The General Solution for a Constant Potential

We have found the
general solution of the Schrödinger Equation in a region in which the potential is constant.
Assume the potential is equal to
and the total energy is equal to
.
Assume further that we are solving the time independent equation.

For
, the general solution is

with positive and real. We could also use the linear combination of the above two solutions.

We should use one set of solutions or the other in a region, not both. There are only two linearly independent solutions.

The solutions are also technically correct for
but
becomes imaginary.
For simplicity, lets write the solutions in terms of
, which again is real and positive.
The general solution is

These are not waves at all, but real exponentials. Note that these are solutions for regions where the particle is not allowed classically, due to energy conservation; the total energy is less than the potential energy. Nevertheless, we will need these solutions in Quantum Mechanics.

Jim Branson 2013-04-22