Absolutely Small - Michael D. Fayer [56]
The solution of the hydrogen atom problem using the Schrödinger Equation is particularly important because it can be solved exactly. The hydrogen atom is a “two-body” problem. There are only two interacting particles, the proton and the electron. The next simplest atom is the helium atom, which has a nucleus with a charge of +2 and two negatively charged electrons. This is a three-body problem that cannot be solved exactly. In classical mechanics, it is also not possible to solve a three-body problem. The problem of determining the orbits of the Earth orbiting the Sun with the Moon orbiting the Earth cannot be solved exactly with classical mechanics. However, in both quantum mechanics and classical mechanics, there are very sophisticated approximate methods that permit very accurate solutions to problems that cannot be solved exactly. The fact that a method is approximate does not mean it is inaccurate. Nonetheless, because the hydrogen atom can be solved exactly with quantum theory, it provides an important starting point for understanding more complicated atoms and molecules.
WHAT THE SCHRÖDINGER EQUATION TELLS US ABOUT HYDROGEN
What does the solution to the Schrödinger Equation for the hydrogen atom give? It gives the energy levels of the hydrogen atom, and it gives the wavefunctions associated with each state of the hydrogen atom. The wavefunctions are the three-dimensional probability amplitude waves that describe the regions of space where the electron is likely to be found. Schrödinger’s solution to the hydrogen atom problem gives energy levels consistent with the empirically obtained Rydberg formula. The energy levels are
where n is the principal quantum number. It is an integer that can take on values ≥1, that is, greater than or equal to 1. The difference in energy between any two energy levels is the Rydberg formula. However, in the Schrödinger solution, RH is not an empirical parameter. In solving the problem, Schrödinger found the Rydberg constant is determined by fundamental constants, with
h is Planck’s constant. e is the charge on the electron. εo is a constant called the permittivity of vacuum. εo = 8.54×10-12 C2/J m, with the units Coulombs squared per Joule - meter. μ is the reduced mass of the proton and the electron. It is
where mp and me are the masses of the proton and electron, respectively. The charge on the electron and the proton and their masses were given above.
While Rydberg took experimental data and developed an empirical formula that described the line spectra of the hydrogen atom, the results of Schrödinger’s solution to the hydrogen atom problem using quantum theory are fundamentally different. We have to spend a moment to marvel at the triumph of quantum theory that emerged in 1925. There are no adjustable parameters in Schrödinger’s derivation of the energy levels of the hydrogen atom. All of the necessary constants are fundamental properties of the particles and the electrostatic interaction that attracts the electron’s negative charge to the proton’s positive charge. Schrödinger did not look at the experimental data and then adjust a constant, RH, until it fit the data. He set up a theoretical formalism and applied it to the hydrogen atom. The application of his theory accurately reproduced the experimental observables, the hydrogen atom line spectra, using only fundamental