# Normalizing wave function

by SoggyBottoms
Tags: function, normalizing, wave
 P: 61 At t = 0 a particle is in the (normalized) state: $$\Psi(x, 0) = B \sin(\frac{\pi}{2a}x)\cos(\frac{7\pi}{2a}x)$$ With $B = \sqrt{\frac{2}{a}}$. Show that this can be rewritten in the form $\Psi(x, 0) = c \psi_3(x) + d \psi_4(x)$ We can rewrite this to: $$\Psi(x, 0) = \frac{B}{2}\left[ c \sin(\frac{4 \pi}{a}x) - d\sin(\frac{3\pi}{a}x)\right]$$ The answer sheet gives $c = -d = \frac{1}{\sqrt{2}}$. I assume you can find this by calculating $A^2 \int \left[ c \sin(\frac{4 \pi}{a}x) - d\sin(\frac{3\pi}{a}x)\right]^2 dx$. I attempted to do it this way, but it becomes a really long calculation and halfway through I just lose track of everything. Is there an easier way to find c and d?