Required math: calculus
Required physics: Schrödinger equation in 3-d
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Sec 4.2.1.
We’ve seen that we can solve the three-dimensional Schrödinger equation by separation of variables, provided that the potential is a function of only. In that case, the angular parts of the equation can be solved in general in terms of spherical harmonics, so the wave function has the form , where the functions are the spherical harmonics, and is the, as yet unsolved, radial function, which satisfies the differential equation
By making the further substitution
we can convert the above equation into a differential equation for :
This equation has the same form as the original Schrödinger equation except that the potential has picked up an extra so-called centrifugal term. We must now solve this equation when is the potential found in the hydrogen atom.
The hydrogen atom consists of a proton and an electron. The proton is, in the first approximation, taken to be fixed, since its mass is more than a thousand times that of the electron. The force between the two particles can be taken as solely electric, since the gravitational force is many orders of magnitude smaller and will have essentially no effect. In this case, the potential is
where is the elementary charge and is the Coulomb constant. The equation to be solved is thus:
The solution of this equation follows a similar method as was used in solving the harmonic oscillator. We first investigate the asymptotic behaviour of the equation for large and small , factor out this behaviour and then use a series to try to find the solution of what’s left.
First, we can introduce a couple of symbol changes. If we define
(note that since for bound states, is real), then we can rewrite 5 as
Since occurs always multiplied by , we can try using a new variable
and this results in
We can simplify the notation a bit more by defining a constant
giving us the equation
Now we can investigate the asymptotic behaviour. First, for large , the two terms in the brackets that depend inversely on become negligible, so we get in this limit:
This has the general solution
and only the first term is acceptable, since the term becomes infinite for large . So for large , we must have
At the other end, when is very small, the term in becomes the largest, so the approximate equation to solve is
[This argument fails if , but all we’re after here is looking at asymptotic behaviour in an attempt to factor this behaviour out of the overall solution. As we’ll see when we finally get the solution, it is valid for as well.]
In this case, the general solution is
This can be verified by direct substitution:
In this case, the term becomes infinite as , so and
So now we know the behaviours at the two extremes, and we can factor both of these out, hoping to solve for what is left over. That is, we can write
where is what we must find. Note that we have absorbed the two constants and into .
Plugging this back into 11 and collecting terms we get
This version of the differential equation may not look any friendlier than the original, but we can now try to solve it by expressing as a series in , which we will do in the next post.