IGNOU MPH 4 2026 - i) Write down the eigenfunctions for the and &nb

Question

i) Write down the eigenfunctions for the $\psi_{300}$ and $\psi_{310}$ states of the hydrogen atom.
ii) Write the eigenvalues of $\hat{L}^2$ and $\hat{L}_z$ for the states $\psi_{300}$ and $\psi_{310}$ .
iii) Calculate the most probable value of r for the hydrogen atom in the state $\psi_{210}$ .

19 Jan 2026

Answer :

Word Count : 188

For the hydrogen atom the stationary states are written as ψₙₗₘ(r,θ,φ)=Rₙₗ(r)Yₗₘ(θ,φ). i) For ψ₃₀₀ we have n=3, l=0, m=0. The angular part is the spherical harmonic Y₀₀(θ,φ)=1/√(4π). The radial part for n=3, l=0 is R₃₀(r)=(\frac{2}{27\sqrt{3}a_0^{3/2}}\left(27-18\frac{r}{a_0}+2\frac{r^2}{a_0^2}\right)e^{-r/3a_0}). Hence ψ₃₀₀(r,θ,φ)=R₃₀(r)·(1/√(4π)). For ψ₃₁₀ we have n=3, l=1, m=0. The angular part is Y₁₀(θ,φ)=(\sqrt{\frac{3}{4\pi}}\cos\theta). The radial part for n=3, l=1 is R₃₁(r)=(\frac{4}{81\sqrt{6}a_0^{3/2}}\frac{r}{a_0}\left(6-\frac{r}{a_0}\right)e^{-r/3a_0}). Thus ψ₃₁₀(r,θ,φ)=R₃₁(r)·(\sqrt{\frac{3}{4\pi}}\cos\theta). ii) The eigenvalues of (\hat L^2) are l(l+1)\hbar² and

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