DOI: 10.1007/s10910-005-4531-8 Journal of Mathematical Chemistry Vol. 38, No. 1, July 2005 (© 2005) Deduction of Heisenberg relations and Schr¨ odinger equation through the structure of N -dimensional parameterized metric vector spaces Ramon Carb´ o-Dorca ∗ Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium E-mail: quantumqsar@hotmail.com Received 31 January 2005; revised 10 February 2005 Here, it is described how N -dimensional parameterized vector spaces, possessing an adapted real metric with the addition of some supplementary axioms, permit the deduc- tion of Heisenberg relations and Schr¨ odinger equation. This specific space structure, proposed as a container of the description of quantum objects, suggests, in this way, that both quantum mechanical cornerstones can be considered as trivial consequences of such a special vector space choice. KEY WORDS: Heisenberg relations, Schr ¨ odinger equation, parameterized vector spaces, real valued metric, inward vector products 1. Introduction Recent papers dealing with the role of Heisenberg relations [1] seems to conclude that uncertainty relationships constitute somehow a quantum theo- retical cornerstone as well as an essential previous step towards setting up Schr ¨ odinger equation. Such an affirmation can be made in this way, as within the most recent paper [2], not only it is clearly shown that Heisenberg relations are deductible from classical considerations, under well defined statistical con- ditions, but Schr¨ odinger equation can be derived from these relationships by means of further elegant theoretical deductions. This situation has inspired a previous study [2], where it was shown Heisenberg relations could be also deduced by means of the definition of param- eterized N -dimensional metric vector space structure. Such space structure, after some straightforward working definitions and a sequence of few ancillary axi- oms, presents the possibility of deducing the well-known uncertainty relationships, through the usual metric related constructs linking two vectors, as Gram matrix, ∗ Permanent address: Institut de Qu´ ımica Computacional/Universitat de Girona/Girona 17071 (Catalonia)/Spain. 89 0259-9791/05/0700-0089/0 © 2005 Springer Science+Business Media, Inc.