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### Space Translation Operator in Quantum Mechanics In quantum mechanics, the space translation operator \(\hat{T}(a)\) for an infinitesimal translation by \(a\) is given by: \[ \hat{T}(a) = e^{-i \frac{a}{\hbar} \hat{p}_x} \] where: - \( \hat{p}_x \) is the momentum operator in the \( x \)-direction. - \( \hbar \) is the reduced Planck's constant. For an infinitesimal translation (\( a \to 0 \)), ____ _______ _______ _____ ________ ___ _______.
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