Q1(a)(i). Most probable distance of the 1s electron

Answer in brief

The most probable distance is r = a₀ = 52.9 pm (0.529 Å), the radius of the first Bohr orbit.

Key points

• For hydrogen in its ground state Z = 1 and n = 1, so ρ = 2r/a₀ and R₁₀ = 2a₀^−3/2 e^−r/a₀.
• The quantity to maximise is the radial probability, not R² alone: P(r) = r²R₁₀² = (4r²/a₀³) e^−2r/a₀.
• Setting dP/dr = 0 gives (4/a₀³) e^−2r/a₀(2r − 2r²/a₀) = 0, so r = a₀.
• R² is largest at the nucleus, but a shell there has no volume; the r² factor pushes the maximum out to a₀.
• The average distance is larger, ⟨r⟩ = 1.5a₀, because the curve has a long tail.