Search results
Results from the WOW.Com Content Network
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 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 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.
The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis ...
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.
A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary.