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I'm learning J.J Sakurai's book Modern Quantum Mechanics and encounter the time dependent evolution section. The author shows a brief induction of Dyson Series, then claiming that the series converges and we can use it to do time dependent perturbation theory. I don't quite know why this simple expansion can be so powerful. Anyone can show me how this is done? e.g. why it converges..

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    $\begingroup$ According to Wikipedia, it doesn't converge, "This series diverges asymptotically, …" $\endgroup$ Commented Nov 2, 2016 at 22:32
  • $\begingroup$ Might be an idea to quote the book in your post, As Sean says, and I have looked for 15 mins, Wikipedia (and everyone else) says it's divergent. $\endgroup$
    – user108787
    Commented Nov 2, 2016 at 23:10
  • $\begingroup$ Yes, you should quote the section on Sakurai that says it converges - it may contain useful qualifiers that you're not seeing clearly, or it may well be wrong. In general the series diverges. As I recall, Fetter and Walecka had a reasonable section on the sort of renormalization procedures that can be used to make it work, so that's one place to look (though admittedly rather more technical than Sakurai if I'm not mistaken). $\endgroup$ Commented Nov 3, 2016 at 0:26
  • $\begingroup$ @EmilioPisanty Oh, thank you for pointing this out. In fact it was my guess that since we can use this expansion as the perturbation expansion, it surely converges. It seems this was wrong. Could you bother to explain why this is so useful even if it diverges? $\endgroup$
    – Jason Tao
    Commented Nov 3, 2016 at 20:23

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