  Recently, the Lorentzian path integral formulation using the Picard-Lefschetz
theory has attracted much attention in quantum cosmology. In this paper, we
analyze the tunneling amplitude in quantum mechanics by using the Lorentzian
Picard-Lefschetz formulation and compare it with the WKB analysis of the
conventional Schr\"{o}dinger equation. We show that the Picard-Lefschetz
Lorentzian formulation is consistent with the WKB approximation for
wave-function and the Euclidean path integral formulation utilizing the
solutions of the Euclidean constraint equation. We also consider some problems
of this Lorentzian Picard-Lefschetz formulation and discuss a simpler
semiclassical approximation of the Lorentzian path integral without integrating
the lapse function.
