A particle of mass $m$ is confined within an infinite, one-dimensional potential well, $U(x)$, of width $a$. $$ U(x) = \begin{cases} \infty & x \leq \frac{-a}{2}, x \leq \frac{a}{2}\\[6pt] 0 & \frac{-a}{2} \leq x \leq \frac{a}{2} \end{cases} $$
I have to find the normalised solutions of the time-independent Schrodinger equation.
I know that if it were $0 \leq x \leq a$ rather than $\frac{-a}{2} \leq x \leq \frac{a}{2}$, the solution would be given by $\psi(x) = \sqrt{\frac{2}{a}} \sin(\frac{n\pi x}{a})$. So to find the solution when it is $\frac{-a}{2} \leq x \leq \frac{a}{2}$, do you just replace $a$ with $a/2$ in the solution?