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When I was in college, the books I was taught from were Sakurai's "Modern quantum mechanics" and Townsend's "A modern approach to quantum mechanics". Though, they are clearly physics books, they both take a non-historical more mathematical approach starting with the two-dimensional vector space of a spin 1/2 particle. They don't assume any classical mechanics, neither Newtonian, Lagrangian or Hamiltonian. The first chapter of Sakurai for example gives a nice intro to the basic ideas of quantum mechanics. If what you're interested in is quantum field theory from a mathematical point of view, I'd suggest Folland's recent book "Quantum field theory". Also, if you decide to go for classical mechanics from an advanced mathematical perspective, you could try Abraham and Marsden's "Foundations of Mechanics".