The Schriodinger Equation
THE SCHRIODINGER EQUATION: De Broglie summarized his work in his doctoral thesis. It is said that his thesis examiners were unsure about giving it their blessing, so they sent it to Einstein for his opinion. [7] Einstein endorsed it, which got de Broglie his PhD, and also led Einstein to promote de Broglie's
idea. Erwin Schriodinger, who had already been working on the quantum theory of electrons in 1922 [8], was exposed to deBroglie's ideas through Einstein. [9] In 1926, Schrodinger published a paper that introduced his famous equation governing the behavior of particle wave functions. The introduction began, The theory which is reported in the following pages is based on the very interesting and fundamental researches of L. de Broglie on what he called "phase waves" and thought to be associated with the motion of material points, especially with the motion of an electron or proton. The point of view taken here... is rather that material points consist of, or are nothing but, wave-systems."
In taking this leap to a fully wave-based view of matter, Schriodinger was compelled by the similarity between Fermat's principle, pertaining to wave paths, and Hamilton's principle, pertaining to particle paths. With this relationship in mind, Schrodinger considered the inadequacy of classical physics to explain atomic emission spectra and asked, \...is one not greatly tempted to investigate whether the non-applicability of ordinary mechanics to micro-mechanical problems is perhaps of exactly the same kind as the nonapplicability of geometrical optics to the phenonema of di raction or interference and may,
perhaps, be overcome in an exactly similar way?" As stated above, the wave-phenomena must in this case be studied in detail. This can only be done by using an \equation of wave propagation." Which one is this to be? In the case of a single material point, moving in an external eld of force, the simplest way is
to try to use the ordinary wave equation..."
The wave equation, already well-known from electromagnetism, is
where is the wave function and u is the phase velocity of the wave. The phase velocity is exactly what de Broglie calculated, so we just need to use de Broglie's results to plug in for u (though Schriodinger actually used a di erent method to obtain the phase velocity that was based on the analogy between Hamilton's principle and Fermat's principle). The only catch is that we can't just plug in the result u = c2=v because v is an unknown variable. Since we only have one equation, we can only allow one unknown, which is ; otherwise the equation won't be solvable. The solution is to express the phase velocity in terms of the total energy of the particle and the external potential, since the total energy is a constant and the external potential is known by assumption. Initially, Schriodinger used the correct relativistic energy and found the relativistic wave equation. However, the predictions that came out of the relativistic wave equation contradicted the experimental data on the ne structure lines of the Hydrogen spectrum. The reason was that the equation neglected the spin of the electron, which he correctly suspected, but he did not know how to x the problem. So instead he swept the issue under the rug by taking the non-relativistic approximation, which chops o the ne structure term. This places the focus on the gross structure predictions, which did match well with experiment. The consequence is that Schriodinger's famous equation is only an approximation, not a true law of physics.
In the non-relativistic approximation, the total energy can be split into the kinetic (K) and potential (V ) components as follows.
Notice we have assumed that the energy in the Planck- Einstein equation is the total energy, even though we have not ruled out the possibility that it should be the kinetic energy since the kinetic energy and total energy of photons are equal. Schriodinger used a di erent method to obtain u, based on the analogy between Hamilton's principle and Fermat's principle that had inspired de Broglie. For more on this issue, see Appendix 1. Plugging this u into the wave equation,