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A Hilbert space is a vector space equipped with an inner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.
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 } .
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.
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot.
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if {} = is a POVM acting on a Hilbert space of dimension , then there exists a PVM {} = acting on a Hilbert space ′ of dimension ′ and an isometry: ′ such that for all ,
Hilbert, with the assistance of John von Neumann, L. Nordheim, and E. P. Wigner, worked on the axiomatic basis of quantum mechanics (see Hilbert space). [5] At the same time, but independently, Dirac formulated quantum mechanics in a way that is close to an axiomatic system, as did Hermann Weyl with the assistance of Erwin Schrödinger.
The physical significance of the projective Hilbert space is that in quantum theory, the wave functions and represent the same physical state, for any .The Born rule demands that if the system is physical and measurable, its wave function has unit norm, | =, in which case it is called a normalized wave function.