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Nice short article:
http://hubcap.clemson.edu/~daw/D_Realism/AmJPhys.v70p288y02.pdf
http://hubcap.clemson.edu/~daw/D_Realism/AmJPhys.v70p288y02.pdf
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The nine formulations of Quantum Mechanics are: wave mechanics, matrix mechanics, Dirac's transformation theory, Heisenberg's matrix mechanics, Schrödinger's wave mechanics, Feynman's path integral, Tomonaga's quantum field theory, Schwinger's quantum action principle, and Bell's formulation.
Wave mechanics describes particles as waves, while matrix mechanics describes particles as discrete packets of energy. Wave mechanics is based on the Schrödinger equation, while matrix mechanics is based on the Heisenberg uncertainty principle.
Feynman's path integral formulation is based on the idea that the path of a particle is not determined by one specific path, but rather by all possible paths. This formulation takes into account the probabilistic nature of quantum mechanics and allows for the calculation of complex systems.
Bell's formulation, also known as Bell's theorem, is a mathematical proof that shows that local hidden variables cannot reproduce all of the predictions of quantum mechanics. This has important implications for our understanding of reality and the role of randomness in the universe.
While the nine formulations of Quantum Mechanics may seem distinct, they are all mathematically equivalent and describe the same physical phenomena. They just provide different ways of understanding and calculating these phenomena. Each formulation has its own advantages and may be more suitable for certain situations.