Calculation Page Notes: Bohr radius

This derivation is taken from a Feynman lecture. Professor Feynman emphasized that the fact that this comes out correct is a bit of an accident (but we'll take it!) It can certainly be thought of as an 'order of magnitude / unitary analysis calculation' that happens to come out right to within about 10%. I am personally really enamored of this derivation because it uses two axioms plus a definition to arrive at a really really basic pair of numbers: The radius and binding energy of a hydrogen atom in its ground state. It's like picking up a chunk of pavement and deriving the existence of St.Paul's cathedral.

The first axiom is the Heisenberg principle: Δx.Δp ≥ hbar.

That is, the uncertainty in position Δx multiplied by the uncertainty in momentum Δp is on the order of 1.1e-34 Joule-seconds. Maybe this is already a zeroth finesse and it should be the Planck constant h, not h / 2pi or hbar. There are a few finesse points in this discussion in fact, this being the least of them!

Here is a diagram of the hydrogen atom     .     where the proton is at rest at r = 0. (I didn't try to draw the electron because it's much smaller and it's respective volume-of-occupation would not fit to scale on this web page. If you happen to be reading this in the Houston Astrodome...)

So we've started with that proton, and never mind that we can't exactly pinpoint its location relative to something else (because of Heisenberg); at this stage we just imagine the proton is sitting somewhere and we need not be definite about where. If I were to insist on putting it in a chamber then it's in there okay but it is drifting to the left or to the right a little bit so Heisenberg's principle is satisfied. The first finesse here is that I can legally write the proton's location in its own inertial reference frame as "proton at the origin", bearing in mind this is a very approximate calculation.

The proton, in order to be a properly electrically neutral hydrogen atom, also has an electron. Now we have a second thing that has to exist in space relative to the first thing, so the Heisenberg principle becomes more relevant and necessary. The electron is attracted to the proton. In fact the electron ought to just fall down and sit right next to the proton due to this attraction but it doesn't.  This is an experimental observation. If the electron were to drop to r = 0 and sit still then we'd have both its position and momentum resolved very precisely. The lack of uncertainty would violate axiom 1. In fact in this derivation it is axiom 1's job to prevent the electron from falling into the proton, thereby providing a theoretical basis for the experimental reality. This is rather a posteriori reasoning but that's theory for you.

On the other hand, the electron does hang around mostly near the proton because of the Coulomb attraction. That force really does tend to pull the two together. In a sense the two ideas oppose each other creating an equilibrium state for this hydrogen atom.

Suppose the electron occupies a spherical region around the proton of radius a. There are a lot of historical analogies for how the electron 'sits' there--a cloud of charge and so on--and they mostly have to do with trying to avoid thinking about the electron as a little tiny planet. I'm going to skip over any attempt at analogies and just push on. I don't want to muddy my thinking by trying to cast the electron in any particular mold at this point. It's just some damn thing.

Here comes the second finesse: I can invoke Heisenberg to derive an average momentum p by starting with the idea that the uncertainty in the electron's position Δx is simply this average radial distance a:
p = hbar/a

Mind the finesse: I went from momentum uncertainty Delta p to a very definitive expression for momentum. This has to be a cheat. If I had to justify this cheat in one sentence it would have to be a run-on sentence like this:

If I were to measure the electron's momentum, supposing the electron behaved like a superball, then it would tend to be rocketing around from left to right or right to left or down to up or up to down, that is, the momentum would be a vector quantity of finite value and zero average... but I can use Heisenberg to constrain it to a typical scalar value as given by the equation above, where I am going for order of magnitude and perhaps waving my arms a little.

Should try and develop the cheat-rationalization a little further, please. Particularly radiation upon acceleration, and not thereof.

By the way I obviously lied up above when I said everything here proceeds from 2 axioms and one definition. Actually to do this properly I'd have to add the Coulomb inverse-square law for the force between two charged particles, so that's another axiom. Ok so three axioms and a definition.

Speaking of axioms, at this point I need to call in the next idea or second axiom to make further progress, and that is to assume that this hydrogen atom has available a minimum energy configuration, or more concisely:
The hydrogen atom assumes a configuration of lowest energy.

That is, the electron in relation to the proton is so configured so as to give an optimally low energy state for the two of them in concert.  This is a profound idea, to me anyway.

The expression for the energy is the definition needed to complete the derivation (plus some details from the third axiom about the Coulomb law.) The energy of the two-particle system can be minimized using simple calculus and from this we get the electron's binding energy also called a Rydberg energy. We find furthermore a definite value for the electron's average distance from the proton, the optimal 'radius' a0 for this minimum energy. This is identified as special number by adding that zero subscript and it is also known as the Bohr radius for the hydrogen atom.

Here first is a complete definition for the energy of the system, neglecting rest masses.

E = Kinetic Energy + Potential Energy = KE + PE.

KE = p2/2m, the electron's momentum squared divided by twice its mass, classically the kinetic energy (1/2)mv2.

PE = (-1/4πε0) x e2/a.

The potential energy is the electrostatic potential. It is related to electrostatic force which is (like gravity) an inverse-square law. That's it, that's all that is needed to derive the result; the rest is simply letting the mill of the gods grind away.

I should give the units and meaning of m, eps0, and e here.

To continue: since I have an expression for the momentum p in terms of the electron's distance from the proton a (from the first axiom) I can substitute this to express E entirely as a function of a (with no p). Then I can choose a value of a, my solution a0, which minimizes the energy E.

E = hbar2 / (2m) x (1/a2) - (e2/4π ε0) x (1/a)

At a = +inf the energy E(a) is 0. This is the reference energy of the electron very far away from the proton. As a gets finite and smaller the function E first becomes slightly less than zero, then reaches a minimum, then arcs upwards, passes through zero again, and for a any smaller the energy asymptotically E climbs towards +inf. This is the expression of the uncertainty principle forbidding the electron to inhabit the region close to a = 0 and mathematically it is because +1/a2 predominates over -1/a for small values of a. This may seem obvious or trivial but I am intensely interested in flipping the idea around, or running the derivation backwards: Suppose all that I knew was that the answer to the minimum energy and optimal radius proceed from finding the minimum of the sum of those two functions. Why should it? Why is this physical? Why is it real? Those two terms are "inverse area" and "inverse distance" so I'll just note this and continue on.

The two competing terms give nice minimum energy at a0 = hbar2 x 4π ε0 / (m e2). The fine structure constant α is ... so this can also be written as ... The associated minimum energy is ...

Some History in the Interest of Broadening My Personal Perspective
Interestingly this solution is with no recourse or reference to the more complex 'complete' machinery of quantum mechanics. It uses the profound idea of quantum mechanics, Axiom 1, together with a sort of pseudo-bunko-classical-mechanics picture to fortuitously arrive at this approximate solution and that's the end of the line. There is a plus and a minus to going through this excercise, therefore. The plus is arriving at a good answer so quickly and easily; it is like an emphasis of the leverage contained in the uncertainty principle. Actually to me it's a lot like the shark theme music in the movie Jaws.

The minus is that the above derivation is not a general method that can be applied to get precise results. In order to do that I must go beyond this simple idea and adopt the 'complete machinery', the more complicated methodology of the physics in quantum mechanics. There are--to the best of my understanding--two more or less equivalent types of this machinery, called respectively matrix mechanics and wave mechanics. Matrix mechanics was published first, by the originator of Axiom 1, Werner Heisenberg, but it was not easily adopted by the physics community, at least not immediately. Second came Schroedinger's wave mechanics and this lent itself immediately to calculations, hence it was initially easier to accept.

The hydrogen atom is the proof of the pudding for wave mechanics. It is generally solved multiple times with increasingly sophisticated versions of the wave mechanics machinery, and the first cut solution seeks to identify the spatial distribution of the electron's existence relative to the proton in terms of a wave function or probability amplitude. It is interesting to me that legally speaking the probability amplitude for every hydrogen atom in the Orion nebula is defined among other places at the tip of my nose. Small but non-zero, this points the way towards bizarre behavior with more certainty than that produces merely by the uncertainty principle. The Jaws theme music builds in intensity the further I go.

The lowest-energy solution to the time-independent Schroedinger equation supposes that the time-dependent part is a unit-magnitude time-varying phase (I like to think of it as a wave function skipping rope) multiplied by the spatial part; that is, separable components. The spatial part falls out of the residual differential equation as a bunch of spherical harmonics with increasingly intricate shapes which become atomic orbitals and are responsible for the chemical properties of the various elements. All of which provides very precise (but still incomplete) values for atom size and binding energy. This Schroedinger equation machinery is in short reasonably complicated mathematics that provides reasonably accurate answers. In fact to put it in terms of history, this is the 1926-era machinery that marks the halfway point between the initiation of quantum mechanics and the end of it's birth period which I set at 1950 with the work of Freeman Dyson in the immediate wake of Quantum Electrodynamics.