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I have used classical Lagrangian mechanics for quite a while, and what I like about it is that everything can be derived from a very small number of geometric principles. There are just three things you need to "take on faith":

  1. That configuration space should be endowed with a Riemannian metric given by mass;

  2. That forces arise from scalar potentials on configuration space;

  3. Hamilton's principle of extremal action.

The last is perhaps most objectionable: why should the universe want to evolve in a way that minimizes the action? I have no idea, and my understanding is that nobody else does either. But if I assume these hypotheses, everything else follows mechanically: equations of motion, Noether's theorem, etc.

What I would like to do is to learn quantum mechanics, starting from similar geometric principles. For example, I might take as a key principle that configurations should be probability distributions over configuration space, rather than single points, but this doesn't lead to quantum mechanics; I also need (for some reason?) to instead take configurations to be complex-valued functions over configuration space. But even then I don't see how to get anything like Schroedinger's equation from a (modification of) Hamilton's principle.

I did some searching and I've found that there is indeed some variational foundation for quantum mechanics (and Feynman seemed to have championed this formulation) but the articles I've seen so far assume I already understand quantum mechanics.

My question: Is there a good reference that builds quantum mechanics from scratch, based on variational and geometric principles? If it's easier to instead start with a more general theory (e.g. quantum field theory) let me know as well.

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It's not possible to derive quantum mechanics from classical mechanics without adding additional hypotheses; this is because many of the results of quantum physics are incompatible with classical physics.

But if you add just a few bits of quantum physics, such as the relations of Planck & de Broglie, and take the resulting "quantum wave" seriously, then one can "derive" the Shroedinger.

Chapter 10 of Notes on Analytical Mechanics, available freely on Researchgate.net is devoted to just such a "derivation", and may serve your need. The earlier chapters should already be familiar to you; these are notes from a seminar I gave.

http://www.researchgate.net/publication/258172792_Notes_on_Analytical_Mechanics

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