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

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

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

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

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

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

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.

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

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:

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