By Fritz Haake
This by way of now vintage textual content presents a very good advent and survey to the constantly increasing box of quantum chaos . the subjects taken care of comprise a close exploration of the quantum features of nonlinear dynamics, quantum standards to differentiate common and abnormal movement, antiunitary symmetries (generalized time reversal), random matrix concept and an intensive account of the quantum mechanics of dissipative structures. every one bankruptcy is followed through a range of difficulties so one can support the newcomer to check and deepen his/her realizing and to procure an energetic command of the tools presented.
In addition to an important variety of smaller revisions and bankruptcy updates, the 3rd version completely expands at the figuring out of common spectral fluctuations, and – to hence make the presentation of all semi-classical advancements self-contained - a brand new bankruptcy on classical Hamiltonian chaos has been inserted.
From the experiences of the 2nd version: In precis, this is often certainly an important reference booklet for the professional. it's going to additionally ably serve a person coming into the sphere for the 1st time who must research the theoretical state-of-the-art in detail." (G. Summy ,University of Oxford, modern Physics 2002, vol. forty three, p 232)
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Additional resources for Quantum Signatures of Chaos
Such a reinterpretation was already sought by Dyson but could be implemented only recently on the basis of a discovery of Pechukas . The fate of the N eigenvalues and eigenvectors of an N × N Hermitian matrix, H = H0 + λV, is in one-to-one correspondence with the classical Hamiltonian dynamics of a particular one-dimensional N -particle system upon changing the weight λ of a perturbation V [32–34]. This fictitious system, now often called Pechukas–Yukawa gas, has λ as a time, the eigenvalues E n of H as coordinates, the diagonal elements Vnn of the perturbation V in the H representation as momenta, and the off-diagonal elements Vnm related to certain angular momenta.
2) The motivation for this definition is that many Hamiltonians of practical importance are invariant under conventional time reversal, [H, T ] = 0. An atom and a molecule in an isotropic environment, for instance, have Hamiltonians of that symmetry. But, as already mentioned in Sect. 1, conventional time reversal is broken by an external magnetic field. In identifying the canonical transformations of Hamiltonians from their symmetries in Sects. 1). 1) is not at all necessary for the above classification of Hamiltonians according to their group of canonical transformations.
The Floquet operator, being unitary, has unimodular eigenvalues (involving eigenphases alias quasi-energies) and mutually orthogonal eigenvectors, FΦν = e−iφν Φν , Φμ |Φν = δμν . 8) I shall in fact be concerned only with normalizable eigenvectors. With the eigenvalue problem solved, the stroboscopic dynamics can be written out explicitly, e−inφν Φν |ψ(0) Φν . 9) ν Monochromatic perturbations are relatively easy to realize experimentally. Much easier to analyse are perturbations for which the temporal modulation takes the form of a periodic train of delta kicks, +∞ H (t) = H0 + λV δ(t − nτ ).
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