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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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