Question

At t = 0 the hydrogen atom is in the superposition state Ψ(𝑟⃗, 0) =1√3𝜓100 + 𝐴𝜓210 where A is a real positive constant. Find A by normalization

Answer 1

Given wave function is

$\psi(r,0) = \frac{1}{\sqrt {3}} \psi _{100}+ A \psi_{210}$

According to Normalization condition,

$\int |\psi(r,0)|^2 dr=1$

$\int |\frac{1}{\sqrt {3}} \psi _{100}+ A \psi_{210}|^2 dr=1$

$\frac{1}{3}\int |\psi _{100}|^2 dr + A^2 \int |\psi _{210}|^2 dr=1$

$\frac{1}{3}+A^2=1$

Since $\int |\psi _{100}|^2 dr =1 and$ $\int |\psi _{210}|^2 dr=1$

On simplifying above equation, we get

$A=\sqrt\frac{2}{3}$