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This site says that "it has recently been proven that the photoelectric effect can be interpreted classically (or at least semi-classically) in non-particle, wavelike terms". Is anyone familiar with this other explanation of the photoelectric effect?

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It may be a reference to the fact that you can reproduce the characteristics of the photoelectron production in a model which treats the incident light classically, but treats the matter in the target quantum mechanically. This is explained in Mandel and Wolf's book (chapter 9), which explains how a simple semiclassical calculation can be used to derive the minimum threshold frequency for photoemission.

It's important to note, however, that this semiclassical calculation in no way provides an argument against the quantum nature of real light.

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