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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(18) This allows considering the infinitesimal variation δf r to be free of any - фото 408

This allows considering the infinitesimal variation δf ( r) to be free of any constraint. The variation картинка 409of the function картинка 410is now the sum of 4 variations, coming from the three terms of (16)and from the integral in (18). For example, the variation of yields 19 which is the sum of a term proportional to e iχand another - фото 411yields:

(19) which is the sum of a term proportional to e iχand another proportional to e - фото 412

which is the sum of a term proportional to e –iχand another proportional to e iχ. This is true for all 4 variations and the total variation Quantum Mechanics Volume 3 - изображение 413can be expressed as the sum of two terms:

(20) Quantum Mechanics Volume 3 - изображение 414

the first being the δf *( r) contribution and the second, that of δf ( r). Now if картинка 415is stationary, картинка 416must be zero whatever the choice of χ, which is real. Choosing for example χ = 0 imposes δc 1+ δc 2= 0, and the choice χ = π/2 leads (after multiplication by i ) to δc 1— δc 2= 0. Adding and subtracting the two relations shows that both coefficients δc 1and δc 2must be zero. In other words, we can impose картинка 417to be zero as just θ *( r) varies but not θ ( r) - or the opposite 1 .

β. Stationary condition: Gross-Pitaevskii equation

We choose to impose the variation картинка 418to be zero as only θ *( r) varies and for χ = 0. We must first add contributions coming from (13)and (14), then from (15). For this last contribution, we must add two terms, one coming from the variations due to θ *( r′), and the other from the variation due to θ *( r′). These two terms only differ by the notation in the integral variable and are thus equal: we just keep one and double it. We finally add the term due to the variation of the integral in (18), and we get:

(21) This variation must be zero for any value of δf r this requires the - фото 419

This variation must be zero for any value of δf *( r); this requires the function that multiplies δf *( r) in the integral to be zero, and consequently that θ ( r) be the solution of the following equation, written for φ( r):

(22) This is the timeindependent GrossPitaevskii equation It is similar to an - фото 420

This is the time-independent Gross-Pitaevskii equation. It is similar to an eigenvalue Schrodinger equation, but with a potential term:

(23) which actually contains the wave function φ in the integral over d 3 r it is - фото 421

which actually contains the wave function φ in the integral over d 3 r ′; it is therefore a nonlinear equation. The physical meaning of the potential term in W 2is simply that, in the mean field approximation, each particle moves in the mean potential created by all the others, each of them being described by the same wave function φ( r′); the factor ( N — 1) corresponds to the fact that each particle interacts with ( N — 1) other particles. The Gross-Pitaevskii equation is often used to describe the properties of a boson system in its ground state (Bose-Einstein condensate).

ϒ. Zero-range potential

The Gross-Pitaevskii equation is often written in conjunction with an approximation where the particle interaction potential has a microscopic range, very small compared to the distances over which the wave function φ ( r) varies. We can then substitute:

(24) Quantum Mechanics Volume 3 - изображение 422

where the constant g is called the “coupling constant”; such a potential is sometimes known as a “contact potential” or, in other contexts, a “Fermi potential”. We then get:

(25) Whether in this form 2 or in its more general form 22 the equation includes - фото 423

Whether in this form 2 or in its more general form (22), the equation includes a cubic term in φ ( r). It may render the problem difficult to solve mathematically, but it also is the source of many interesting physical phenomena. This equation explains, for example, the existence of quantum vortices in superfluid liquid helium.

δ. Other normalization

Rather than normalizing the wave function φ ( r) to 1 in the entire space, one sometimes chooses a normalization taking into account the particle number by setting:

(26) Quantum Mechanics Volume 3 - изображение 424

This amounts to multiplying by Quantum Mechanics Volume 3 - изображение 425the wave function we have used until now. At each point rof space, the particle (numerical) density n ( r) is then given by:

(27) Quantum Mechanics Volume 3 - изображение 426

With this normalization, the factor ( N — 1) in (25)is replaced by ( N — 1)/ N , which can generally be taken equal to 1 for large N . The Gross-Pitaevskii equation then becomes:

(28) As already mentioned we shall see in 4a that μ is simply the chemical - фото 427

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