1. Schrödinger’s Equation in a Central Potential Field
The quantum mechanics of the hydrogen atom
Steve Peurifoy
Math 467
2006/04/25

2. Schrödinger’s Equation
The entire state of a system is encapsulated in its wave function
which is a complex-valued function of position and time.
The probability of an experiment detecting a particular value for an
observable can be determined from the wave function. For example,
the probability that our electron will be found in a given region of
space R is
The wave function evolves according to the equation
where is the Hamiltonian operator for the system.
In the case where all forces derive from a conservative scalar
potential (independent of time), Schrödinger’s equation becomes

3. Separating the space and time variables
Assume that the wave function can be written in the form
Then we can separate the time dependent and position dependent parts as
The time dependent solution is quickly found to be
The spatial solution satisfies
Based on physical as well as dimensional arguments, the separation
constant is identified as the total energy of the system in a given
eigenstate. From this, we see that the frequency of the time dependent
solution satisfies the familiar relationship

4. Spherical Coordinates and Further Separation
A system with a central (spherically symmetric) potential is an
excellent approximation to the hydrogen atom since the proton at
the nucleus is almost 2000 times as massive as the electron.
For such a system, it is natural to recast the spatial equation using
the spherical form of the Laplacian:
Assume that can be written as the product of a radial part and
an angular part:
If we further assume that the potential is a function only of radial distance
(in keeping with the spherical symmetry noted previously), substitution and
division by the above product solution yields

5. Spherical Coordinates and Further Separation
Multiplying through by completes the radial/angular separation:
The operator may be defined as
So we can write the angular eigenvalue problem as
Based on the Hamiltonian dynamics underlying the system (and
confirmed by dimensional analysis), the eigenvalue can be
identified as the square of the total angular momentum.
The radial part of the solution then satisfies

6. Separation of the Angular Components
If we let
and multiply the angular eigenvalue equation through by , we
can accomplish the final separation:
We can readily solve for azimuthal component:
Because must be periodic, we conclude that must
be an integer, so

7. Solution of the Remaining Angular Equation
The remaining angular component is the solution to
This equation is most readily solved by a return to the
Cartesian coordinate (normalized to unit radius):
After simplification, the resulting equation is
For the special case a power series solution can be found for this
equation. The resulting solution diverges at unless the power series
terminates after a finite number of terms. This will be the case when
The resulting functions are called Legendre polynomials of the first
kind and are denoted Legendre polynomials of the second
kind are also solutions to the above equation but they diverge at
and are thus excluded here.

8. Solution of the Remaining Angular Equation
For the case, substitution shows that the function
is a solution. This is referred to as an associated Legendre function.
Here again, is required for a physically meaningful solution.
In addition, must be greater than or equal to in order for the
function to be nonzero.
The angular part of our solution then consists of terms of the form

9. Solution of the Radial Equation
Substituting for , the radial equation becomes
At this point it is necessary to specify the form of the potential:
Here, is the nuclear charge (the charge on one proton in the case
of hydrogen) and the zero point for the potential is chosen by convention
as the completely ionized state (electron infinitely far away).
A solution to the radial equation is of the form
The constraint that integrate to 1 over all space requires
that the power series terminate after a finite number of terms. This gives
rise to the principal quantum number , which must be an integer
greater than or equal to
Thus in this (non-relativistic) problem, there are three quantized
eigenvalues which define the state of the electron:

10. References R.L. White, Basic Quantum Mechanics, McGraw-Hill, Inc.
New York, 1966
Richard Haberman, Applied Partial Differential Equations,
4th ed., Pearson/Prentice Hall, Upper Saddle River, New Jersey,
2004