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We show that the time-dependent Schr\"odinger equation (TDSE) is thephenomenological dynamical law of evolution unraveled in the classical limitfrom a timeless formulation in terms of probability amplitudes conditioned bythe values of suitably chosen internal clock variables, thereby unifying theconditional probability interpretation (CPI) and the semiclassical approach forthe problem of time in quantum theory. Our formalism stems from an exactfactorization of the Hamiltonian eigenfunction of the clock plus systemcomposite, where the clock and system factors play the role of marginal andconditional probability amplitudes, respectively. Application of the VariationPrinciple leads to a pair of exact coupled pseudoeigenvalue equations for theseamplitudes, whose solution requires an iterative self-consistent procedure. Theequation for the conditional amplitude constitutes an effective "equation ofmotion" for the quantum state of the system with respect to the clockvariables. These coupled equations also provide a convenient framework fortreating the back-reaction of the system on the clock at various levels ofapproximation. At the lowest level, when the WKB approximation for the marginalamplitude is appropriate, in the classical limit of the clock variables theTDSE for the system emerges as a matter of course from the conditionalequation. In this connection, we provide a discussion of the characteristicsrequired by physical systems to serve as good clocks. This development is seento be advantageous over the original CPI and semiclassical approach since itmaintains the essence of the conventional formalism of quantum mechanics,admits a transparent interpretation, avoids the use of the Born-Oppenheimerapproximation, and resolves various objections raised about them.

Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory