- Active and Passive Transformations

1. Basis Change ↔ Vector & Operator(Matrix) Transformation #

$${|base\rangle} \to {U^\dagger|base\rangle}$$ $$\Leftrightarrow \quad |\psi\rangle \to U|\psi\rangle, A \to UAU^\dagger (|a\rangle \to U|a\rangle)$$

2. Expectation Value $\langle A \rangle$ transformation #

$$\langle\psi|A|\psi\rangle \to \langle\psi’|A|\psi’\rangle \quad (|\psi’\rangle = U|\psi\rangle)$$

or

$$\langle\psi|A|\psi\rangle \to \langle\psi|U^\dagger AU|\psi\rangle$$

Hence,

Active: $|\psi\rangle \to U|\psi\rangle, \quad A \to A$

$\Leftrightarrow$

Passive: $|\psi\rangle \to |\psi\rangle, \quad A \to U^\dagger AU$