We construct approximate solutions to the time–dependent Schrödinger equation i ¯ h ∂ψ ∂t = − ¯ h 2 2 ∆ ψ + V ψ for small values of ¯ h. If V satisfies appropriate analyticity and growth hypotheses and |t| ≤ T , these solutions agree with exact solutions up to errors whose norms are bounded by C exp { − γ/¯ h } , for some C and γ > 0. Under more restrictive hypotheses, we prove that for sufficiently small T ′ , |t| ≤ T ′ | log(¯ h)| implies the norms of the errors are bounded by C ′ exp − γ ′ /¯ h σ , for some C ′ , γ ′ > 0, and σ > 0.