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The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. [25] Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space.
A quantum description normally 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.
In quantum mechanics a state space is a separable complex Hilbert space.The dimension of this Hilbert space depends on the system we choose to describe. [1] [2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors.
A state of the quantum system is a unit vector of , up to scalar multiples; or equivalently, a ray of the Hilbert space . The expectation value of an observable A for a system in a state ψ {\displaystyle \psi } is given by the inner product ψ , A ψ {\displaystyle \langle \psi ,A\psi \rangle } .
In quantum mechanics, the Hilbert space is the space of complex-valued functions belonging to (,), where the simple is the classical configuration space of free particle which has finite degrees of freedom, and is the Lebesgue measure on .
In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics. [ 19 ] [ 20 ]
In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space of a complex Hilbert space is the set of equivalence classes [] of non-zero vectors , for the equivalence relation on given by
In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space and are linearly ...
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