Is work done by the external agency when an adiabatic change in the Hamiltonian is performed on some State ψ?

Yes. We don't even need to restrict ourselves to adiabatic changes here. The change in energy of the system is the work done on the system, in classical or in quantum mechanics, essentially by definition.

Yes What I am confused about is if the state is superposition of energy levels, we have no adequate definition of Energy of the system.

I have thought of a specific situation to illustrate my problem.
Consider the case of a Particle in a infinite well, If the walls are compressed slowly and state is in a superposition of first and second excited levels, What is the work done on the walls?
One can say that the work done is the change in expectation value of the energy, But why should it be so?

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