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Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue of an operator ^ on some quantum state | is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues.
with the real eigenvalues n 2 π 2; the well-known orthogonality of the sine functions follows as a consequence of A being symmetric. The operator A can be seen to have a compact inverse, meaning that the corresponding differential equation Af = g is solved by some integral (and therefore compact) operator G.
At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. [16] He was the first to use the German word eigen , which means "own", [ 6 ] to denote eigenvalues and eigenvectors in 1904, [ c ] though he may have been following a related usage by Hermann von Helmholtz .
Consider the Hermitian operator D with eigenvalues λ 1, λ 2, … and corresponding eigenfunctions f 1 (t), f 2 (t), …. This Hermitian operator has the following properties: Its eigenvalues are real, λ i = λ i * [4] [6] Its eigenfunctions obey an orthogonality condition, , = if i ≠ j [6] [7] [8]
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.
This condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when x = y is an eigenvector. (Recall that an eigenvector of a linear map A is a non-zero vector v such that A v = λv for some scalar λ. The value λ is the corresponding eigenvalue.
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 [, †] = †.
The momentum operator can be described as a symmetric (i.e. Hermitian), unbounded operator acting on a dense subspace of the quantum state space. If the operator acts on a (normalizable) quantum state then the operator is self-adjoint. In physics the term Hermitian often refers to both symmetric and self-adjoint operators. [7] [8]