We have the equation.
Remember that is an independent variable in the above equation while and are constants to be determined in the solution.
, there are solutions
We could also use the linear combinations of the above two solutions
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
Lets write the solutions in terms of
The solutions are
Jim Branson 2013-04-22