Is \(x^k p_x^m\) Hermitian?

[KBriggs]

Homework Statement

Show that the operator x^kp_x^m is not hermitian, whereas \frac{x^kp_x^m+p_x^mx^k}{2} is, where k and m are positive integers.

The Attempt at a Solution

Is this valid?

<x^kp_x^m>^*=\left(\int_{-\infty}^\infty\Psi^*x^k(-i\hbar)^m\frac{\partial^m\Psi}{\partial x^m} dx\right)^*
=\int_{-\infty}^\infty\Psi^*(i\hbar)^m\frac{\partial^m(x^k\Psi^*)}{\partial x^m} dx \neq <x^kp_x^m>

That is, can you conjugate an integral by conjugating its integrand? Can you conjugate a derivative by conjugating the function you are differentiating?

And assuming that you can, did I carry out the conjugation correctly?

 

[Mindscrape]

You can conjugate the integrand, however, you have to conjugate all of the terms. Basically what happens is that the bra and the ket switch, and the operator is conjugated.

\langle \phi | A | \psi \rangle^* = \langle \psi | A^* | \phi \rangle