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I read about the quantum quench problem in condensed matter physics. But what does really mean? Has anybody a good explanation about the origin of quantum quench problem?

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I don't know if this is what you mean to ask, but what I know as a quantum quench is a sudden change in the potential, sufficiently fast that it can be considered instantaneous. In that case the state doesn't change instantaneously, but obviously its time evolution does: from that point it evolves according to the new Hamiltonian. If originally you were in an eigenstate, generally you will be in a superposition of states for the new Hamiltonian after the quench.

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I searched for "Quantum Quench" with Google, found http://www.physics.rutgers.edu/~eyuzbash/papers/PhysRevLett_105_135701.pdf (PDF link, but freely available) and looked at the first paragraph and found

The shattering of cold glass in hot water is but one of many spectacular effects that can be induced by a rapid thermal quench in classical media. What happens when an isolated quantum phase of matter is subject to a sudden, violent deformation of its system Hamiltonian (a ‘‘quantum quench’’)?

(emphasis added)

Now, that is not a lot of detail, but it sets the stage. Further, it is the kind of thing that you should have been able to find yourself before asking here.

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