Normalization of wave equation

Question

I am a beginner in the field of Quantum Mechanics. I was reading the book of SP KUILA (Vol-II) on Engineering Physics. There I came across this line:

But since according to the condition of normalisation, $$ \int_{-\infty}^{\infty}ψ^*ψdx=1,$$ $$ψ^*=0$$ at x=+∞ and x=-∞.

I did not understand why it is zero. Please explain.

Answer 1

That $$\lim_{x \rightarrow \pm \infty} f(x) = 0$$ is a necessary condition for the integral $$ \intop_{-\infty}^{\infty} dx f(x) $$ to converge.

Apply this to $f(x) = |\Psi(x)|$ to get $|\Psi(\pm \infty)| = 0$, which can only be true if $\Psi(\pm \infty) = \Psi^{*}(\pm \infty) = 0$.