Quantum Mechanical Unitary Transformations

$\hat{U} | ψ ⟩ = | ψ^{'} ⟩$ and $⟨ ψ^{'} | = ⟨ | {\hat{U}}^{†}$

Definition

${\hat{U}}^{†} \hat{U} = I$ .

Consider the Unitary Transformation

$A | ψ ⟩ = | φ ⟩ \Rightarrow^{Unitary Transformation} {\hat{A}}^{'} | ψ^{'} ⟩ = | φ^{'} ⟩$ . ${\hat{A}}^{'} \hat{U} | ψ ⟩ = \hat{U} | φ ⟩$ . Then ${\hat{U}}^{†} {\hat{A}}^{'} \hat{U} | ψ ⟩ = | φ ⟩$ . Thus, ${\hat{A}}^{'} = \hat{U} A {\hat{U}}^{†}$ . ${\hat{A}}^{'}$ and $\hat{A}$ are unitarily equivalent observables.

Properties

Let $\hat{A}$ be Hermitian. Then, ${\hat{A}}^{'} = \hat{U} \hat{A} {\hat{U}}^{†}$ , ${\hat{A}}^{' †} = \hat{U} {\hat{A}}^{†} {\hat{U}}^{†} = {\hat{A}}^{'}$ .
Let $\hat{A} | ψ ⟩ = a | ψ ⟩$ . Then, ${\hat{A}}^{'} | ψ^{'} ⟩ = \hat{U} \hat{A} {\hat{U}}^{†} \hat{U} | ψ ⟩ = a | ψ^{'} ⟩$ .
Consider $⟨ ψ | \hat{A} | χ ⟩$ .Then, $⟨ ψ^{'} | {\hat{A}}^{'} | χ^{'} ⟩ = ⟨ ψ | {\hat{U}}^{†} \hat{U} \hat{A} {\hat{U}}^{†} \hat{U} | χ = ⟨ ψ | \hat{A} | χ ⟩$ .
So, in general, $⟨ ψ^{'} | χ^{'} ⟩ = ⟨ ψ | χ ⟩$ . Then, the norm and expectation values are invariant under unitary transformations.