  We construct general solutions of the time-dependent Dirac equation in (1+1)
dimensions with a Lorentz scalar potential, subject to the so-called Majorana
condition, in the Majorana representation. In this situation, these solutions
are real-valued and describe a one-dimensional Majorana single particle. We
specifically obtain solutions for the following cases: a Majorana particle at
rest inside a box, a free (i.e., in a penetrable box with the periodic boundary
condition), in an impenetrable box with no potential (here we only have four
boundary conditions), and in a linear potential. All these problems are treated
in a very detailed and systematic way. In addition, we obtain and discuss
various results related to real wave functions. Finally, we also wish to point
out that, in choosing the Majorana representation, the solutions of the Dirac
equation with a Lorentz scalar potential can be chosen to be real but do not
need to be real. In fact, complex solutions for this equation can also be
obtained. Thus, a Majorana particle cannot be described only with the Dirac
equation in the Majorana representation without explicitly imposing the
Majorana condition.
