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We know that Millikan ended up proving Einstein's photoelectric equation instead of disproving it. Since he aimed at disproving it, what did he have in mind? In other words, what did he expect to see?

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Millikan indeed "ended up" confirming Einstein's photoeffect formula, but that he was aiming to disprove it in his 1916 experiment seems to be a later gloss from his Nobel lecture:

This work resulted, contrary to my own expectation, in the first direct experimental proof… of the Einstein equation and the first direct photo-electric determination of Planck’s $h$.

In an earlier paper from January 1916 he wrote something else:

"Einstein’s photoelectric equation… cannot in my judgment be looked upon at present as resting upon any sort of a satisfactory theoretical foundation... [even though] it actually represents very accurately the behavior."

Millikan could hardly have been aiming to experimentally disprove a "very accurate" equation. It appears that his goal was simply to measure the Planck constant. Here is from Holton's Centennial Focus:

"Ironically, it had been Millikan’s experiment which convinced the experimentalist-inclined committee in Stockholm to admit Einstein to that select circle in 1922. One final irony: In 1950, at age 82, Millikan published his Autobiography, with Chapter 9 entitled simply “The Experimental Proof of the Existence of the Photon–Einstein’s Photoelectric Equation.” By then, Millikan had of course come to terms with the photon. Moreover, he had evidently changed his mind about what he had done around 1916, for now he wrote that as the experimental data became clear in his lab, they “proved simply and irrefutably, I thought, that the emitted electron that escapes with the energy hf gets that energy by the direct transfer of hf units of energy from the light to the electron, and hence scarcely permits of any other interpretation than that which Einstein had originally suggested, namely that of the semi-corpuscular or photon theory of light itself.”

In the end, Millikan re-imagined the complex personal history of his splendid experiment to fit the simple story told in so many of our physics textbooks."

This was not Millikan's first time "re-imagining" his experiments, see Is Millikan's famous oil drop experiment a fraud?

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