An excellent book which does more or less what you ask for is Asher Peres' "Quantum theory:concepts and methods". It starts from the Stern-Gerlach experiments and logical reasoning to develop the basic principles of quantum mechanics. From there, it develops the necessary algebra.

Another interesting book for an approach of the conceptual side of quantum mechanics is "Quantum Paradoxes" by Aharonov and Rohrlich. But to fully appreciate this one, I think you will need to go through a standard curriculum first.

Then, there is "Quantum computation and Quantum Information" by Nielsen and Chuang, which is meant as an introduction to the ideas of QM as applied to information theory for people with an informatics background mostly. So it also starts from an algebraic and conceptual approach.

This post imported from StackExchange Physics at 2014-03-30 15:58 (UCT), posted by SE-user Raskolnikov