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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.
Equivalently, it is the number of involutions of an n-element set with precisely k fixed points, or in other words, the number of matchings in the complete graph on n vertices that leave k vertices uncovered (indeed, the Hermite polynomials are the matching polynomials of these graphs).
Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N −1; Hermitian operators (i.e., self-adjoint operators): N* = N; skew-Hermitian operators: N* = −N; positive operators: N = MM* for some M (so N is self-adjoint).
Here is the nabla operator, a vector operator consisting of first derivatives. The well-known commutation relations for the p and q operators follow directly from the differentiation rules. Classically the electrons and nuclei in a molecule have kinetic energy of the form p 2 /(2 m ) and interact via Coulomb interactions , which are inversely ...
If A is Hermitian and Ax, x ≥ 0 for every x, then A is called 'nonnegative', written A ≥ 0; if equality holds only when x = 0, then A is called 'positive'. The set of self adjoint operators admits a partial order, in which A ≥ B if A − B ≥ 0. If A has the form B*B for some B, then A is nonnegative; if B is invertible, then A is positive.
The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Because the Fock operator is a one-electron operator, it does not include the electron correlation energy. The Fock matrix is defined by the Fock operator.
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 [, †] = †.
We also define the total number operator for the field which is a sum of number operators of each mode: ^ = ^ The multi-mode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes