Normalize a wavefunction

Question

This is a little beginner level, but how do we normalize $\Psi(x)= x(a-x)$ over the interval $0\leq x\leq a$. I do not understand how separate the a from the equation.

Answer 1

If I understand you correctly, then you have some parameter $a>0$ and a function \begin{align*} \psi:[0,a]&\to\mathbb R\\ x&\mapsto x(a-x) \end{align*} which should be the wave function of some quantum system. For this, however, it has to be normalized which is why you are looking for $C=C(a)>0$ such that the modified function $\Psi(x):=\frac1C\psi(x)$ satisfies $\int_0^a|\Psi(x)|^2\,dx=1$, that is, \begin{align*} 1=\int_0^a|\Psi(x)|^2\,dx&=\frac1{C^2}\int_0^a|\psi(x)|^2\,dx\\ &=\frac1{C^2}\int_0^ax^2(a-x)^2\,dx\\ &=\frac1{C^2}\Big[ \frac{a^2x^3}3-\frac{ax^4}2+\frac{x^5}5 \Big]_0^a=\frac1{C^2}\frac{a^5}{30}\,. \end{align*} Now this is equivalent to $C^2=\frac{a^5}{30}$ or $C=\sqrt{\frac{a^5}{30}}$. Therefore the normalized wave function is given by \begin{align*} \Psi:[0,a]&\to\mathbb R\\ x&\mapsto \sqrt{\frac{30}{a^5}}x(a-x)\,. \end{align*}