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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 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.
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 quantum field theory, it is expected that the Hilbert space is also the space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined. However, it is often hard to define a concrete Borel measure on the classical configuration space, since the integral theory on infinite ...
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 } .
The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures.
In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system.The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable".
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.