What is the average radial position of the particle as a

Question

1. a. How many radial nodes arc present in the radial eigenfunction associated with the 2p atomic orbital for hydrogen-like systems? What about the number of nodes in the corresponding spherical harmonics?

b. For the hydrogen-like Mercury ion (Min, at what values of r will the radial nodes appear?

Note: R₂¹(r) = {1 / 4(6)^1/2}{Z/a_a}^3/2 ρe^-ρ/4

Where ρ = 2Zr / a_a

2. Using the atomic wavefunctions given in class and in your textbook, plot BY HAND the probable location of the electron contained in the 2p] orbital of a hydrogen atom on the graph below. Explain with the help of equations, how you achieved this plot?

3. The normalized eigenfunctions for a particular hydrogen-like atom whose radius is R_o are

Ψ(r, θ, Φ) = R_n¹(r) Y₁¹(θ, Φ) for r ≤R₀

and Ψ(r, θ, Φ) = 0 for r>R₀

Y₁¹(θ, Φ) is a spherical harmonic equal to (3/8 π)^1/2 sin θe^{i θ}, and R_n⁰(r) = (2/R₀)^1/2[(sin [nπr/R₀])/R₀]

What is the average radial position of the particle as a function of the quantum number n.