  We consider in detail the quantum-mechanical problem associated with the
motion of a one-dimensional particle under the action of the double-well
potential. Our main tool will be the euclidean (imaginary time) version of the
path-integral method. Once we perform the Wick rotation, the euclidean equation
of motion is the same as the usual one for the point particle in real time,
except that the potential at issue is turned upside down. In doing so, our
double-well potential becomes a two-humped potential. As required by the
semiclassical approximation we may study the quadratic fluctuations over the
instanton which represents in this context the localised finite-action
solutions of the euclidean equation of motion. The determinants of the
quadratic differential operators are evaluated by means of the zeta-function
method. We write in closed form the eigenfunctions as well as the energy
eigenvalues corresponding to such operators by using the shape-invariance
symmetry. The effect of the multi-instantons configurations is also included in
this approach.
