  Supersymmetry plays a main role in all current thinking about superstring
theory. Indeed, many remarkable properties of string theory have been explained
using supersymmetry as a tool. In this dissertation, we review the basic
formulation of supersymmetric quantum mechanics starting with introducing the
concepts of supercharges and superalgebra. We show that, if there is a
supersymmetric state, it is the zero-energy ground state. If such a state
exists, the supersymmetry is unbroken otherwise it is broken. So far, there has
been no unbroken supersymmetry observed in nature, and if nature is described
by supersymmetry, it must be broken. In fact, supersymmetry may be broken
spontaneously at any order of perturbation theory, or dynamically due to
non-perturbative effects. The goal of this dissertation is to study the methods
of supersymmetry breaking. For this purpose, special attention is given to
discuss the normalization of the ground state of the supersymmetric harmonic
oscillator. Then we explain that perturbation theory gives us incorrect results
for both the ground state wave function as well as the energy spectrum and it
fails to give an explanation to the supersymmetry breaking. Later in the
dissertation, a review of the uses of instantons in quantum mechanics is given.
In particular, instantons are used to compute the tunneling effects within the
path integral approach to quantum mechanics. As a result, we give evidence that
the instantons, which are a non-perturbative effect in quantum mechanics and
can not be seen in perturbation theory, leads to calculate the corrections to
the ground state energy and provides a possible explanation for the
supersymmetry breaking.
