DOI: 10.14704/nq.2014.12.4.774

Homogeneity of Space-time Implies the Free Schrödinger Equation

Gao Shan


The free Schrödinger equation is shown to be a consequence of space-time homogeneity in the non-relativistic domain. First, we show that space-time translation invariance of natural laws, which is a conseqiuence of space-time homogeneity, entails that the state of a free particle with definite momentum and energy assumes the plane wave form when the time evolution of the state is linear. Next, we show that the conservation of energy and momentum, which is a consequence of space-time translation invariance, may further determine the energy-momentum relation In the non-relativistic domain. These results then lead to the free Schrödinger equation naturally. The new integrated analysis may help understand the origin of the wave equations in quantum theory.


Schrödinger equation; spacetime homogeneity; wave equations; quantum theory

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Duff MJ, Okun LB, and Veneziano G. Trialogue on the number of fundamental constants. Journal of High Energy Physics 2002; 03: 023.

Feynman RP, Leighton RB, and Sands M. The Feynman Lectures on Physics, vol. I Reading: Addison-Wesley, 1963. pp. 16/6-16/7.

Grabert H, Hänggi P, and Talkner P. Is quantum mechanics equivalent to a classical stochastic process? Phys Rev A 1979; 19: 2440-2445.

Greiner W. Quantum Mechanics: An Introduction. New York: Springer, 1994.

Landau L and Lifshitz E. Quantum Mechanics. Oxford: Pergamon Press,1977.

Lévy-Leblond JM. Nonrelativistic particles and wave equations. Commun Math Phys 1967; 6: 286.

Musielak ZE. and Fry JL. Physical theories in Galilean space-time and the origin of Schrödinger-like equations. Annals of Physics 2009; 324; 296-308.

Nelson E. Derivation of the Schrödinger equation from Newtonian mechanics. Phys Rev 1966; 150: 1079-1085.

Nelson E. The mystery of stochastic mechanics. nelson/papers/talk.pdf. Accessed date: August 8, 2014.

Pal PB. Nothing but relativity. Eur J Phys 2003; 24: 315-319.

Schiff L. Quantum Mechanics. New York: McGraw-Hill, 1968.

Shankar R. Principles of Quantum Mechanics, 2nd ed. New York: Plenum, 1994.

Wallstrom T. Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations. Phys Rev A 1994; 49: 1613-1617.

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The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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