mmwave
When studying Thermodynamics, we make much use of adiabatic processes where a change is made so fast that no significant heat can flow. For example, the compression stroke of a diesel engine.

In quantum mechanics, it seems they turn the definition on its head, adiabatic is defined as a change to the system that occurs gradually compared to the evolution of the unperturbed system. An example is a pendulum of period T with its length changing slowly compared to T.

Are these two definitions somehow the same? Are they even related? I can't make sense of this.

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No significant heat (energy) can flow. In the quick thermo process the time is to short for a flow rate of any size to transfer significant energy. In the adiabatic condition in QM the rate is controlled to be arbitrarily low, so that no significant amount is transferred in the long slow process..