## A geometrical interpretation of the time evolution of the Schrödinger equation for discrete quantum systems.(English)Zbl 0618.35099

The time evolution of a quantum state in the Schrödinger picture of quantum mechanics is presented as a curve in $${\mathbb{C}}P^ n$$. An $$(n+1)$$- state quantum system is considered semi-classically as a $${\mathbb{C}}P^ n$$ bundle over three dimensional space, with structure group $$SU(n+1)/{\mathbb{Z}}_{n+1}$$.

### MSC:

 35Q99 Partial differential equations of mathematical physics and other areas of application 35J10 Schrödinger operator, Schrödinger equation 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 35A30 Geometric theory, characteristics, transformations in context of PDEs
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### References:

 [1] T. Eguchi , P. Gilkey and A. Hanson , Phys. Rep. , t. 66 , 1980 , p. 213 - 393 . MR 598586 [2] P.A.M. Dirac , The Principles of Quantum Mechanics , Oxford . University Press , 1981 , 4 th Edition. · Zbl 0012.18104 [3] N. Steenrod , The Topology of Fibre Bundles , Princeton University Press , 1951 . MR 39258 | Zbl 0054.07103 · Zbl 0054.07103
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