Chapter
2

The radial distribution function

Many textbook plots of orbitals and wavefunctions can be confusing, and it is instructive to compare the probability density function, ψ², with the radial distribution function, P = 4πr²ψ².  ψ²dτ gives the probability of finding the electron in a small volume element, dτ, whereas Pdr is the probability of finding the electron anywhere within a spherical shell of thickness dr at distance r from the nucleus (the volume of the shell is 4πr² dr).

Exercise 3.15

Attempt problem 20.20 on p.452 of E&R.

Fig. 20.9 and 20.10 of the text illustrate the difference between ψ² and P = 4πr²ψ² for the ground state wavefunction of the H atom. The maximum in P gives the most probable radius at which the electron will be found; for a 1s orbital in hydrogen, r = a₀, precisely the Bohr radius! For a 2s orbital, r = 5.2a₀ (check that this is the case on the appropriate plot in Fig. 20.10, p.449 of E&R).