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?