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The central limit theorem has been found to apply to random vectors incomplex Hilbert space. This amounts to sufficient reason to study the complexvalued Gaussian, looking for relevance to quantum mechanics. Here we show thatthe Gaussian, with all terms fully complex, acting as a propagator, leads toSchrodinger nonrelativistic equation including scalar and vector potentials,assuming only that the norm is conserved. No prior physical laws need to bepostulated. It thereby presents as a process of irregular motion analogous tothe real random walk but executed under the rules of the complex number system.Inferences are 1. There is a standard view that Schrodinger equation isdeterministic, while wavefunction collapse is probabilistic by Born's rule.This is opposed by the now demonstrated linkage to the central limit theorem,indicating a stochastic picture for the foundation of Schrodinger equationitself. 2. This picture is also consistent with the dynamic origin ofprobabilities suggested for the Born rule in the de Broglie Bohm pilot wavetheory. Reasons for the primary role of C are open to discussion. The presentderivation is compared with recent reconstructions of the quantum formalism,which have the aim of rationalizing its obscurities.

Schr\"odinger's equation as a consequence of the central limit theorem without prior assumption of physical laws