Delve into advanced concepts of quantum mechanics in this 59-minute lecture by Francesco Becattini from the Galileo Galilei Institute. Explore two distinct methods for understanding global equilibrium and thermodynamic relations, with a focus on why gradients reach zero. Examine translation operators, Wigner functions, and auxiliary functions, while learning about intermediate steps in quantum calculations. Investigate the decomposition of spin density matrices and identify common pitfalls in formula derivation. Gain a deeper understanding of complex quantum mechanical principles and their applications in theoretical physics.