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Physical Review C

Center-of-Mass Motion in Many-Particle Systems

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

Published Mar 1, 1973• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.7.925

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arXiv:1106.2022•Jun 10, 2011•1 citation
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.
An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.
We examine the Gartenhaus and Schwartz (G-S) transformation, which generates a translationally invariant wave function from one which does not possess this property. A clarification of the G-S procedure for the calculation of matrix elements is achieved if one takes two independent limits of the parameters $\ensuremath{\Lambda}$, ${\ensuremath{\Lambda}}^{\ensuremath{'}}$ which appear in the expression for a transformed operator ${\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}}^{\ensuremath{'}}=({\mathrm{lim}}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}}^{\ifmmode\dagger\else\textdagger\fi{}})\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}({\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}})$. It is shown that the G-S operator, $U={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}}={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}\mathrm{exp}[\ensuremath{-}\frac{1}{2}i\ensuremath{\Lambda}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}+\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})]$, is equivalent to the operator $U={(2\ensuremath{\pi})}^{3}\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}})\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})$. This operator is seen to be a specific example of the more general class of operators presented earlier.
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.
It is shown that one can find an orthogonal transformation that will enable one to split the motion of a many-body system to a center-of-mass motion and an internal motion. A particular orthogonal transformation has been chosen which retains the independent-particle aspect of a harmonic-oscillator shell-model Hamiltonian. It is suggested that one could easily study and eliminate the states with spurious motions of the center of mass by a direct transformation of the shell-model wave function into the new coordinate system.
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.
If one assumes a translationally invariant motion of the nucleons relative to the c. m. position in single particle mean fields a correlated single particle picture of the nuclear wave function emerges. A single particle product ansatz leads for that Hamiltonian to nonlinear equations for the single particle wave functions. In contrast to a standard not translationally invariant shell model picture those single particle s-, p- etc states are coupled. The strength of the resulting coupling is an open question. The Schroedinger equation for that Hamiltonian can be solved by few- and many -body techniques, which will allow to check the validity or non-validity of a single particle product ansatz. Realistic nuclear wave functions exhibit repulsive 2-body short range correlations. Therefore a translationally invariant single particle picture -- if useful at all -- can only be expected beyond those ranges. Since exact A = 3 and 4 nucleon ground state wave functions and beyond based on modern nuclear forces are available, the translationally invariant shell model picture can be optimized by an adjustment to the exact wave function and its validity or non-validity decided.