1. Quantum statistics: a formal approach Reminder to axioms of quantum mechanics  state vector of the Hilbert space describes quantum mechanical system  Observables are represented by hermitian operator   expectation value  System in state with and Eigenstates of  Measurement of A yields a n with probability with First consider a truly isolated system

2. We stated: Matrix representations: Closer inspection of the density operator of a pure state with Let’s show that in fact

3. Now let‘s consider a quantum system interacting with the external world Instead of coherent superposition for isolated system Coupling with the external worlddecoherence State of a system in equilibrium is an incoherent superposition of eigenstates We can recall the idea of a Gibbs ensemble, an infinite collection of systems each in an N-particle eigenstate Simple example for the difference between coherent superposition & statistical mix interference term

4. coherent superposition & statistical mix with Next we see that in the canonical ensemble Equivalent considerations can be made for the micro- and grandcanonical ensamble and with is our thermal average in the canonical ensemble

5. The notation gets meaning via So far we wrote: Let’s reconsider the Boltzmann factor Eigenstates of N-particle Schroedingder eq. From basis independent

6. Let’s consider internal energy U In general