For one thing, you haven't defined any of your variables, so it's hard to provide a very specific answer. You can find various derivations here
. The usual approach is to solve for the respective variances for the two observables of interest in vector space using your formalism of choice and then applying the Cauchy inequality.
Your second question doesn't make a whole lot of sense. You say, "In a given stationary state, it can be expressed as...." What
can be expressed as...? Is this a hydrogen atom? Quantization comes about due to confinement of small particles (e.g., electrons) within a potential field. The nature of the quantization depends on how the boundary conditions are defined (i.e., what the field is: particle in a box, harmonic oscillator, point charge, whatever). Using the boundary conditions the steady state wave equation is solved, giving rise to the eigenfunctions and quantized observables. So, the proof depends on how the system is set up. Since you haven't described how the system is set up, no proof can be offered other than a general strategy, which is the same for all types of boundary conditions.