Hermitian Operator

Hermitian Operator

If an operator has

1. Real Eigenvalues
2. Orthogonal Eigenfunctions and distinct Eigenvalues for these Eigenfunctions,
3. Eigenfunctions that form a complete set: i.e can express any function as a superposition of its constituent eigenfunctions. Then the operator is known as a hermitian operator. N.B: Observational operators are hermitian.

Example:

$${\int }_{-\infty }^{\infty }{\phi }^{\ast }(A\Psi )dx={\int }_{-\infty }^{\infty }(A\phi {)}^{\ast }\Psi dx={\int }_{-\infty }^{\infty }{\phi }^{\ast }{A}^{†}\Psi dx$$