Orbital Azkaban

We’ve been thinking a lot about orbitals in my creative cluster. My students brought up this topic as one that was murky in their minds when they took first semester general chemistry. They had no problem drawing the shapes of orbitals, ordering them according to energy, and writing ground state electron configurations. However, the finer points – thinking about nodes, probability distributions, how all these orbitals are ‘stacked’ (superimposed) on each other, what the electrons are even doing – these they thought were rather confusing. Since they’re sophomores they haven’t taken Quantum Chemistry yet. (They are looking forward to taking Quantum with me next semester, or at least that’s my impression!) Is there anything we can do to help students at the general chemistry level gain a better conceptual understanding of orbitals?

As we started discussing nodes, someone in the group (most likely me) used the analogy of teleportation to describe how an electron ‘travels’ between the two lobes of a p-orbital while avoiding the nodal plane. We talked about how it is difficult to pinpoint exactly where an electron is located – perhaps it disappears and reappears from view. Inevitably Harry Potter makes an appearance as we discuss Apparition as an analogy. My memory is hazy at this point, but possibly the Room of Requirement came up in the context of the Marauder’s Map. Several moments later, one of the students was sketching the Quantum Marauder’s Map on the white board.

We started thinking about how rooms in a castle might represent orbital volumes. Given our limitation of whiteboard-flatland, we first worked in two dimensions. Footprints signify an electron (a quantum wizard!) and the floor-area of the room bounded by the walls represented orbital boundaries. Each color represents the same individual apparating between different ‘allowed’ rooms. Our simple floor plan has no overlap between rooms. Individuals never meet each other. They’re sort of like prison cells. Orbital Azkaban. Voila!

How might we conceive Orbital Azkaban in 3-D? We decided that some rooms must be circular to represent s-orbitals, possibly with concentric rings since the larger s-orbitals have circular nodes within. (Note we’re still working with partial 2-D constructs, as technically these should be spheres rather than circles.) The planar nodes of p-orbitals could be different levels in a multistory building where individuals apparate between levels. You have to jump between levels in quantum world. We then came up with what looks like a multi-level, multi-ring Bundt cake (see the lower right drawing in green), a panopticon prison of sorts, where the guards hover in a nucleus at the center (similar to the Kyln interstellar prison in Guardians of the Galaxy).

But orbitals overlap each other in an actual atom. In these superimposed mess, the ‘walls’ of each cell are different for each prisoner even though they occupy shared space. Electrons also ‘avoid’ each other for both electrostatic and symmetry (Pauli principle) reasons. How does one keep a prisoner within a certain space without actual physical walls? The first thing that jumped to mind was an electronic ‘collar’ akin to dog collars that prevent them from wandering beyond certain confines. A morbid idea when applied to prisoners. To make the idea less repulsive, we started thinking about Orbital Azkaban as a zombie trap (hence the title of bundt-layer-cake structure) except that this prison would not have physical walls. Different types of zombies with different levels of danger could represent different orbitals. The zombie collars confine these creatures to certain spaces although the zombies can teleport into allowed areas.

If I were to return to the Quantum Marauder’s Map picture, what might it look like as we start to conceive of orbitals in the same space?

The 1s orbital is a small circular room confining the prisoner.

The 2s orbital is a larger circular room. The prisoner spends most of the time in the larger outer ring, but occasionally teleports into the inner circle.

The 2p orbital is two adjoining circular rooms that ‘touch’ at a single point. The prisoner spends an equal amount of time in each ‘lobe’, apparating between them.

And here’s what it looks like when the three different prisoners are put together. There are no actual physical walls. And in fact the boundaries represent only a 90% chance where the zombie-prisoner is found. (Orbital boundaries are typically defined in the same way: There is a probability of 0.9 that the electron is within the enclosed volume.)

Each ‘orbital’ can take up to two prisoners. (Only one prisoner is shown in each orbital.) These two prisoners in the same orbital will mildly avoid each other as they move around in their respective areas/volumes. This represents the electrostatic repulsion between negatively-charged particles. Prisoners from different orbitals with the same gender will avoid each other in a different but more significant way representing the Pauli Exclusion Principle. (An electron may be either spin up or spin down. I’m going to represent it by a gender binary for simplicity’s sake even if it might be less PC.) It’s a little tricky to describe what exactly this avoidance means without going into the mathematics. Technically, the overlap integral (calculus!) of the two orbitals must go to zero. The Pauli avoidance only applies to electrons of the same spin. I haven’t attempted to represent these complications pictorially, because I think it would make things more confusing.

This exercise reminded me about the limits of models and analogies. They help to a certain extent… until they don’t. Mathematics allows for a level of abstraction beyond any picture we can draw or ‘see’ with our mind’s eye. I tell my students that math is their friend in this regard, and that’s why it’s worth learning. Every time I teach Quantum, one or more students will tell me: “I understand the concepts, I just can’t handle the math.” My response is that: “If you don’t understand the math, you don’t actually have a good grasp of the concepts.” That’s the nature of quantum mechanics. Feynman was right – no one really understands it. But the math helps. A picture of Orbital Azkaban only gets you so far. The mathematical framework gets you further and beyond.

P.S. If you’re wondering who the dolphin-whale is on the right side of the white board, it’s a student rendition of one of my wild ideas earlier this semester.