Hermitianoperatori

Hermitian operatori is the term typically used for a Hermitian (or self-adjoint) linear operator in a complex inner product space, such as a finite-dimensional vector space or a Hilbert space. An operator A is Hermitian if it equals its adjoint, A = A†, where the adjoint is defined via the inner product. In matrix form this means A is equal to its conjugate transpose.

Key properties include that Hermitian operators have real eigenvalues. Eigenvectors corresponding to distinct eigenvalues are orthogonal, and in finite dimensions a Hermitian operator is diagonalizable by a unitary transformation. The spectral theorem provides a decomposition A = ∑ λi |ψi><ψi|, with real eigenvalues λi and an orthonormal eigenbasis {|ψi>}. This structure underlies many practical computations in linear algebra and physics.

In quantum mechanics, Hermitian operators represent observables. The possible measurement outcomes are the eigenvalues, and expectation values are real. The probability of obtaining an eigenvalue λi when the system is in state |ψ> is |<ψi|ψ>|^2. The Hamiltonian H, a central Hermitian operator, generates unitary time evolution via U(t) = exp(-iHt/ħ).

Domain considerations are important in infinite-dimensional spaces. While every Hermitian operator on a finite-dimensional space is self-adjoint, densely defined operators in Hilbert spaces require careful specification of their domain to ensure self-adjointness and the existence of a unitary evolution. Common examples of Hermitian operators include the Pauli matrices and the position and momentum operators on appropriate domains.

External references: Hermitian operators, self-adjointness, spectral theorem, quantum observables.