The general solution to the time independent Shrodinger Equation
is
Notice that solutions where the total energy (EV) is positive will be harmonic as was the case inside the infinite square well.
Solutions where the total energy is negative result in a complex value of k and will become a negative exponential in 'x'.
In the classical case negative energies are not allowed and a particle could never enter to a region where the total energy was less than zero. For instance if a marble was contained in a spherical depression and had less total energy than was required to overcome the potential energy at an elevation equal to the the rim of the depression, it would remain contained within the depression indefinitely... it would never be able to escape without external assistance.
In the quantum case the particle will have some, exponentially declining probability of entering a region where its total energy is less than zero. Particles do not usually appear within their negative energy region but they do have a finite probability of appearing on the other side of a potential energy barrier which would be insurmountable in the classical case.
This phenomena of appearing magically on the other side of a barrier is known as tunnelling and has important applications in semiconductor electonics where tunneling devices are used to control currents. Wikipedia contains the following explanation of important applications of the tunnelling phenomena.
