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So I understand the concept of "measurement" in quantum mechanics (at least I do as much as I think I'm going to). But I don't know how to interpret the same concept in the case that I am working with a quantum statistical mechanical system i.e. one where you are working with a statistical ensemble given by a density matrix. In this situation you still have observables with eigenvalues and eigenstates and so on. So from ordinary Quantum Mechanics I feel like when you "measure" the result should be an eigenvalue of the operator. Does the concept of "measurement" even make sense here? Perhaps I am confusing myself because the concept does not even make sense? I have been told that the expectation value is what you should expect to actually measure, is this correct?

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    $\begingroup$ To clarify: Are you asking how to express the measurement axioms in terms of a density matrix? (If so, then have you considered a pure state expressed as a density matrix?) Or are you asking why that expression is still valid when the density matrix represents an ensemble? (Keep in mind that in the real world, we never know the state of a big complicated system well enough to require using a pure state, if we're using a model that's rich enough to represent such a system. We never know the details of the state in other galaxies, for example, so we might as well always be using a mixed state.) $\endgroup$ Sep 28 '21 at 13:40
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It is not different from conventional quantum mechanics, apart from the facts that:

  • One works mostly in second quantization
  • Density matrix is used a lot more than a wave function
  • Measured quantities and the methods of measurement are often more complex than in the toy QM problems (electric current, dielectric response, optical absorption, etc.)

There is no change to QM principles, but rather to the methods of calculating the averages.

Perhaps, it is more instructive to start with zero-temperature formalism, since this is where one still uses the wave function. E.g., Fetter & Walecka discuss in details calculating the ground state energy - first via conventional perturbation theory and then using the many-body formalism.

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