  We introduce a concept of ridge-lines to investigate the semi-classical
prediction from wave-packets with arbitrary width in conventional quantum
mechanics and the Wheeler--DeWitt quantum cosmology. Two primary approaches are
applied to the exact calculation of the ridge-lines, namely the contour and the
stream approach. Moreover, aspects of these are discussed and compared to other
scenarios and approaches, i.e. the narrow WKB wave-packets and the
first-derivative test. As the main result, we show that the semi-classical
predictions in toy models have more abundant solutions than in the classical
theory, and most interestingly they may deviate from classical solutions due to
the quantum corrections.
