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Solution to the Schrödinger Equation in a Constant Potential

Assume we want to solve the Schrödinger Equation in a region in which the potential is
constant and equal to
.
We will find two solutions for each energy
.
We have the equation.

Remember that
is an independent variable in the above equation
while
and
are constants to be determined in the solution.

For
, there are solutions

and

if we define
by the equation
.
These are waves traveling in opposite directions with the same energy (and magnitude of momentum).
We could also use the linear combinations of the above two solutions

and

There are only two linearly independent solutions.
We need to choose either the exponentials or the trig functions, not both.
The sin and cos solutions represent states of definite energy but contain
particles moving to the left and to the right.
They are not definite momentum states.
They will be useful to us for some solutions.

The solutions are also technically correct for
but
becomes imaginary.
Lets write the solutions in terms of
The solutions are

and

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.
We will use these solutions in Quantum Mechanics.

Jim Branson
2013-04-22