Also it's one of the most useful equations in all of QM and QFT
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Time Evolution in Quantum Mechanics •In general, a quantum state |Ψ〉changes in time according to the *time ordered* exponential of the integral of the Hamiltonian (1) •If the Hamiltonian doesn't depend on time, then the time evolution is much simpler (2)pic.twitter.com/grKF4MMMDW
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Do you know if there is any rigorous way to interpret \mathcal{T} as an operator? Clearly it does not act on H, the Hilbert space of the system states. Maybe it acts on the (Banach?) space of the linear operators on H? This notation had been low-key bothering me for years now.
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Yea it just means that you order things based on what comes first
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In what cases wouldn’t it depend on time?
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Suppose you initially have a free electron and then turn on an electric field at some later time
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I built a module in QuTiP that will do (1) for you! It's designed to compute photon scattering in arbitrary systems but can be applicable to more general dissipative driven quantum systems https://nbviewer.jupyter.org/github/qutip/qutip-notebooks/blob/master/examples/temporal-photon-scattering.ipynb …
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If I just exp[-iHt] myself, will I evolve to a state where everything I need to gets done is just finished? Like |<thad done with stuff|exp[-iHt]|thad staring at computer>|^2 = 1?
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? System dynamics 
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