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Is there a book or other resource that compiles how the theory & mathematical equipment of quantum mechanics was developed from various experiments? Something that shows clearly the relationships between real experimental data and the mathematical models of QM?

For instance, a chapter on how the Stern-Gerlach experiment is designed/works in high detail, and exactly how the data is measured and represented by theory.

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Yes there is. The canonical "tribal story" of the field is the classic Inward Bound: Of Matter and Forces in the Physical World (1986) by Abraham Pais. Oxford University Press; ISBN-10: 0198519710 ; ISBN-13: 978-0198519713 .

Pais, a leading phenomenologically inclined theorist and pioneer in his generation, puts it all together as only an experienced researcher could, but no historian of science. The major difference is that this author has a professional understanding of avoiding the inessential, sorely lacking among historians in this specific field.

Bram does just what you asked and illustrates the logic of the developments. But you must be a physicist aware of the answers. (Very often, the historical approach of entering a subject is the absolutely worst entry method.) If you posted this question in the History of science SE you'd get very different answers: probably cool and astounding stories, but no instinctive understanding of the unforgiving logical flow and chain of evidence, as you get here.

You did not ask this, but if Pais' book intrigued you to seek the theoretical developments (no serious experimental coverage here!) from the horses' mouths, the classic source-book is B. L. van der Waerden's Sources of Quantum Mechanics (Dover) ISBN-10: 048645892X ; ISBN-13: 978-0486458922 . Again vdW was a superb mathematician, and proficient in avoidance of the inessential, the critical requirement in these waters.

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To start, try Segre's book From X-Rays To Quarks. It's a good historical overview which will point you to more detailed (mathematical) treatments if you desire.

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