By Maurice A De Gosson
ISBN-10: 1848161425
ISBN-13: 9781848161429
ISBN-10: 1860942741
ISBN-13: 9781860942747
This paintings offers with the rules of classical physics from the ''symplectic'' standpoint, and of quantum mechanics from the ''metaplectic'' viewpoint. The Bohmian interpretation of quantum mechanics is mentioned. part house quantization is completed utilizing the ''principle of the symplectic camel'', that is a lately came upon deep topological estate of Hamiltonian flows. The mathematical instruments constructed during this booklet are the speculation of the metaplectic staff, the Maslov index in an actual shape, and the Leray index of a couple of Lagrangian planes. the concept that of the ''metatron'' is brought, in reference to the Bohmian idea of movement. an actual type of Feynman's fundamental is brought in reference to the prolonged metaplectic illustration.
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14. Concepts, past and present, IOP Newsletter, Spring 2001, 32 FROM KEPLER TO SCHRODINGER... AND BEYOND function H. ). What really interests the physicist is the evolution of the system, and this evolution is obtained from H by solving Hamilton's equations of motion. The situation is similar in quantum mechanics: to every Hamiltonian H one associates the linear space ri of all solutions \I> of the corresponding Schrodinger equation However, no more than the function H alone describes the effective motion of the classical system, does the space ri yield a complete description of the quantum system.
2) can be written: z = JVzH(z,t). 12) There is a fundamental relation between the symplectic form ft, the Hamilton vector field XJJ and H itself. 13) Action and Hamilton-Jacobi's Theory 11 for all z, z'. This relation is fundamental because it can be written very simply in the language of intrinsic differentiable geometry as ixHn + dH = 0. : iXHn(z)(z') = n(xH(z),z'). i5) This "abstract" form of Eq. 13) is particularly tractable when one wants to study Hamiltonian mechanics on symplectic manifolds (this becomes necessary, for instance, when the physical system is subjected to constraints).
There is, of course, nothing wrong a priori with such a positivistic attitude (which amounts to identify "perception" and "existence"), as far as it does not lead to a systematic rejection of any phase-space theory just because it is a phase space theory! It is not because Heisenberg's inequalities indicate that there is a "limitation" in the accuracy of the way we perceive our physical world that we should be forbidden to consider a more clear-cut mathematical world where things have precise positions and momenta.
The Principles of Newtonian & Quantum Mechanics by Maurice A De Gosson
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