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Birkhäuser Basel eBooks

On a Classical Limit of Quantum Theory and the Non-Linear Hartree Equation

Jürg Martin Fröhlich, Tai‐Peng Tsai, Horng‐Tzer Yau

Published Jan 1, 2010• 38 citations
Publisher

Abstract

Historically, the description of Nature in theoretical physics began with an encoding of rather special, simple natural phenomena exhibiting high approximate symmetry in special, unstable mathematical structures. For example, the shape of the orbits of planets around the sun was conceived to be circular. As the ambition to describe more general classes of natural phenomena grew those special mathematical structures had to be deformed and enlarged into more general, more stable mathematical structures. Kepler discovered that the idea of elliptic orbits, with the sun in one of the foci, provides a better model for the motion of planets. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0346-0422-2_3

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Historically, the description of Nature in theoretical physics began with an encoding of rather special, simple natural phenomena exhibiting high approximate symmetry in special, unstable mathematical structures. For example, the shape of the orbits of planets around the sun was conceived to be circular. As the ambition to describe more general classes of natural phenomena grew those special mathematical structures had to be deformed and enlarged into more general, more stable mathematical structures. Kepler discovered that the idea of elliptic orbits, with the sun in one of the foci, provides a better model for the motion of planets. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
So far, this book has dealt almost exclusively with classical physics, which is a theoretical structure based on many brilliant and imaginative experiments performed on macroscopic objects, that is on objects of about our size or a few orders of magnitude smaller or larger. Classical physics was found to give good results when extrapolated downward to molecules and upward to planets, stars, and galaxies, and it was seriously suggested ate in the nineteenth century that all the major discoveries in physics had been made — that the physical basis of the universe was understood and that only minor tidying-up remained.
The equation of motion of a quantum particle(s) in a potential field will be introduced. The equation of motion should determine drift fields a (t, x) and the movement X t of quantum particles, accordingly. However, the interrelation between the equation of motion and the movement X t of a particle(s) is not so direct as in classical mechanics, and we must go into deeper mathematical structures of diffusion processes to clarify the relation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The mathematical background required for the study of geometric phases in classical and quantum mechanics is rather extensive. The aim of this introductory chapter is to provide a background of some basic notions of classical differential geometry and topology. Classical differential geometry is now a well established tool in modern theoretical physics. Many classical theories like mechanics, electrodynamics, Einstein’s General Relativity or Yang-Mills gauge theories are well known examples where the geometrical methods enter in the natural and very effective way. As we shall see throughout this book, also quantum physics shows its intricate beauty when one applies an appropriate geometric framework. All this proves Wigner’s celebrated statement about the “unreasonable effectiveness” of mathematics in natural sciences. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Schrödinger’s equation and the “wave function” that satisfies it are the first bold brushstrokes of a new portrait of Nature that physicists gradually filled in during the middle half of the twentieth century. The emerging pattern was strange, but not entirely alien to science. It was drawn, to paraphrase Galileo, in a mathematical idiom “without some knowledge of which it is humanly impossible to understand.” While nonmathematical accounts are conceivable, I would rather at this point say more about the mathematics because it will deepen our view of the whole enterprise of modern physics. Most people never get beyond the “math is numbers and numbers are boring” barrier that hides behind it a limitless universe of beautiful things. So let us delve briefly into the nonboring side of mathematics.
Usually one uses completely different mathematical descriptions to formulate classical and quantum mechanics. Classical theory may be nicely formulated in terms of symplectic geometry, and the quantum one in terms of algebraic objects related to a complex Hilbert space. However, it turns out that standard, nonrelativistic quantum mechanics possesses natural geometric structure that is even richer than that found in classical mechanics. This section reveals the beauty of the geometric approach to quantum theory and stands as a basis for the elegant geometrical ideas of Pancharatnam and, later on, of Aharonov and Anandan. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.