Erwin Schrodinger and the Quantum Revolution by John Gribbin

Author:John Gribbin
Language: eng
Format: epub
Publisher: Wiley
Published: 2013-03-03T16:00:00+00:00

Back to the future

This new speculation stemmed from Schrödinger’s early fascination with thermodynamics and the way the world is governed by statistical laws. He noticed that his wave equation had a similar structure to the equation used to describe diffusion processes, such as the way molecules of perfume from an open bottle spread through the air. Thanks to his work in statistical mechanics, he was also aware of a curious property of the diffusion equation. Such an equation can be run in reverse, to describe a world in which molecules of perfume come together out of the air and congregate in an open bottle, even though we do not see events like that going on in the everyday world. This kind of reversibility lies at the heart of the statistical understanding of thermodynamics developed by Boltzmann and others. Now, you might think that if you had all the information about the distribution of scent molecules through the air at a certain time, and the equivalent information for a later time, you could calculate the distribution of the molecules for any time in between either by working forward from the earlier time or backward from the later time. But, as Schrödinger realized, you would be wrong. The way to find the distribution for intermediate times is to combine the solution for the equation going forward in time with the solution to the equation going backwards in time—in effect, multiplying the two equations, or their solutions, together.

The connection with quantum mechanics—what Schrödinger described in his paper as “the most interesting thing about our result”—comes from the way the square of Schrödinger’s wave function is used to calculate probabilities in the Copenhagen Interpretation. The wave function is usually denoted by the Greek letter psi (ψ), but the square involved in the calculations is not simply ψ × ψ. Because the equations, like all good wave equations, involve the square root of minus one (i), and are therefore “complex” in the mathematical meaning of the term, the wave function has to be multiplied by something known as its complex conjugate, which can be denoted by ψ*. So the probability of finding an electron at a particular place depends on ψ × ψ*. But the complex conjugate is, in effect, the same as the wave function running backwards in time. Like the solution to the diffusion problem, the probabilities in the Copenhagen Interpretation depend on combining two equations, one describing processes proceeding forward in time and one describing processes proceeding backwards in time.

At this point Schrödinger ran into a brick wall, and rather lamely concluded: “I cannot foresee whether the analogy will prove useful for the explanation of quantum mechanical concepts.” Nor could anyone else in the 1930s, and Schrödinger’s paper stirred so little interest that, half a century later, when the American physicist John Cramer (b. 1934) did find a way to interpret the complex conjugate to provide a new understanding of quantum mechanics he did so in complete ignorance of Schrödinger’s 1931 paper.

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