Wikipedia:Reference desk/Archives/Science/2017 December 2

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December 2

Who first analyzed the 'particle in a box' model?

This is a question about the history of quantum mechanics. Certain famous physical models are attributed to or associated with the physicists who discovered (or otherwise pioneered) them: blackbody spectrum (Planck), photoelectric effect (Einstein), hydrogen atom (Bohr).

The particle in a box model is an important model in quantum mechanics... My question is: who was the first to study the (quantum mechanical) particle in a box? Or, who was the first to promulgate/publish on/popularize it?— Preceding unsigned comment added by 72.221.67.126 (talk • contribs)

It's hard to find a definitive source, but many references I am finding in Google give a strong relationship between Paul Dirac and the problem, though it is hard to interpret if Dirac himself dealt with the problem, or if it is merely others using Dirac's mathematics in doing such problems.--Jayron32 15:54, 2 December 2017 (UTC)

This is a great question about the history of quantum mechanics - but it will be hard to find a really rigorously-defined answer unless we dive into a lot of really early 20th-century publications searching for the earliest use of the term!

I do not know who first used the name "particle in a box." But the problem itself is an excellent bridge between classical- and quantum- physics, because the fixed boundary-value wave equation had been studied for a long time before it was applied to atomic physics.

In the old days of physics education - that is, before modern quantum theory became part of the standard curriculum - physicists and physics students would spend many years studying classical mathematical methods. In specific, when studying the classical wave equation, one would practice the Dirichlet condition as a solution to a partial differential equation in one or more dimensions.

Mathematically, the Dirichlet problem is identical to the particle-in-a-box in quantum mechanics: it is a boundary-value problem with zero-values at the boundary, and can be solved by an infinite series of sinusoidal functions. With even more mathematical formalism, we can generalize the boundary condition to zero- or non-zero-, on the zeroth- or other-order derivative; and then we have the Neumann boundary condition and its standard solutions; and we have many other problem-variants, whose names are less familiar or universally-agreed-upon.

So, it is almost certain that the very first quantum mechanics theorists immediately recognized this as a useful simplified model identical to the models of classical physics, solved using standard mathematical tools. We might say that mathematicians like Dirichlet - who lived and died before most of quantum physics was empirically established - were the first to model systems in this fashion. Equally, we might blame the particle-in-a-box on any of the other great scientists who studied potential theory: Hilbert, Laplace, Cauchy and others... We can even attribute it to Fourier, whose infamous Fourier series allows us to solve the particle-in-the-box. It is his mathematical formalism that allows us to transform from a function in the domain of the continuum into a series of discrete functions over the domain of integers - in other words, Quantization - and so he solved the "particle in a box" many centuries before other scientists realized that this was an adequate model of the probability-distributions for the variation of the wave-equation that is usefully applied in atomic-scale physics.

For your reading pleasure, here is Quantization as a Eigenvalue problem, (1926, Schrödinger).

Nimur (talk) 01:53, 4 December 2017 (UTC)