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In order to really be able to follow quantum mechanics you're going to need to understand some calculus, since a good portion of it is some fairly dense mathematics. That said, you might try some popularizations if you're willing to just trust that the author's math is good. In particular, you might want to try to find the book "Mr. Tompkins Explores the Atom" by George Gamow (it's included in the book "Mr. Tompkins in Paperback" if you can find that). If you're in the US, you can almost certainly get it through the interlibrary loan program; ask at your local public library. "Thirty Years that Shook Physics," also by Gamow, is another good book; it's sort of a history of the development of quantum mechanics. For those who aren't afraid of a little calculus and want something a bit more rigorous but not totally impenetrable, "The Feynman Lectures on Physics, Vol. III" by Richard Feynman is excellent.
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