  Picard--Lefschetz theory is applied to path integrals of quantum mechanics,
in order to compute real-time dynamics directly. After discussing basic
properties of real-time path integrals on Lefschetz thimbles, we demonstrate
its computational method in a concrete way by solving three simple examples of
quantum mechanics. It is applied to quantum mechanics of a double-well
potential, and quantum tunneling is discussed. We identify all of the complex
saddle points of the classical action, and their properties are discussed in
detail. However a big theoretical difficulty turns out to appear in rewriting
the original path integral into a sum of path integrals on Lefschetz thimbles.
We discuss generality of that problem and mention its importance. Real-time
tunneling processes are shown to be described by those complex saddle points,
and thus semi-classical description of real-time quantum tunneling becomes
possible on solid ground if we could solve that problem.
