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1. Identity Transformation

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   The Identity Transformation and Quantum State
   In Quantum Mechanics, the identity operator ($\hat{I}$ or $\hat{\mathbf{1}}$) acts on a Hilbert space $\mathcal{H}$ such that for any state vector $|\psi\rangle \in \mathcal{H}$:
   $$ \hat{I} |\psi\rangle = |\psi\rangle $$
   The key physical interpretation arises when considering the measurement process. If a system is known to be in a certain state, applying the identity operator before or after measuremen…

2. Parity Inversion

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   Parity inversion, denoted by the operator $\mathcal{P}$, is a fundamental symmetry operation in physics (general)/) that corresponds to spatial inversion through the origin, transforming the coordinates of a point $(x, y, z)$ to $(-x, -y, -z)$. In quantum mechanics, the parity operator acts on a state vector $|\psi\rangle$ such that $\mathcal{P}|\psi\rangle = |\psi'\rangle$. If a system's Hamiltonian $H$ commutes with $\mathcal{P}$ (i.e., $[H, \mathcal{P}] = 0$), the system possesses…