Search results
Results from the WOW.Com Content Network
The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. [25] Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space.
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 ...
In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space of a complex Hilbert space is the set of equivalence classes [] of non-zero vectors , for the equivalence relation on given by
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 space is a fixed complex Hilbert space of countably infinite dimension. The observables of a quantum system are defined to be the (possibly unbounded ) self-adjoint operators A {\displaystyle A} on H {\displaystyle \mathbb {H} } .
In quantum mechanics, quantum states are described by density matrices, which are certain trace class operators. [ 1 ] Trace-class operators are essentially the same as nuclear operators , though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator ...
In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert 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.