  A microscopic understanding of the thermodynamic entropy in quantum systems
has been a mystery ever since the invention of quantum mechanics. In classical
physics, this entropy is believed to be the logarithm of the volume of phase
space accessible to an isolated system [1]. There is no quantum mechanical
analog to this. Instead, Von Neumann's hypothesis for the entropy [2] is most
widely used. However this gives zero for systems with a known wave function,
that is a pure state. This is because it measures the lack of information about
the system rather than the flow of heat as obtained from thermodynamic
experiments. Many arguments attempt to sidestep these issues by considering the
system of interest coupled to a large external one, unlike the classical case
where Boltzmann's approach for isolated systems is far more satisfactory. With
new experimental techniques, probing the quantum nature of thermalization is
now possible [3, 4]. Here, using recent advances in our understanding of
quantum thermalization [5-10] we show how to obtain the entropy as is measured
from thermodynamic experiments, solely from the self-entanglement of the
wavefunction, and find strong numerical evidence that the two are in agreement
for non-integrable systems. It is striking that this entropy, which is closely
related to the concept of heat, and generally thought of as microscopic chaotic
motion, can be determined for systems in energy eigenstates which are
stationary in time and therefore not chaotic, but instead have a very complex
spatial dependence.
