# Definition of invariance of a QM operator under a transformation

In the Sakurai book "Modern quantum mechanics" (pg. 263) an operator $$S$$ is said to be invariant under a unitary transformation $$T$$ if:

$$T^\dagger S T = S.$$

Where that definition come from?

My guess is that it comes from the similarity transformation of matrix in liner algebra:

$$M^{\prime}=T^{-1} M T.$$

So if $$M= T^{-1}M T$$ then I know that the matrix doesn't change when I change the basis of the vector space. If $$T$$ is unitary then $$T^{-1} = T^{\dagger}$$

You could also derive that expression in the following way. Let us consider a state $$\rvert \Psi \rangle$$. The expectation value of the operator $$S$$ in that state is $$$$\langle\Psi\lvert S\rvert\Psi \rangle$$$$ Now let us suppose we act with the operator $$T$$ on the state $$\rvert \Psi \rangle$$. The new expectation value is $$$$\langle\Psi\lvert T^\dagger S T\rvert \Psi \rangle$$$$ If we require that a measurement of the observable $$S$$ be independent of wether we perform it on the state $$\rvert\Psi \rangle$$ or $$T\rvert\Psi \rangle$$, then the two expectation values must be the same, i.e. $$$$\langle\Psi\lvert S\rvert\Psi \rangle=\langle\Psi\lvert T^\dagger S T\rvert \Psi \rangle$$$$ From which $$$$T^\dagger S T = S$$$$ If $$T$$ is unitary, then $$T^\dagger=T^{-1}$$.