On the origins of the Schrödinger equation

Physicists have obtained the Schrödinger equation (shown here) from a mathematical identity. Their approach shows that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.

Physicists have obtained the Schrödinger equation (shown here) from a mathematical identity. Their approach shows that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.

One of the cornerstones of quantum physics is the Schrödinger equation, which describes what a system of quantum objects such as atoms and subatomic particles will do in the future based on its current state.

The classical analogies are Newton’s second law and Hamiltonian mechanics, which predict what a classical system will do in the future given its current configuration. Although the Schrödinger equation was published in 1926, the authors of a new study explain that the equation’s origins are still not fully appreciated by many physicists.

In a new paper published in PNAS, Wolfgang P. Schleich, et al., from institutions in Germany and the US, explain that physicists usually reach the Schrödinger equation using a mathematical recipe. In the new study, the scientists have shown that it’s possible to obtain the Schrödinger equation from a simple mathematical identity, and found that the mathematics involved may help answer some of the fundamental questions regarding this important equation.

Although much of the paper involves complex mathematical equations, the physicists describe the question of the Schrödinger equation’s origins in a poetic way: “The birth of the time-dependent Schrödinger equation was perhaps not unlike the birth of a river.

Often, it is difficult to locate uniquely its spring despite the fact that signs may officially mark its beginning.

Usually, many bubbling brooks and streams merge suddenly to form a mighty river. In the case of quantum mechanics, there are so many convincing experimental results that many of the major textbooks do not really motivate the subject [of the Schrödinger equation’s origins].

Instead, they often simply postulate the classical-to-quantum rules….The reason given is that ‘it works.'” Coauthor Marlan O.

Scully, a physics professor at Texas A&M University, explains how physicists may use the Schrödinger equation throughout their careers, but many still lack a deeper understanding of the equation.

“Many physicists, maybe even most physicists, do not even think about the origins of the Schrödinger equation in the same sense that Schrödinger did,” Scully told Phys.org. “We are often taught (see, for example, the classic book by Leonard Schiff, ‘Quantum Mechanics’) that energy is to be replaced by a time derivative and that momentum is to be replaced by a spatial derivative.

And if you put this into a Hamiltonian for the classical dynamics of particles, you get the Schrödinger equation.

It’s too bad that we don’t spend more time motivating and teaching a little bit of history to our students; but we don’t and, as a consequence, many students don’t know about the origins.”….

Read more at: http://phys.org/news/2013-04-schrodinger-equation.html#jCp

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