Hermitian Operator

#Physics

Topics

• Hermitian Matrix
• Antihermatian Matrix

$\displaystyle H=H^{^{\dagger}}$

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

$\displaystyle \braket{ \Psi | H\Psi }=\braket{ H\Psi | \Psi}$

• $\displaystyle H$ 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