  In this paper, we study the quantum dynamics of a one degree-of-freedom (DOF)
Hamiltonian that is a normal form for a saddle node bifurcation of equilibrium
points in phase space. The Hamiltonian has the form of the sum of kinetic
energy and potential energy. The bifurcation parameter is in the potential
energy function and its effect on the potential energy is to vary the depth of
the potential well. The main focus is to evaluate the effect of the depth of
the well on the quantum dynamics. This evaluation is carried out through the
computation of energy eigenvalues and eigenvectors of the time-independent
Schr\"odinger equations, expectation values and position uncertainties for
position coordinate, and Wigner functions.
