Question

9. According to quantum mechanics, we must describe the position of electron in the hydrogen atom in terms of probabilities. (a) What is the difference between the probability density as a function of r and the radial probability function as a function of r?(2 pts) (b) What is the significance of the term 4nr2 in the radial probability functions for the s orbitals?(2 pts) (c) Make sketches of what you think the probability density as a function of r and the radial probability function would look like for the 4s orbital of the hydrogen atom.(2 pts)

Answer 1

a)

For the 1s level of a hydrogen atom, the eigenfunction is

ψ(r,θ,ϕ)=12πa−3/20exp(−r/a0)ψ(r,θ,ϕ)=12πa0−3/2exp⁡(−r/a0)

and there is no angular dependence.

But when you want to work out a probability density P(r)P(r) for the electron to be found between rr and r+drr+dr, then you need to consider an integral of the square of the modulus of the eigenfunction over the volume enclosed by the spherical shell between rr and r+drr+dr and this volume is 4πr2 dr4πr2 dr.

In other words, the (unnormalised) probability density as a function of radius

P(r)=4πr2ψ(r)ψ∗(r)P(r)=4πr2ψ(r)ψ∗(r)

.

So whilst ψ(r)ψ(r) peaks at the origin, P(r)P(r) is zero at the origin. To work out at what radius the electron is most likely to be you look for a maximum in P(r)P(r).

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