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Hello I am currently studying introductory QM and am confused about bases and operators. If I have an operator $\hat{Q}$, does this represent a change of basis matrix? In other words, does $\hat{Q} | \psi \rangle$ for any state $ | \psi \rangle$ simply represent the expansion of $|\psi \rangle$ in the eigenbasis of $\hat{Q}$?

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No, an arbitrary operator does not represent a change of basis. And even those that can be used to perform changes of basis should not always be interpreted as such.

A "change of basis" in a Hilbert space is usually meant to be a change from one orthonormal basis to another. The operators that map orthonormal systems to orthonormal systems are precisely the unitary operators. But to speak of a change of basis for a unitary operator you have to simultaneously transform all states by $\lvert\psi\rangle\mapsto U \lvert \psi \rangle$ and all operators by $O\mapsto UOU^\dagger$. Just applying a unitary operator to a state is not a "change of basis", it is transforming the state.

As an example, all symmetry operators are unitary operators. So when you rotate a state, you apply a unitary operator to it. The "proper" physical interpretation here is not that you changed you basis, it is that you rotated the state. What pointed into the $x$-direction before now points into the $y$-direction, and unless you applied the rotation to all states and operators, you may not take the passive view and declared that you just switched the basis labels $x$ and $y$.

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