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The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups. [22] The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. [23]
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.
Von Neumman formalized quantum mechanics using the concept of Hilbert spaces and linear operators. [3] He acknowledged the previous work by Paul Dirac on the mathematical formalization of quantum mechanics, but was skeptical of Dirac's use of delta functions. He wrote the book in an attempt to be even more mathematically rigorous than Dirac. [4]
In mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John von Neumann in 1932.
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.
The mathematical structure of quantum mechanics is based in large part on linear algebra: Wave functions and other quantum states can be represented as vectors in a complex Hilbert space. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" |ψ .
Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group.
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 ]
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