Question

What is the probability of finding a particle at centre of box in it's first excited state in a infinite potential well ??

Answer 1

The wave function of particle in the first excited state is

$\psi(x)=\sqrt{\frac{2}{L}}sin(\frac{2\pi x}{L})$

The probability of finding a particle at centre of box  is

$P=|\psi(x)|^2(L)$

$=|\sqrt{\frac{2}{L}}sin(\frac{2\pi (x/2)}{L})|^2(L) =0$ since $sin(\pi) =0$