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AlexChandler
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Homework Statement
This is something I've been trying to prove for a bit today. My quantum mechanics book claims that the following two definitions about hermitian operators are completely equivalent
my operator here is Q (with a hat) and we have functions f,g
[tex] \langle f \mid \hat Q f \rangle = \langle Q f \mid \hat f \rangle [/tex] for any function f in hilbert space
and
[tex] \langle f \mid \hat Q g \rangle = \langle Q f \mid \hat g \rangle [/tex] for any functions f,g in hilbert space
2. Related formulas
[tex] \langle f \mid g \rangle = \int f^{\ast} g dx [/tex]
The Attempt at a Solution
Clearly the second definition implies the first, but I'm having trouble showing that the first implies the second.
My quantum mechanics book has this as an exercise and as a hint it suggests to let f=g+h
and then let f=g+ih with i being the square root of -1. I have done both of these things, expanding the inner products in terms of integrals. If i assume the first definition and let f=g+h, i can get
[tex] \langle f \mid \hat Q g \rangle + \langle g \mid \hat Q f \rangle = \langle \hat Q f \mid g \rangle + \langle \hat Q g \mid f \rangle [/tex]
doing a similar thing with f=g+ih i get the same result, and not sure where to go from here. Anybody have a better way to prove it or any ideas? thanks
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