The inverse square force law was also the first really interesting test of quantum mechanics: it describes the motion of an electron around a proton in a hydrogen atom. A high school teacher named Balmer was the one who guessed the spectrum of hydrogen, based on tables of empirical data. Bohr came up with a rough-and-ready approach to quantum mechanics that explained Balmer's formula, but Schrödinger was the first to give a really detailed explanation.
In both cases the solution is "better than it needs to be". What I mean is this. In classical mechanics we can solve for the motion of a particle in any central force by doing an integral; if the force is attractive we'll get orbits that go round and round... but usually the orbits will precess. The magical thing about the inverse square force law
$$ F = -k/r^2 $$
is that they don't precess: we get closed orbits! This is also true for the harmonic oscillator, where the force is described by Hooke's other law:
$$ F = -kr $$
Even better, in both cases the motion is in an ellipse! It was an eerie stroke of luck for Newton that the Greeks—especially Apollonius—"just so happened" to have spent a lot of time studying conic sections just for their intrinsic beauty. That let Newton invent a proof using Euclidean geometry that the planets go around in ellipses, given classical mechanics and an inverse square force law. He probably figured this out using calculus, but in the Principia he hid his tracks, since calculus wasn't rigorous, while Euclid's Elements was regarded as the pinnacle of rigor.
In quantum mechanics we find that a hydrogen atom has $(n+1)^2$ bound states in the $n$th energy level, if we start counting at $n = 0$. Again this is "better than it needs to be". For a typical central force we expect bound states of different total angular momentum to have different energies; but for the inverse square force law something magical happens: there are states of different total angular momenta having the same energy! The $n$th energy level has:
$$1 + 3 + 5 + \cdots + 2n+1 = (n+1)^2$$
states. Here I'm not talking about subtleties involving spin, which double the count of states and split some of these energy levels. I'm just talking about Schrödinger's original calculation.
The reason for both these "magical effects" is the Runge-Lenz vector: an extra conserved quantity besides energy and angular momentum, which is special to the inverse square force law! The formula for it looks a bit funny: $$ \frac{v \times J}{k} - \frac{q}{|q|} $$ where:
To understand the meaning of the Runge-Lenz vector, you need to know two things about it:
So, the fact that it's conserved means the orbit doesn't precess, and doesn't get more skinny or round as time passes.
For a proof of these facts, a proof that the Runge-Lenz vector is conserved, and an argument that uses it to deduce that the orbits in an inverse square force law are conic sections, try these homework problems of mine:
For a medley of more elegant proofs that an inverse square force law gives elliptical orbits, and conversely that elliptical orbits must come from an inverse square force law, see this webpage by Greg Egan:
Both classically and quantum mechanically, Noether's theorem relates conserved quantities and symmetries, so the fact that the Runge-Lenz vector is conserved means the inverse square force law has more symmetry than your average central force. But it's a rather sneaky "hidden symmetry", which changes the eccentricity of the orbits! My friend Michael Weiss says this book has a lovely diagram of this symmetry at work in the quantum case, turning an s-orbital into a p-orbital:
In fact, the angular momentum and Runge-Lenz vector (rescaled by a function of the energy) give a total of 6 conserved quantities, which generate a 6-dimensional group of symmetries. What this group is depends on whether we are looking at solutions with negative energy (bound states, where the particle moves in an ellipse) or positive energy (scattering states, where it moves in a hyperbola).
In the case of bound states, the 6-dimensional symmetry group you get is SO(4), the group of rotations in 4d space!
In the case of scattering states, the 6-dimensional symmmetry group you get is SO_{0}(3,1), the connected component of the Lorentz group! This group is famous in special relativity... who'd have thought it was lurking in Newtonian gravity?
For each $n$, SO(4) has an irreducible representation of dimension $(n+1)^2$ which explains why the hydrogen atom has this many states in its $n$th energy level, discounting spin.
Now, you've probably seen those cute pictures where people draw an atom like a tiny little solar system, with electrons racing around in elliptical orbits. This is silly because it neglects the big difference between classical and quantum mechanics. The uncertainty principle means that electrons don't have well-defined orbits!
But, the analogy between the atom and the solar system becomes rather deep if relate the two using geometric quantization. If you've got just one planet orbiting the sun in an ellipse, or just one electron orbiting your nucleus, you've got a system where the Runge-Lenz vector, you've got a system with SO(4) symmetry!—and quantizing the first gives the second.
If you want more details, read this:
Unfortunately, I still don't feel I know the "real reason" why the Kepler problem has a hidden SO(4) or SO_{0}(3,1) symmetry. Guillemin and Sternberg's book shows that if we only consider the bound states of the Kepler problem, we get a physics problem that is secretly the same as the motion of a free point particle on the unit sphere in 4d space! A bit more precisely, they're the same via a "generalized canonical transformation", where we reparametrize time as well as changing the other variables. This is very beautiful, because mathematically this means we're looking at geodesic motion on S^{3}, which is the same as SU(2), the double cover of the rotation group.
However, Guillemin and Sternberg need a few yucky calculations to reach this conclusion, so I don't feel the subject has been completely demystified. Perhaps this book will make it clearer:
It sounds good, but I haven't read it yet.
I also regard it as mysterious that an object moving in an inverse square force law traces out a conic section. There are lots of ways to prove it, of course. Newton did it using Euclidean geometry. My homework problems above give two other ways. The one using the Runge-Lenz vector is pretty... but I'm still looking for the truly beautiful way, where you leave the room saying: "Inverse square force law... conic sections... of course! Now the connection is obvious!"
If you want to think more about why the inverse square law leads to elliptical orbits, try the argument in Feynman's "lost lecture":
In any event, the book is definitely worth reading. I didn't listen to the CD that came with it, but I'm sure that's fun too.
Once my friend Minhyong Kim lent me this book by the famous Russian mathematical physicist Arnol'd:
This idea of reparametrizing time be related to the "generalized canonical transformation" that I mentioned above. Unfortunately, I haven't put the puzzle pieces together yet. So, if you have profound insights on these issues, let me know!
Here are some more references which may help you find these insights:
Moreover, there is a generalized version of the Kepler problem for every formally real Jordan algebra, and the Kepler problem we know and love is related to the Jordan algebra of $2 \times 2$ hermitian complex matrices, which can also be thought of as 4-dimensional Minkowski spacetime! For more, try:
The SUSY QM approach is how we solved the hydrogen atom in our undergrad quantum class at MIT.For more, see:
© 2012 John Baez
baez@math.removethis.ucr.andthis.edu