I was wondering about temperatures and pure quantum states. I'm currently working on thermalization of isolated quantum systems, which can be described by pure quantum states (kets). How do we define a temperature for these?
Normally you work within the density matrix formalism so you can define an ensemble, which is a mixed state, for a certain temperature. In my scenario we prepare a system in a lab in a pure state, that is isolated, and let it thermalize to its equilibrium state. This must be (in very good approximation) still a pure state if the system is strongly isolated, because of the unitarity of evolution operator.
We can then probably assign a certain temperature to this equilibrium system, as we could bring this system in contact with a heath bath with a certain temperature and monitor if energy flows out of the system or into the system (aka it has a higher or lower temperature). But from a theoretical viewpoint I have no idea how to assign a temperature to a pure quantum state, so I'm quite dazzled by this.
I know the ground state is supposed to have $T = 0 K$, but that's all I could think of. The Internet is not giving me much more information.
Edit: Ofcourse I am talking about a certain many-body system with a corresponding Hamiltonian, which should be non-integrable and chaotic. This way the system will thermalize (in most cases) following the Eigenstate Thermalization Hypothesis. You could take the 1D spin chain non-integrable Ising model as example.