[QUE/QM-06024]

If \(\psi(x,y,z)\)is the wave function of a particle in three dimension   write expression for probability that

1. \(x\) is between \(x_1\) and \(x_2\);
2. \(p_z\) lies between \(p_1\) and \(p_2\);
3. Assume that \(\psi(\vec{r}) =  R(r) Y_{\ell m}(\theta,\phi)\). What is the probability that the particle will be found inside a sphere of radius \(R\)? What is the probability that the particle will be found outside the sphere of radius \(R\)?