We study a quantum particle conned to a one-dimensional box with walls at the positions ±1. The Hilbert-space of this system in the Schrödinger representation is again given by $L_2(-1,1)$. In this Hilbert space, we consider the two functions $g_0(x):=(1+i)exp(i \pi x)$ and $g_1(x):=exp(i 2 \pi x)$. I shown that $g_0$ and $g_1$ are orthogonal in $L_2([-1,1])$ and that the orthonormal states $\phi_0=\frac{g_0(x)}{2}$ and $\phi_1=\frac{g_1(x)}{\sqrt{2}}$.
Now we now consider a general superposition $\Psi=\alpha \phi_0+\beta \phi_1$ of the two states $\phi_0=\frac{g_0(x)}{2}$ and $\phi_1=\frac{g_1(x)}{\sqrt{2}}.$ Now I have to show that in order for $\Psi$ to be normalized we need that $|\alpha|^2+|\beta|^2=1$. But I'm not sure how to show that. Can anyone help me. What will an argument be? Do I need to proof it the both way?