Time evolution (Hamiltonian/Energy) vs Space/Position Translation (Momentum)

# Time evolution (Hamiltonian/Energy) vs Space/Position Translation (Momentum) 작성중인 글... 왜 양자역학에서 time evolution 과 space translation 이 Hamiltonian/Energy 과 Momentum 으로 표현되는지를 설명할 예정. 상대론 (Energy-Momentum tensor) 과도 관련이 있고, 특정 coodinate (space-time 통합해서 한 event/position) 의 quantum state |x^{i} \vec{e}_{i}\rangle 의 Taylor expansion 과도 관련 있음. ## PH

• 2023-02-11 : First posting.

## TOC ## Space/Position Translation \hat{T} (d\vec{x}') |\vec{x}\rangle = 1 - i \frac{\hat{\vec{P}} \cdot d\vec{x}'}{\hbar} |\vec{x}\rangle = |\vec{x}+d\vec{x}'\rangle ## Time evolution (or Time translation) \hat{T} (d t') |\psi ; t\rangle = 1 - i \frac{\hat{H} d t'}{\hbar} |\psi ; t\rangle = |\psi ; t + d t'\rangle ## Relativity: Space-time translation (Combination of space and time) For the notation, refer . \hat{T} (\frac{d \vec{x}_{\text{C}}(\tau)}{d \tau}^{A} \nabla_{A}) |x^{i} \vec{e}_{i}\rangle = 1 - i \frac{\hat{H} d t'}{\hbar} |x^{i} \vec{e}_{i}\rangle = |(x^{i} + d x'^{i}) \vec{e}_{i}\rangle ## RRA

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