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Two Hilbert spaces V and W may form a third space V ⊗ W by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.
It yields a complex number. With the inner product, the function space is an inner product space. The explicit appearance of the inner product (usually an integral or a sum of integrals) depends on the choice of representation, but the complex number (Ψ, Φ) does not. Much of the physical interpretation of quantum mechanics stems from the Born ...
Self-adjoint operators are used in functional analysis and quantum mechanics. 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
where , is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian [1] after Charles Hermite. It is often denoted by A † in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics.
A real inner product space is defined in the ... The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a ...
The bounded observables of the quantum mechanical system are defined to be the self-adjoint elements of the C*-algebra. The states of the quantum mechanical system are defined to be the states of the C*-algebra (in other words the normalized positive linear functionals ).
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 applications to quantum statistical mechanics, the operator has the form =, where is the Hamiltonian of the quantum system and is the inverse temperature. With these notations, the Bogoliubov inner product takes the form