Wavefunction for even potential

[sspitz]

In his textbook, Griffiths claims that solutions to the TISE for even potentials for a given energy can always be written as a linear combination of even and odd functions. That I understand. However, I do not see why that fact justifies only looking for even or odd solutions, as he does later in the chapter.

This is how I see the steps going for a hypothetical simple case:
(1) There is an even potential
(2) Leads to a 2nd order ode (TISE). Write down the general solution. Let's say it's sine and cosine.
(3) Look for even solutions that satisfy boundary conditions (cosine). Look for odd solutions that satisfy bc (sine).

Is it not possible that there could be a psi that is:
(1) A linear combination of sine and cosine
(2) Satisfies the boundary conditions
(3) While sine and cosine don't satisfy the boundary conditions individually

Looking at even and odd functions separately would miss this?

 

[jbsteele]

Yes, that is possible. Griffiths' point is that when looking for solutions to the TISE for an even potential, it is much easier to look for even and odd solutions separately than to try to find a linear combination of both that satisfies the boundary conditions. This is because the boundary conditions for even potentials are often symmetric about the origin, which means that only the even (or odd) parts of the wavefunction need to be considered. It is therefore more efficient to just look for even (or odd) solutions.