  We describe a further development of the stochastic state selection method,
which is a kind of Monte Carlo method we have proposed in order to numerically
study large quantum spin systems. In the stochastic state selection method we
make a sampling which is simultaneous for many states. This feature enables us
to modify the method so that a number of given constraints are satisfied in
each sampling. In this paper we discuss this modified stochastic state
selection method that will be called the constrained stochastic state selection
method in distinction from the previously proposed one (the conventional
stochastic state selection method) in this paper. We argue that in virtue of
the constrained sampling some quantities obtained in each sampling become more
reliable, i.e. their statistical fluctuations are less than those from the
conventional stochastic state selection method. In numerical calculations of
the spin-1/2 quantum Heisenberg antiferromagnet on a 36-site triangular lattice
we explicitly show that data errors in our estimation of the ground state
energy are reduced. Then we successfully evaluate several low-lying energy
eigenvalues of the model on a 48-site lattice. Our results support that this
system can be described by the theory based on the spontaneous symmetry
breaking in the semiclassical Neel ordered antiferromagnet.
