If we simply redefine the position of the box so that
then our problem has symmetry under the Parity operation.
This means that
is an eigenfunction of
with the same energy eigenvalue.
If we operate twice with parity, we get back to the original function,
The boundary conditions are
This gives two types of solutions.
This is an example of a symmetry of the problem, causing an operator to commute with the Hamiltonian. We can then have simultaneous eigenfunctions of that operator and . In this case all the energy eigenfunctions are also eigenstates of parity. Parity is conserved.
An arbitrary wave function can be written as a sum of the energy eigenfunctions
recovering the Fourier series in its standard form.
Jim Branson 2013-04-22