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Schrödinger’s Equation

Bruce Rosenblum, Fred Kuttner

Published Jun 29, 2006• 0 citations
Publisher

Abstract

Abstract By the early 1920s physicists had accepted the fact that, depending on the experiment chosen, matter as well as light could be displayed either as compact lumps or as widely spread-out waves. Few pretended to understand this seeming contradiction. The significance of this paradox came a few years later with the Schrödinger equation. But Erwin Schrödinger wasn’t looking for significance. At the time he saw de Broglie’s matter waves only as a way to get rid of Bohr’s “damn quantum jumps.”

DOI
10.1093/oso/9780195175592.003.0008

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Abstract By the early 1920s physicists had accepted the fact that, depending on the experiment chosen, matter as well as light could be displayed either as compact lumps or as widely spread-out waves. Few pretended to understand this seeming contradiction. The significance of this paradox came a few years later with the Schrödinger equation. But Erwin Schrödinger wasn’t looking for significance. At the time he saw de Broglie’s matter waves only as a way to get rid of Bohr’s “damn quantum jumps.”
Where did we get that [equation] from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. Richard Feynman De Broglie's matter waves The early struggles of physicists towards a quantum theory were mostly concerned with attempts to understand the nature of light. The traditional picture of light as a wave motion had been challenged by Planck and Einstein. They had shown that certain experimental results that were impossible to understand in terms of a wave picture could be easily explained if light was thought of as a stream of particles, now called photons. William Bragg, who, with his son, won the 1915 Nobel Prize for studies of crystal structure using X-rays, summarized this dilemma for physics by exclaiming in despair that he was teaching the corpuscular theory of light on Mondays, Wednesdays and Fridays, and the undulatory theory on Tuesdays, Thursdays and Saturdays! Physicists were still wrestling with these apparently contradictory properties of light when, in 1924, Prince Louis de Broglie (pronounced ‘de Broy’) suggested that all matter, even objects that we usually think of as particles – such as electrons – should also display wavelike behaviour! This revolutionary idea was completely unexpected and, what is more, was included by de Broglie in his Ph. D. thesis. Like most people, physicists are generally reluctant to accept any wild new idea, especially if there is not a shred of evidence to support it.
Abstract We have seen that it took about two decades to fully realize that light or, generally, electromagnetic radiation has both wave and particle properties. The final proof came with the discovery of the Compton effect (Episode 31). Even Compton himself needed years before he saw that the particle picture of radiation was unavoidable in view of the experimental evidence. Still in the year of Compton’s publications, Louis de Broglie [1] on purely theoretical grounds proposed that the particle–wave duality not only holds for radiation but also for material particles, i.e., for particles with a rest mass such as the electron.
Bohr’s model of the hydrogen atom was successful in that it gave us a radically new way to look at atoms. However, it has serious shortcomings. It could not be used to explain the spectra of He or any multielectron atom. It could not predict the intensities of the H absorption and emission lines. With de Broglie’s hypothesis that matter was also wavelike,1 there arose a question at the 1925 Solvey conference: What is the wave equation? De Broglie could not answer this; however, over the next year Erwin Schro¨dinger, working in Vienna, published a series of papers in which he deduced the general form of the equation that bears his name and applied it successfully to the hydrogen atom.2,3 What emerged was a new set of postulates, much like Newton’s, that laid the foundations of quantum theory.
In 1924, a young PhD student submitted a thesis containing the germ of an idea that would revolutionize our understanding of the material world. The student was Louis De Broglie, and for a while, he had been meditating on the wave/particle nature of light. De Broglie’s ideas were greeted rather sceptically at first. His PhD supervisor even asked for a second copy so he could send it to Einstein for his opinion. Einstein gave the work his seal of approval. News of De Broglie’s thesis spread around the physics community from 1924 to 1925. In autumn 1925, Erwin Schrodinger was visiting colleagues in Switzerland and was asked to present a summary of De Broglie’s work. Having studied De Broglie’s thesis in some detail he became an enthusiastic supporter. During the Christmas and New Year of 1925/1926, Schrodinger holidayed in the Alps with his mistress. On his return, he had a functioning wave equation worked out.
Toward the end of the nineteenth century, it became clear that some experiments cannot be explained on the basis of known physics, now referred to as classical physics. In 1900, Max Planck was able to correctly describe the distribution of blackbody radiation energy as a function of wavelength λ or frequency ν = c/λ. He had to accept the existence of energy packets (quanta) with energy E = hν. A few years later, Albert Einstein found that this unexpected relation between energy and frequency is also necessary in order to understand photoemission from metallic surfaces. A decade later, Niels Bohr arrived at the correct equation for the electronic spectrum of the hydrogen atom by applying “quantum conditions” on the motion of the electron around the nucleus and the Planck’s relation E = hν on the energy difference between the energy levels (1916). A decade later, Heisenberg, Born, and Jordan introduced “quantum mechanics” and de Broglie and Schrödinger (1925-1926) introduced “wave mechanics.”