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Few-Body Systems

Translationally Invariant Single Particle Picture of the Three-Nucleon System

Hiroyuki Kamada, W. Glöckle, J. Golak

Published Apr 4, 2012• 0 citations
Publisher
DOI
10.1007/s00601-012-0426-y

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arXiv:2004.12020•Journal of Physics G Nuclear and Particle Physics•Jun 16, 2020•15 citations
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.
Abstract The need to enforce fermionic antisymmetry in the nuclear many-body problem commonly requires use of single-particle coordinates, defined relative to some fixed origin. To obtain physical operators which nonetheless act on the nuclear many-body system in a Galilean-invariant fashion, thereby avoiding spurious center-of-mass contributions to observables, it is necessary to express these operators with respect to the translational intrinsic frame. Several commonly-encountered operators in nuclear many-body calculations, including the magnetic dipole and electric quadrupole operators (in the impulse approximation) and generators of U(3) and Sp ( 3 , R ) symmetry groups, are bilinear in the coordinates and momenta of the nucleons and, when expressed in intrinsic form, become two-body operators. To work with such operators in a second-quantized many-body calculation, it is necessary to relate three distinct forms: the defining intrinsic-frame expression, an explicitly two-body expression in terms of two-particle relative coordinates, and a decomposition into one-body and separable two-body parts. We establish the relations between these forms, for general (non-scalar and non-isoscalar) operators bilinear in coordinates and momenta.