  There is a well-known analogy between statistical and quantum mechanics. In
statistical mechanics, Boltzmann realized that the probability for a system in
thermal equilibrium to occupy a given state is proportional to exp(-E/kT) where
E is the energy of that state. In quantum mechanics, Feynman realized that the
amplitude for a system to undergo a given history is proportional to exp(-S/i
hbar) where S is the action of that history. In statistical mechanics we can
recover Boltzmann's formula by maximizing entropy subject to a constraint on
the expected energy. This raises the question: what is the quantum mechanical
analogue of entropy? We give a formula for this quantity, which we call
"quantropy". We recover Feynman's formula from assuming that histories have
complex amplitudes, that these amplitudes sum to one, and that the amplitudes
give a stationary point of quantropy subject to a constraint on the expected
action. Alternatively, we can assume the amplitudes sum to one and that they
give a stationary point of a quantity we call "free action", which is analogous
to free energy in statistical mechanics. We compute the quantropy, expected
action and free action for a free particle, and draw some conclusions from the
results.
