Hermitian Operator

Topics

• Hermitian Matrix
• Antihermatian Matrix

$H=H^{{}^{†}}$

• $H$ is considered a hermitian operator if it satisfies the above
• $H^{{}^{†}}$ is the Hermitian conjugate of $H$
• $(H^{\top }{)}^{\ast }$ is the complex conjugate of the transpose of the matrix

$⟨\Psi |H\Psi ⟩=⟨H\Psi |\Psi ⟩$

• $H$ is considered a hermitian operator if it satisfies the above
• See Braket Notation

Properties

• Eigenvalues of Hermitian operators are real
• Eigenvectors of Hermitian operators are orthogonal if they have different eigenvalues
• The set of eigenvectors of a Hermitian operator can be used as a basis