Geodesic
Derivation of the Schr\"odinger equation from scale invariance
Teoretičeskaâ i matematičeskaâ fizika, Tome 26 (1976) no. 3, pp. 414-418

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It is shown that ff one postulates an equation of nonrelativistic quantum mechanics in the form $\partial\psi/\partial t=f(V,\psi,\psi^*)$, where $V$ is the potential energy, and requires that this equation, like the action (of the usual form) be invariant under the group of transformations $\mathbf x\to\lambda\mathbf x$, $t\to\mu(\lambda)t$, and also imposes certain other weak restrictions, these conditions uniquely lead to the Schrödinger equation. 
@article{TMF_1976_26_3_a14,
     author = {V. Ya. Kreinovich},
     title = {Derivation of the {Schr\"odinger} equation from scale invariance},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {414--418},
     publisher = {mathdoc},
     volume = {26},
     number = {3},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1976_26_3_a14/}
}
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V. Ya. Kreinovich. Derivation of the Schr\"odinger equation from scale invariance. Teoretičeskaâ i matematičeskaâ fizika, Tome 26 (1976) no. 3, pp. 414-418. http://geodesic.mathdoc.fr/item/TMF_1976_26_3_a14/