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I'm sort of struggling to understand the difference between active and passive transformations. I'm working with two books, and both say that in one case the vectors are transformed (active) and in the other the operators are transformed (passive). That's fine and dandy, but in terms of being able to actually identify a transformation as one or the other, I am at a loss.

For example, in Boas's Mathematical Methods in the Physical Sciences, she uses two 2x2 rotation matrices as examples. The first has a -sinΘ in the upper right and the sinΘ in the lower left, and is the vector rotation, and the second is the same but with sine terms swapped, and is claimed to be "axes" rotated.

Now, if you just switch the direction of rotation, one effectively becomes the other... i.e., change the sign of Θ and suddenly you have the other matrix. So what make one passive and the other active, really?

In my other book, Shankar's Principles of Quantum Mechanics, the treatment is a little different. He says,

And then he goes on to talk about how it would be the same story if we just went straight to subjecting all operators to the change U Suppose we subject all the vectors |V> in a space to a unitary transformation (analogous to a rotation, I understand)

|V> → U|V>

Under this transformation the matrix elements of any operator Ω are modified as follows:

<V'|Ω|V> → <UV'|Ω|UV> = <V'|U^{+}ΩU|V>^{+}ΩU which is fine. What I don't understand is something pretty elementary - why is the matrix element of the operator Ω = <V'|Ω|V>. Where the hell did the <V'| come from... why not just <V|?

Thanks for reading my novel. Any help would be extremely appreciated.

Edit: So I think I figured out what the <V'| is... just any other arbitrary row vector in the space... but why isn't it specified as a basis vector? Or is this implied? Like if <V'| was a row basis vector, and |V> was another column basis vector, then I could see how it would be selecting an element of the matrix... but if they're both just some arbitrary vectors, couldn't they have some scaling effect on the matrix element?

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# Active and Passive Transformations Confusion

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