physics•May 1, 2016•0 citations

Centre-Of-Mass Motion in Many-Particle Systems. Application to Atoms and to Nuclei

Roger F. Prat

The centre-of-mass position and momentum of an ( N + 1)-particle system are treated as superselection rules. As a consequence, particle positions and momenta do not obey Heisenberg commutation rules. A Schrödinger equation for the internal state is obtained in terms of the ( N + 1)-particle original positions. Application to atoms justifies the fixed-nucleus approximation as consistent with the classical CM premises. Application to nuclear structure gives at once the Schrödinger equation for the intrinsic states. The present theory allows system states to be altogether localized, translation invariant with respect to a laboratory frame, and with a nonzero total momentum.

Europhysics Letters (EPL)•Feb 14, 1992•0 citations

Center-of-Mass Motion in Many-Particle Systems

David J. Ernst, Carl M. Shakin, Roy M. Thaler

We have reexamined the problem of constructing a nuclear ground-state wave function consistent with the requirement of translational invariance, beginning with a wave function which does not have this property. A class of transformations is constructed which transforms the initial-state vector into an eigenstate of total momentum. The particular transformation of this class which leads to the lowest intrinsic energy is given.

Physical Review C•Mar 1, 1973•18 citations
A method is presented for the calculation of the one-body and two-body density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance, in the case of a nucleus (a finite self-bound system). We restore translational invariance by using the so-called fixed center-of-mass approximation for constructing an intrinsic nuclear ground state wavefunction by starting from a non-translationally invariant wavefunction and applying a projection prescription. We discuss results for the one-body and two-body momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator orbitals, as the initial non-translationally invariant wavefunction. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.
A method is presented for the calculation of the one‐body (1DM) and two‐body (2DM) density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance (TI), in the case of a nucleus (a finite self‐bound system). We restore TI by using the so‐called fixed center‐of‐mass (CM) approximation for constructing an intrinsic nuclear ground state wavefunction (WF) by starting from a non‐translationally invariant (nTI) WF and applying a projection prescription. We discuss results for the one‐body (OBMD) and two‐body (TBMD) momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator (HO) orbitals, as the initial nTI WF. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.
The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wavefunctions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators a^+ and a for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set {a^+} times other exponential of the set {a}. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of "Tassie-Barker" factors, and point out other model-independent results. The intrinsic wavefunction of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the 4He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wavefunction. We find that the TI introduces important effects in the MDs.
Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.
The properties of one-body and two-body density matrices of a nucleus as a nonrelativistic system are studied. Unlike the usual procedure applied in the theory of infinite systems, for a nucleus of A nucleons these quantities are determined as the expectation values of A-body multiplicative operators that depend on the relative coordinates and momenta (Jacobi variables) and affect the intrinsic wave functions. The translational invariance of these operators is thus ensured. An algebraic technique based on the Cartesian representation is developed for handling such operators. Within this technique, the coordinate and momentum operators are linear combinations of production and annihilation operators $$\hat \vec a^ + $$ and $$\hat \vec a$$ with commutation relations for bosons (oscillators). Each of the multiplicative operators reduces to the normally ordered product of two exponents, one of which depends on set $$\left\{ {\hat \vec a^ + } \right\}$$ only and is located to the left of the second, which depends on $$\left\{ {\hat \vec a} \right\}$$ only. This procedure offers a new view of the origin of the so-called Tassie–Barker factors and other model-independent results. In addition, the proposed method yields a fair description of the available experimental data and can easily be extended to investigate the role of short-range nucleon-nucleon correlations within these translationally invariant calculations of nucleon density and momentum distributions in light nuclei.
The centre-of-mass position and momentum of an ( N + 1)-particle system are treated as superselection rules. As a consequence, particle positions and momenta do not obey Heisenberg commutation rules. A Schrödinger equation for the internal state is obtained in terms of the ( N + 1)-particle original positions. Application to atoms justifies the fixed-nucleus approximation as consistent with the classical CM premises. Application to nuclear structure gives at once the Schrödinger equation for the intrinsic states. The present theory allows system states to be altogether localized, translation invariant with respect to a laboratory frame, and with a nonzero total momentum.