Schrodinger's equation applied to a single electron Hydrogen atom takes
the following form:
Written in terms of angular and radial portions, Schrodinger's
equation is given by
The eigenfunctions of the angular momentum operator are a set of complex functions called the spherical harmonics
, and they have the eigenvalues
Using separation of variables, it is assumed that since the potential
depends only on r, the solution may be written as
the solution to the radial portion of the equation.
Leaving the form of the spherical harmonics for a different discussion,
it is this radial solution that is now considered.
The primary force between the electron and the proton in Hydrogen is
the Coulomb force, which describes the electric force between two
point charges. The electron is not observed to be in the nucleus, where
strong and weak nuclear forces would dominate, nor are the particles
massive enough for gravitational forces to dominate. This gives
us a potential of .
Using a suitable separation of variables,
While Schrodinger's original equation was in three dimensions, it
has now been reduced to a single dimension. What's more, while the
equation as a whole required a solution using complex numbers, the
radial term can be written in terms of real numbers.
But what does this represent? The wave function given by Schrodinger's
equation has been linked to the probability distribution of the electron.
The actual probability density is given by
Now, consider a sphere of radius r centered on the nucleus of the
hydrogen atom. If you increase the radius of the sphere, you also
increase the surface area of the sphere. So, if you were to ignore
for a second the dependence of on and , and just
add up the contribution due to , you would get that the
probability of finding the electron between and
We can then interpret as the square root of the radial
probability density of the electron in the Hydrogen atom.
Where is small, we are less likely to find the electron.
Where is large, we are more likely to find the electron.
The equation for can be solved with the usual techniques, but
like many problems in physics has many solutions that will not satisfy
the boundary conditions. We want the specific eigenfunctions and
eigenvalues that will satisfy the boundary conditions, which are that
as you get very far away from the atom it should be less likely to find
the electron and that as you get very close to the center of the atom
it should be hard to find the electron.
You can try building your own model of this, but be careful how you
set up your solution to the radial equation. There are solutions
to the equation that do not satisfy the boundary conditions, and in
fact get very large as you get far away from the atom, so if you
integrate the problem from the inside out, numerically it is
likely that you will "jump" to these solutions which do not
satisfy the boundary conditions.
It is customary to integrate from the outside in, which lets you
rule out this instability in the numerical solution and force the
outer boundary condition. Forcing the outer boundary condition does
not do anything about the inner boundary condition, however, and
it is up to you to find out what values of and will
allow for a solution that satisfies your inner boundary condition.