Hermitian Operators

A physical variable must have real expectation values (and eigenvalues). This implies that the operators representing physical variables have some special properties.

By computing the complex conjugate of the expectation value of a physical variable, we can easily show that physical operators are their own Hermitian conjugate.

\begin{displaymath}\bgroup\color{black} \langle\psi\vert H\vert\psi\rangle^*=\le...
...fty\psi(x)(H\psi(x))^* dx=\langle H\psi\vert\psi\rangle \egroup\end{displaymath}


\begin{displaymath}\bgroup\color{black} \langle H\psi\vert\psi\rangle=\langle\psi\vert H\psi\rangle=\langle H^\dagger\psi\vert\psi\rangle\egroup\end{displaymath}


\begin{displaymath}\bgroup\color{black}H^\dagger=H \egroup\end{displaymath}

Operators that are their own Hermitian Conjugate are called Hermitian Operators.

\bgroup\color{black}$\displaystyle H^\dagger=H$\egroup



Jim Branson 2013-04-22