2.6.3 Solution herm-c
Show that the operator is a Hermitian operator, but is not.
By definition, corresponds to multiplying by 2, so is simply the function . Now write the inner product and see whether it is the same as for any and :
since the complex conjugate does not affect a real number like 2. So is indeed Hermitian.
On the other hand,
so is not Hermitian. An operator like that flips over the sign of an inner product if it is moved to the other side is called
skew-Hermitian. An operator like is neither Hermitian nor skew-Hermitian.