Mathematical Structure of Quantum Mechanics
A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.
Read more about this topic: Mathematical Formulation Of Quantum Mechanics
Famous quotes containing the words mathematical, structure, quantum and/or mechanics:
“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (1817–1862)
“What is the most rigorous law of our being? Growth. No smallest atom of our moral, mental, or physical structure can stand still a year. It grows—it must grow; nothing can prevent it.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“the moderate Aristotelian city
Of darning and the Eight-Fifteen, where Euclid’s geometry
And Newton’s mechanics would account for our experience,
And the kitchen table exists because I scrub it.”
—W.H. (Wystan Hugh)