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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 } .
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R.
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 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 ...
It states that any orthomodular form that has an infinite orthonormal set is a Hilbert space over the real numbers, complex numbers or quaternions. [ 1 ] [ 2 ] Originally proved by Maria Pia Solèr , the result is significant for quantum logic [ 3 ] [ 4 ] and the foundations of quantum mechanics .
The total wave functions are in the tensor product space of the Hilbert space of the spatial part (which is spanned by the position eigenstates) and the Hilbert space for the spin. Wave function The word "wave function" could mean one of following: A vector in Hilbert space which can represent a state; synonymous to "ket" or "state vector".