We consider the evolution of biological populations under the
joint action of mutation and selection. The
genotypes are linear arrangements of $N$ sites, each of which
may take the value +1 or -1; the evolution of an infinite
population is then described by a dynamical system for a
probability distribution on $\{-1,+1\}^N$.
We first show that this model is exactly equivalent to an Ising quantum
chain. In this picture, the fitness of a configuration corresponds to
the interaction energy of the spins within the chain,
whereas mutation corresponds to interaction of the spins with a
transversal field. This equivalence is exploited to obtain exact
equilibrium solutions for representative fitness landscapes,
with special emphasis on phase transitions.
We then reinterpret the approach in the framework of branching processes.
This admits a plausible biological interpretation
of the quantum-mechanical observables that appeared `enigmatic'
in the previous approach. The key to the connection is the time-reversed
Markov process and its stationary distribution (i.e., the
type distribution of the individuals that are ancestral to
today's population).