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 derivitive 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.

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