You’re mistaken about the figure, unless you’re viewing the web page on a device that distorts the aspect ratio; the dashed arc is perfectly circular and it is centred on the sun. I checked the image file itself, and all four points’ centres are 231 pixels from the centre of the sun.

I have no idea why you think the two orbits should have the same area (or why you think one of them is circular). The two orbits have the same semi-major axis (as required) and different semi-minor axes, so they do not have the same area. You *could* choose the new position for the planet so that its orbit was circular, and then the dashed arc would overlay it, but that would make for a far more confusing image because the generality of the construction would not be apparent.

This is not correct according to the figure drawn. The four points shown are not at the same distance. All you have to do is draw a straight line from the Sun through the points on the ellipse, and you can clearly see that the points on the ellipse are much closer. Don’t worry though, the error is only in the drawing because they didn’t draw the ellipse and the circle with the same exact total area. At that point the dotted arc would overlay the circle and should be eliminated because it would be redundant. Sorry to be picky like that but … you know … it’s mathematics.

]]>• Eyal Subag, Symmetries of the hydrogen atom and algebraic families.

Abstract.We show how the Schrödinger equation for the hydrogen atom in two dimensions gives rise to an algebraic family of Harish-Chandra pairs that codifies hidden symmetries. The hidden symmetries vary continuously between SO(3), SO(2,1) and the Euclidean group O(2)⋉R2. We show that solutions of the Schrödinger equation may be organized into an algebraic family of Harish-Chandra modules. Furthermore, we use Jantzen filtration techniques to algebraically recover the spectrum of the Schrödinger operator. This is a first application to physics of the algebraic families of Harish-Chandra pairs and modules developed in the work of Bernstein et al. [Int. Math. Res. Notices, rny147 (2018); rny146 (2018)].

One interesting thing about this paper is that it constructs a ‘space of 2d hydrogen atom state’ for any *complex* energy. The physical meaning of these seems open for exploration.

I looked up the source: It’s “Galileo’s finger – The ten great ideas of science” by Peter Atkins, chapter six “Symmetry” and Fig. 6.7 on page 175. Unfortunately, since it is really a text addressed to laymen, it does neither contain any mathematics nor references where this idea would have been made more precise. I can only guess that Atkins might have written it elsewhere, e.g. in his book on “Molecular Quantum Mechanics”, but I did not checked it so far.

]]>Here’s a fun thing that’s sort of related. It’s true that irreducible unitary representation of a group doesn’t usually carry any unit vector to any other. But sometimes the group acts transitively on the unit sphere of its irreducible spinor representations; this happens up to but not beyond. One exciting case is , where

is the group of 2 × 2 quaternionic unitary matrices, and the spinors are the quaternionic unitary matrices act transitively on the unit sphere in Another is , where

is the group of 4 × 4 complex unitary matrices with determinant 1, and the spinors are these matrices act transitively on the unit sphere in When you get beyond there are pure spinors which are different, and ‘better’, than the rest.

]]>Okay, great! I think you’re right: as a function of the point moves around a great circle on the unit 4-sphere at unit speed. So, its velocity vector does the same, and in the same 2-plane.

]]>You’re right; I was being pretty sloppy.

It would be fun to dig into this issue, and see exactly which vectors you *can* get by applying an SO(4) element to an *n*s state, but I have too many other puzzles on my mind to tackle this one!

I’m confident of two things: (1) the semi-minor axis of an ellipse that is the orthogonal projection of a great circle is proportional to the cosine of the angle between the plane of the great circle and the plane it is projected on, and (2) in the usual stereographic projection of the velocity circle from a great circle, there is the same relationship between the angle of the plane of the great circle and the semi-minor axis of the elliptical orbit.

As far as I understand Göransson’s approach, both the point and the velocity wrt move in great circles, and unless I’m confused, the two planes will necessarily be parallel. So the great circle traced out by will also be parallel to the great circle in the standard approach, and up to some choice of scaling, both ought to project to the same circles in velocity space.

]]>John wrote:

What this means is that the hidden 4-dimensional rotation symmetries of the hydrogen atom can do things like take a 4s state to a 4p state or 4d state or 4f state.

I’m not sure if I’m taking you more literally than you intended, or if I’m nitpicking unnecessarily, but although a generic element of SO(4) will certainly take a 4s state to a superposition of states that includes a 4p state (and a 4d state, and a 4f state), I’m not convinced that any element of SO(4) can take a 4s state to a 4p state.

That is, given an eigenfunction of the 3D orbital angular momentum operator with , I don’t believe there is an element of SO(4) that takes it to an eigenfunction of with . It’s not inconceivable that there could be such an element, but it seems like it would take something much stronger than having an irreducible representation of SO(4) to make this true.

]]>Wolfgang wrote:

As a chemist I wonder: Is this a symmetry taking one orbital type into another one?

Yes, exactly! All states of a given energy lie in a single irreducible representation of SO(4). What this means is that the hidden 4-dimensional rotation symmetries of the hydrogen atom can do things like take a 4s state to a 4p state or 4d state or 4f state. So, all 16 states here are related by 4-dimensional rotation symmetries:

while 3d rotation symmetries only suffice to relate states in a given row.

I don’t understand the pictorial approach you’re describing well enough to see if it’s secretly a lower-dimensional analogue of what I’m talking about. I *do* know that what I’m talking about works in every dimension—if we posit atoms in other dimensions that are still governed by the inverse square force law, which is a bit odd.