Question

The particle of mass m in the infinite square well (of width a) starts out of the left half of the well, and is (at 1-0) equally likely to be found at any point in that region, what is the initial wave function Ψ(0)? Assume it is real, do not forget to normalize it.

Answer 1

Probability of finding the particle is same at any point in the left part of the region. So the probability density is a constant in the left part of the region.Hence wave function is

$\Psi(x,0)=B \text{ for } [0\leq x\leq \frac{a}{2}]$ and 0 elsewhere.

On normalization we get

$\int_0^\frac{a}{2} |B|^2=1\\ \implies B=\sqrt{\frac{2}{a}}$ .

So

wave function is

$\Psi(x,0)=\sqrt{\frac{2}{a}} \text{ for } [0\leq x\leq \frac{a}{2}]$ and 0 elsewhere.