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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1. Time evolution

We assume that the ket describing the physical system of N bosons can be written using relation (7)of Complement C XV:

(1) but we now suppose that the individual ket θ is a function of time θ t - фото 497

but we now suppose that the individual ket | θ 〉 is a function of time | θ ( t )〉. The creation operator Quantum Mechanics Volume 3 - изображение 498in the corresponding individual state is then time-dependent:

(2) Quantum Mechanics Volume 3 - изображение 499

We will let the ket | θ ( t )〉 vary arbitrarily, as long as it remains normalized at all times:

(3) Quantum Mechanics Volume 3 - изображение 500

We are looking for the time variations of | θ ( t )〉 that will yield for картинка 501variations as close as possible to those predicted by the exact N -particle Schrodinger equation. As the one-particle potential V 1may also be time-dependent, it will be written as V 1( t ).

1-a. Functional variation

Let us introduce the functional of |Ψ( t )〉:

(4) It can be shown that this functional is stationary when Ψ t is solution of - фото 502

It can be shown that this functional is stationary when |Ψ( t )〉 is solution of the exact Schrodinger equation (an explicit demonstration of this property is given in § 2 of Complement F XV. If |Ψ( t )〉 belongs to a variational family, imposing the stationarity of this functional allows selecting, among all the family kets, the one closest to the exact solution of the Schrodinger equation. We shall therefore try and make this functional stationary, choosing as the variational family the set of kets картинка 503written as in (1)where the individual ket | θ ( t )〉 is time-dependent.

As condition (3)means that the norm of картинка 504remains constant, the second bracket in expression (4)must be zero. We now have to evaluate the average value of the Hamiltonian H ( t ) that, actually, has been already computed in (34)of Complement C XV:

(5) The only term left to be computed in 4contains the time derivative This term - фото 505

The only term left to be computed in (4)contains the time derivative.

This term includes the diagonal matrix element:

(6) For an infinitesimal time dt the operator is proportional to the difference - фото 506

For an infinitesimal time dt , the operator Quantum Mechanics Volume 3 - изображение 507is proportional to the difference Quantum Mechanics Volume 3 - изображение 508, hence to the difference between two creation operators associated with two slightly different orthonormal bases. Now, for bosons, all the creation operators commute with each other, regardless of their associated basis. Therefore, in each term of the summation over k , we can move the derivative of the operator to the far right, and obtain the same result, whatever the value of k . The summation is therefore equal to N times the expression:

(7) Now we know that 8 Using in 6the bra associated with that expression - фото 509

Now, we know that:

(8) Using in 6the bra associated with that expression multiplied by N we get - фото 510

Using in (6)the bra associated with that expression, multiplied by N , we get:

(9) Regrouping all these results we finally obtain 10 1b Variational - фото 511

Regrouping all these results, we finally obtain:

(10) 1b Variational computation the timedependent GrossPitaevskii equation We - фото 512

1-b. Variational computation: the time-dependent Gross-Pitaevskii equation

We now make an infinitesimal variation of | θ ( t )〉:

(11) Quantum Mechanics Volume 3 - изображение 513

in order to find the kets | θ ( t )〉 for which the previous expression will be stationary. As in the search for a stationary state in Complement C XV, we get variations coming from the infinitesimal ket eiχ | δθ ( t )〉 and others from the infinitesimal bra e –iχ〈 δθ ( t )|; as χ is chosen arbitrarily, the same argument as before leads us to conclude that each of these variations must be zero. Writing only the variation associated with the infinitesimal bra, we see that the stationarity condition requires | θ ( t )〉 to be a solution of the following equation, written for | φ ( t )〉:

(12) The mean field operator is defined as in relations 45and 46of Complement C - фото 514

The mean field operator is defined as in relations 45and 46of Complement C XVby a partial trace - фото 515is defined as in relations (45)and (46)of Complement C XVby a partial trace:

(13) Quantum Mechanics Volume 3 - изображение 516

where P φ(t)is the projector onto the ket | φ ( t )〉:

(14) Quantum Mechanics Volume 3 - изображение 517

As we take the trace over particle 2 whose state is time-dependent, the mean field is also time-dependent. Relation (12)is the general form of the time-dependent Gross-Pitaevskii equation.

Let us return, as in § 2 of Complement C XV, to the simple case of spinless bosons, interacting through a contact potential:

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