##

Hermitian Conjugate of an Operator

First let us define the **Hermitian Conjugate** of an operator
to be
.
The meaning of this conjugate is given in the following equation.

That is,
must operate on the conjugate of
and give the same result for
the integral as when
operates on
.
The definition of the **Hermitian Conjugate of an operator** can be simply written in Bra-Ket notation.

Starting from this definition, we can prove some simple things.
Taking the complex conjugate

Now taking the Hermitian conjugate of
.

If we take the Hermitian conjugate twice, we get back to the same operator.
Its easy to show that

and

just from the properties of the dot product.
We can also show that

* Example:
Find the Hermitian conjugate of the operator .*

* Example:
Find the Hermitian conjugate of the operator
.*

Jim Branson
2013-04-22