Question

At time t=0, a hydrogen atom is in the superposition state:

ψ(r,0)=(4/(2a₀)^{3/2}))*[((e^{-r/a₀})/(√(4π)))+A(r/(a₀))e^{-r/a₀}(-iY₁¹+Y₁⁻¹+√7Y₁⁰)]

a) what is the probability density P_r(r)[=∫|ψ(r,0)|²r²dΩ] that the electron is found in the shell of thickness dr about the proton at the radius r?

b) at what value of r is P_r(r) maximum?

c)Given the inital state ψ(r,0), what is ψ(r,t)?

I know using the orthonormalization condition makes it easier but i dont know how to incorporate it properly. Can you please show the steps so i can understand the process. Thank you