Valid Hermitian Operators

  • Thread starter KBriggs
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  • #1
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Homework Statement


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




The Attempt at a Solution



Is this valid?

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

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?
 

Answers and Replies

  • #2
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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.

[tex]\langle \phi | A | \psi \rangle^* = \langle \psi | A^* | \phi \rangle[/tex]
 

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