Schrödinger's Cormorant

6.8 That Reminds Me of Something

Making the Connection to Electron Wavefunctions in Atoms

Cormorant, who has studied a bit of Chemistry in his time, remarks “You know, those wavefunctions we've discovered remind me a bit of atomic orbitals. I mean like s-orbitals seem a bit like the n=1 function and p-orbitals are like n=2. Is that cool, or what”.

“What” replies Nefertiti. “I think you’re comparing apples to oranges there. But, we may still learn something if we pick apart your confused notion”. Cormorant considers this concession to be a minor triumph.

Let’s take a quick look at this issue. In Chemistry, we learn that electrons in atoms are confined – or, at least, are most likely to be found - in regions of space, surrounding the nucleus, which we call orbitals. Cormorant is correct that these are, in reality, based on wavefunctions that describe the probability density for the position of the electron within the atom. They were discovered not by experiment but by Erwin himself, solving his own equation - in three dimensions - with the electrostatic potential field arising from the positive charge of the nucleus. The solutions are three-dimensional wavefunctions.

Because of the spherical symmetry of the potential field surrounding the nucleus, it is mathematically extremely helpful to abandon the usual Cartesian x,y,z coordinates and instead use polar coordinates, which means these wavefunctions are of the form ψ(r,θ,φ), where r is the distance from the nucleus and θ,φ are angles, relative to the x- and z-axes. This has the effect of making the wavefunction separable, so ψ(r,θ,φ) = ψr(r)ψθ(θ)ψφ(φ). We already know, from what we've been able to do by separating the time and spatial parts of the overall wavefunction, just how precious this separability is.

An electron in an atom is - in a sense - confined in a “box”, just like our model particle, but that box is an infinite sphere, with the nucleus in the middle and infinity at the outside. The three-dimensional wavefunction must, therefore, go to zero at the edges of the box (in other words, as r tends to infinity), to meet the smoothness requirement that led to the same thing in our one-dimensional box and also to allow the function to be normalised. There are more boundary conditions too, resulting from the need for the wave function along a line at an angle θ relative to the coordinate system to be the same at an angle θ+2π, or θ+4π etc, because these physically represent the same line. As a result, the physically possible wavefunctions are limited by quantum numbers and the possible energies are, correspondingly, limited to certain possible values. The detail of this is mathematically fearsome but we can see that - in general terms - what we foresaw back in chapter 1 is indeed the case: the quantisation of energy in an atom is a consequence of the need for a stable standing wavefunction for an electron. Hurrah!

We should be careful to be clear what we are looking at in the familiar pictures of s- and p-orbitals we learn in Chemistry:

Remember that we are talking about 3-dimensional wavefunctions - so we can't easily visualise them in the same way we have been doing for the 1-dimensional particle in a box. A pictures like those above can be thought of as representing a slice, passing through the nucleus and cutting through the 3-dimensional probability density distribution, dependent on ψ2. The boundary drawn typically represents the 90% probability contour: meaning the intersection of our slice with the volume within which there is a 90% chance that the electron would be found, if it were pinned down.

In the case of an s-orbital, the probability density distribution, ψ2, is identical along any line that passes through the nucleus (any θ,φ), so it depends only on r, the distance from the nucleus. As a result, the orbital has spherical symmetry. For a p-orbital, however, this r-dependent part of the wavefunction is multiplied by a variable term that depends on the angular part of the wavefunction, ψθψφ. In the case of the p-orbital shown above, this multiplier will be maximal along the vertical axis and zero along the horizontal axis, giving the characteristic "dumbell" shape.

Now we can see the similarity that Cormorant was alluding to (so he insists) with the particle in a box. The n=2 particle in a box wavefunction has a node at the midpoint of the box: ψ2 is zero, so there is a near-zero chance of the particle being located in the vicinity. The n=1 function, by contrast, has no such node. A point in a 1-dimensional wavefunction is equivalent to a plane in a three-dimensional wavefunction. So a p-orbital is, in this sense, analogous to the n=2 particle in a box wavefunction: there is a nodal plane, in which the wavefunction goes to zero, so there is negligible chance of finding the electron in the vicinity of this plane. An s-orbital, by contrast, has no such nodal plane.

Within the 90% probability contour, the probability density is not constant but varies with distance from the nucleus, according to the radial (r-dependent) part of the wavefunction, ψr. By extension of what we already know, ψr2 gives the relative probability density of finding the electron close to a given r, along any particular direction (although the absolute probability will, in general, also depend on ψθ2ψφ2).

It is interesting to look at the radial probability density function for s- and p-orbitals. These, unlike the familiar orbital diagrams like those above, are analogous to the ψ2 versus r diagrams that we constructed for the particle in a box. Here are the calculated functions for 2s and 2p orbitals:

Let's look at some slightly random but, hopefully, interesting issues arising from this diagram:

Question 6-10

We've barely made even the slightest of scratches on the surface of what we can understand about atoms by exploring these wavefunctions. To go further, however, requires a great deal of maths, some of which is, realistically, beyond us at this point. Hopefully you may feel motivated to go on and study quantum mechanics at a higher level, so that you can go on through the door that we've started to open. Nefertiti has already signed up. For now, though, let's get back on to safer ground and the particle in a 1-dimensional box.

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