[Physics] the expectation value of the position times momentum operator

Question

Should I write the expectation of the position times momentum operator as:

$$\langle xp\rangle = \langle \psi|x (-i\hbar \partial_x) |\psi \rangle$$

or

$$\langle xp\rangle = \langle \psi| (-i\hbar \partial_x x) |\psi\rangle$$

Answer 1

In fact $xp$ is not self-adjoint, it can have non-real expectation values. But its symmetrized form $D=(1/2)(xp+px)$ is better behaved (it has a self-adjoint extension). It is the generator of dilatations which scales momenta and coordinates. The complexification of ${\exp}[iDa]$ (i.e. $a$ becomes complex) is important for the study of the spectrum of a class of Schrödinger operators.