Search results
Results from the WOW.Com Content Network
The operator is said to be positive-definite, and written >, if , >, for all {}. [ 1 ] Many authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator.
Because k is a non-negative integer, for every even n we have ℓ = 0, 2, …, n − 2, n and for every odd n we have ℓ = 1, 3, …, n − 2, n. The magnetic quantum number m is an integer satisfying −ℓ ≤ m ≤ ℓ, so for every n and ℓ there are 2ℓ + 1 different quantum states, labeled by m.
In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum ...
If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
where a is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable A has a measured value a. If ψ is an eigenfunction of a given operator ^, then a definite quantity (the eigenvalue a) will be observed if a measurement of the observable A is made on the state ψ.
The operator C can be defined by C(Bh) = Ah, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement of Ran(B). The operator C is well-defined since A*A ≤ B*B implies Ker(B) ⊂ Ker(A). The lemma then follows. In particular, if A*A = B*B, then C is a partial isometry, which is unique if Ker(B*) ⊂ Ker(C).
The off-diagonal elements are complex conjugates of each other (also called coherences); they are restricted in magnitude by the requirement that () be a positive semi-definite operator, see below. A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system.
The operator X is a raising operator for N if c is real and positive, and a lowering operator for N if c is real and negative. If N is a Hermitian operator, then c must be real, and the Hermitian adjoint of X obeys the commutation relation [, †] = †.