By Maurice A De Gosson

The second one variation of this ebook offers, because the first, with the rules of classical physics from the 'symplectic' perspective, and of quantum mechanics from the 'metaplectic' standpoint. now we have revised and augmented the subjects studied within the first variation within the gentle of recent effects, and additional numerous new sections. The Bohmian interpretation of quantum mechanics is mentioned intimately. section area quantization is completed utilizing the 'principle of the symplectic camel', that's a deep topological estate of Hamiltonian flows. We introduce the inspiration of 'quantum blob', that are considered because the primary part area unit. The mathematical instruments constructed during this e-book are the speculation of the symplectic and metaplectic team, the Maslov index in a rigorous shape, and the Leray index of a couple of Lagrangian planes. the idea that of the 'metatron' is brought, in reference to the Bohmian concept of movement. The short-time habit of the propagator is studied and utilized to the quantum Zeno impression.

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**Example text**

Gromov’s non-squeezing theorem opens new perspectives in both Hamiltonian and quantum mechanics. It raises many interesting questions (most of them still unanswered) because it highlights the fact that the existence and properties of closed orbits is in some mysterious way related to the property of the symplectic camel (see Hofer and Zehnder’s book [89]). We will use Gromov’s result in Chapter 3 to eﬃciently quantize phase space in “quantum blobs”, which generalize the usual notion of quantum cell.

It is in fact Hamilton-Jacobi’s equation, not for the Hamiltonian function H, though, but rather for H Ψ = H + QΨ where we have set QΨ = − ∇2r R . 51) Bohm called that function QΨ (which has the dimension of an energy) the quantum potential associated to Ψ. The quantum potential is actually very unlike a usual potential. First, it does not arise from any external source, and secondly it is intrinsically non-local. It is a “self-organizing” potential, in fact a response to the environment in which the quantum process takes place.

20) we can only determine the motion corresponding to “locked” initial values of the momentum, corresponding to the “constraint” p0 = ∇r Φ0 (r0 ). However, in principle, we can use the method to determine the motion corresponding to an arbitrary initial phase space point (r0 , p0 ) by choosing one function Φ0 such that p0 = ∇r Φ0 (r0 ), then to solve the Hamilton-Jacobi equation with Cauchy datum Φ0 and, ﬁnally, to integrate Eq. 22). Of course, the solutions we obtain are a priori only deﬁned for short times, because Φ is not usually deﬁned for large values of t.

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