Claude Cohen-Tannoudji - Quantum Mechanics, Volume 3

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This new, third volume of Cohen-Tannoudji's groundbreaking textbook covers advanced topics of quantum mechanics such as uncorrelated and correlated identical particles, the quantum theory of the electromagnetic field, absorption, emission and scattering of photons by atoms, and quantum entanglement. Written in a didactically unrivalled manner, the textbook explains the fundamental concepts in seven chapters which are elaborated in accompanying complements that provide more detailed discussions, examples and applications.<br> <br> * Completing the success story: the third and final volume of the quantum mechanics textbook written by 1997 Nobel laureate Claude Cohen-Tannoudji and his colleagues Bernard Diu and Franck Laloë<br> * As easily comprehensible as possible: all steps of the physical background and its mathematical representation are spelled out explicitly<br> * Comprehensive: in addition to the fundamentals themselves, the books comes with a wealth of elaborately explained examples and applications<br> <br> Claude Cohen-Tannoudji was a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris where he also studied and received his PhD in 1962. In 1973 he became Professor of atomic and molecular physics at the Collège des France. His main research interests were optical pumping, quantum optics and atom-photon interactions. In 1997, Claude Cohen-Tannoudji, together with Steven Chu and William D. Phillips, was awarded the Nobel Prize in Physics for his research on laser cooling and trapping of neutral atoms.<br> <br> Bernard Diu was Professor at the Denis Diderot University (Paris VII). He was engaged in research at the Laboratory of Theoretical Physics and High Energy where his focus was on strong interactions physics and statistical mechanics.<br> <br> Franck Laloë was a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris. His first assignment was with the University of Paris VI before he was appointed to the CNRS, the French National Research Center. His research was focused on optical pumping, statistical mechanics of quantum gases, musical acoustics and the foundations of quantum mechanics.<br>

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We must however keep in mind that all these calculations derive from the mean field approximation, in which we replaced the exact equilibrium density operator by an operator of the form (32). In many cases this approximation is good, even excellent, as is the case, in particular, for a long-range interaction potential: each particle will interact with several others, therefore enhancing the averaging effect of the interaction potential. It remains, however, an approximation: if, for example, the particles interact via a “hard core” potential (infinite potential when the mutual distance becomes less than a certain microscopic distance), the particles, in the real world, can never be found at a distance from each other smaller than the hard core diameter; now this impossibility is not taken into account in (32). Consequently, there is no guarantee of the quality of a mean field approximation in all situations, and there are cases for which it is not sufficient.

1 1 They are not simply the juxtaposition of that complement’s equations: one could imagine writing those equations independently for each energy level, and then performing a thermal average. We are going to see (for example in § 2-d-β) that the determination of each level’s position already implies thermal averages, meaning that the levels are coupled.

2 2 Contrary to what is usually the case for a density operator, the trace of this reduced operator is not equal to 1, but to the average particle number — see relation (44). This different normalization is often more useful when studying systems composed of a large number of particles.

3 3 We have changed the notation and of Chapter XVinto and to avoid any confusion with the distribution functions fβ.

4 4 For fermions, and when the temperature approaches zero, the distribution function included in the definition of ρI(1) becomes a step function and ρI(1) does indeed coincide with PN(1).

5 5 The definition of partial traces is given in § 5-b of Complement EIII. The left hand side of (71)can be written as Σi, j 〈1 1: θi; 2: θj〉 dρI(1)O(1, 2) |1 : θi; 2 : θj〉. We then insert, after dρj(1), a closure relation on the kets |1 :θk;2 : θk′), with k′ = j since dρI(1) does not act on particle 2. This yields: Σi, k 〈1 :θi| dρI (1)|1: θk〉 Σj 〈1 : θk; 2: θj|O(1, 2)|1 : θi; 2: θj〉, where the sum over j is the definition of the matrix element between 〈1 : θk| and |1 : θi〉 of the partial trace over particle 2 of the operator O(1,2). We then get the right-hand side of (71).

6 6 This was expected, since this choice does not lead to any variation of the trial density operator.

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