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what is mathematical difference/relation between an hermitian operator $\hat A$ and a vector $\vec A$?

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Your choice of letter A and symbols around it tells me nothing about the operator and the vector in question. So it sounds to me as "what is the difference/relation between a Hermitian operator and a vector?" Well, you can apply operators to vectors - that's about it. – user31373 Jun 8 '12 at 18:43
I'm having a hard time guessing why you think they are similar enough you have to ask this question. Can you include your thoughts leading up to the question? – rschwieb Jun 8 '12 at 18:57

These are quite different objects. You can apply an operator (Hermitian or not) to a vector to obtain another vector, in fact Hermitian operators are often represented by matrices.

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