Working through a text (self-learning, not homework), I reached this problem:
Prove that the set of functions $\psi_n(x)=a^{-1/2}e^{i\pi nx/a}$ is orthonormal for integer $n$.
So that's a fairly straightforward problem, but I feel like I'm losing my mind. It's particularly clear that if, for two such functions $\psi_m, \psi_n$, the sum $n+m$ is odd, then: $$ \int _{[0,a]}dx\;\psi^*_m \psi_n\ne 0,\;\; m\ne n $$
Does the author mean to make $\psi_n=a^{-1/2}e^{i2\pi nx/a}$? I wish to believe so, but the previous definition is stated in the next two problems in the same way (as first written).
I seriously hope I'm not making any stupid mistakes, but I have a feeling it might be the case.
It's a typo! We need the $2$, for $m,n \in \mathbb{N}$ with $m\neq n$ we have
$$\int_0^a\psi_n \psi^*_m dx = \frac{1}{a}\int_0^a \exp \left( 2 \frac{ i \pi ( m-n) x}{a} \right) = \frac{1}{2 i \pi ( m-n)}\exp \left ( \frac{ 2i \pi ( m-n)x }{a} \right) \Big |_0^a =\frac{\exp[2 i \pi (m-n) ] -1}{i \pi ( m-n)}= 0 $$
If $m = n$, we have that
$$ \int_0^a\psi_n \psi^*_n dx = \frac{1}{a} \int_0^a dx = 1$$
So it indeed is an orthonormal set of functions.