Hermitian Operator

What does it mean for an operator to be Hermitian?

If the operator A is Hermitian then we can do the following:

<f|Ag> = <Af/g>

The notation <f|g> is showing the inner product of two functions, f and g, using Bra-Ket notation.

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Another explanation provided the following:

The operator \hat A is called hermitian if

\int (\hat A \psi)^*\psi dx = \int \psi^* \hat A \psi dx

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We might also look at this from the standpoint of using matrices:

In the linear algebra of real matrices, Hermitian operators are symmetric matrices.

In the example below, \hat O is a Hermitian Operator and this makes the identity below true:

\int dx \phi_n^*(x) \hat O \psi_m(x) = \begin{bmatrix} \int dx \psi_m^*(x) \hat O \phi_n(x) \end{bmatrix}^*

Yes, it is bewildering, and we’d like to explain why.

First, for an integral you are accustomed to seeing \int f(x) dx but they changed it to \int dx f(x) because everywhere else the operator is shown to the left of the function on which it operates.