Quantum Resolution Entropy?

 The concept of thermodynamic entropy originates in classical physics, in particular in the first law which is related to energy distribution. For example, even though particle states are often defined through p and x, in many cases it is the p (or pp/2m) distribution which is key as in a Maxwell-Boltzmann gas or collective kinetic energy converted to heat.

   In quantum mechanics, defining a precise energy state, e.g. one associated with p momentum or j angular momentum, involves a second variable associated with a mathematical generator e.g. d/dx or d/dtheta. A high value of p or j suggests a sharp resolution function, i.e. high information in this second variable. Thus, we argue that there is a new type or entropy introduced, namely resolution entropy, which does not exist in classical physics. For example, exp(ipx) represents a classically sharp p and corresponding pp/2m, but there is uncertainty in x. The question we ask is whether there are any properties associated with this new entropy. We suggest that like in the classical case, a spontaneous change should lead to an increase in this new entropy even if in terms of energy it seems that one has usual dynamics with no entropic considerations. In particular, we apply this idea to a quantum bound state.