Recent results in the theory of the three-nucleon systems Page: 4 of 17
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detailing results of our calculations I will attempt to provide simple physical
pictures for these results, where possible.
A wide variety of ca'culational techniques exist for solving the Schrodinger
equation, including the Raleigh-Ritz variational technique/3-7/, the hyperspherical
harmonic expansion/8-9/, the Green's function Monte Carlo method (GFMC)/7'10/, and
the Faddeev procedure . All of these have been used for the trinucleon bound
states and for the a-particle. The august variational technique provides a rigorous
upper bound for the energy, but does not generate a wave function whose quality
matches that bound. Moreover, it is often not applied in a constructive manner;
that is, it is not always clear how to systematically improve the wave function in
order to lower the energy. For simple potentials, those without spin and isospin
dependence (i.e., identical central forces for all nucleons), this technique :an be
extremely effective. This remark applies to all of the above mentioned methods/4/.
The introduction of spin-dependence in general, and a strong tensor force in
particular, can significantly alter the convergence properties of the first three
methods, which must be evaluated on a case by case basis. Nevertheless, their
implementation for the a-particle is not much more difficult than for the triton,
which contrasts strongly with the Faddeev approach./14/
The GFMC method is an old technique which is enjoying a resurgence 3 t I
is "exact" in principle within the Monte Carlo sampling errors. The "wave function"
generated by this procedure is a random sample of points over the nuclear Hilbert
space, which can be viewed as the representation of the exact wave function by a set
of 6-functions. The Schrodinger equation in (imaginary) time, r, is integrated
forward with small time steps from an initial sample, an, components of excited
state wave functions (energy, En) decay exponentially with increasing time
(-exp(-(En-E0)r) , leaving only the ground state component with energy E0 This
procedure converges to the lowest energy state of the Hamiltonian irrgspectivI of
the symmetry of that state under particle interchange. The general presumption is
that (for Fermione) the totally symmetric state, rather than the totally
antisymmetric state, lies lowest in energy, because Pauli Principle constraints
increase the wave function complexity By projecting the Monte Carlo distribution
of 6-functions on a totally antisymmetric trial function, this problem vanishes "in
the mean"; unfortunately this trick does not control the variance, %hereby the
"noise" in the energy distribution can grow uncontrollably and eventually will swamp
the signal as one goes forward in time.
Recently, it has been found that In the A-3 and 4 systems (onl/), the tntally
symmetric ground states lie higner in energy than the antisymmetric ones and he ncen
do not hinder the convergence. This unexpected result is simple to understand, It
we note that isospin ig the degree of freedom which adjusts itself to acLommndta
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Friar, J.L. Recent results in the theory of the three-nucleon systems, article, June 1, 1987; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1105016/m1/4/: accessed April 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.