(6-13)   ①   We have:

Looking at the two halves of the box (left and right), the n=2 wavefunction has a positive segment on the right and an equal but negative segment on the left. These will be multiplied, in the product function by the equal, positive values of the n=1 function. So the negative area on the left will be exactly cancelled out by the positive area on the right and so the total - which is the definite integral - will be zero. 

We can see this visually here:

⑤  Since V(x) = 0 within the box, the Hamiltonian is just:

The form of this allows us to evaluate it quite simply because we can take the coefficients (c) outside the derivative, to get:

Comparing to our original expression for ys , the coefficient of ψ1 has ended up multiplied by c1E1 while that of ψ2 is multiplied by c2E2 and – since these numbers are not the same – this means that operation of Ĥ on ψs  does not yield a simple multiple of ψs.

So ψs does not satisfy the Schrödinger equation: it’s not an eigenfunction.