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We introduce a concept of ridge-lines to investigate the semi-classicalprediction from wave-packets with arbitrary width in conventional quantummechanics and the Wheeler--DeWitt quantum cosmology. Two primary approaches areapplied to the exact calculation of the ridge-lines, namely the contour and thestream approach. Moreover, aspects of these are discussed and compared to otherscenarios and approaches, i.e. the narrow WKB wave-packets and thefirst-derivative test. As the main result, we show that the semi-classicalpredictions in toy models have more abundant solutions than in the classicaltheory, and most interestingly they may deviate from classical solutions due tothe quantum corrections.

Semi-classical predictions of cosmological wave-packets from ridge-lines