  Numerical calculations for Majorana zero modes on a one-dimensional chain are
performed using the technique of block diagonalization for general parameter
settings. It is found that Majorana zero modes occur near the ends of the chain
and decay exponentially away from the ends. The phase diagrams show that
Majorana zero modes of a long-enough chain indeed have a parameter domain of
$2t>|\mu|$ as predicted from the bulk property of the chain, but a short chain
has a much smaller parameter domain than the prediction. Through a numerical
simulation Majorana zero modes are found to be robust under the disturbance of
noise. Finally the reversion of the parity of the ground states is studied by
applying a bias voltage on the quantum dot at an end of the chain. It is found
that for a weak coupling between a chain and a quantum dot the parity of the
ground states can be reversed through adiabatically tuning the bias voltage.
