Quantum Mechanics, Volume 2

Angular Momentum, Spin, and Approximation Methods

Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë

• Publisher: Wiley-VCH
• Year: 2020
• ISBN:
  9783527822737
• Edition: 2
• Topic: Scienze fisiche
• Subtopic: Teoria quantistica

This new edition of the unrivalled textbook introduces concepts such as the quantum theory of scattering by a potential, special and general cases of adding angular momenta, time-independent and time-dependent perturbation theory, and systems of identical particles. The entire book has been revised to take into account new developments in quantum mechanics curricula. The textbook retains its typical style also in the new edition: it explains the fundamental concepts in chapters which are elaborated in accompanying complements that provide more detailed discussions, examples and applications. * The quantum mechanics classic in a new edition: written by 1997 Nobel laureate Claude Cohen-Tannoudji and his colleagues Bernard Diu and Franck Laloë
* As easily comprehensible as possible: all steps of the physical background and its mathematical representation are spelled out explicitly
* Comprehensive: in addition to the fundamentals themselves, the book contains more than 170 worked examples plus exercises 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. 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. 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.

Chapter XIII
Approximation methods for time-dependent problems

1. A Statement of the problem
2. B Approximate solution of the Schrödinger equation
   1. B-1 The Schrödinger equation in the {|φ_n〉} representation
   2. B-2 Perturbation equations
   3. B-3 Solution to first order in λ
3. C An important special case: a sinusoidal or constant perturbation
   1. C-1 Application of the general equations
   2. C-2 Sinusoidal perturbation coupling two discrete states: the resonance phenomenon
   3. C-3 Coupling with the states of the continuous spectrum
4. D Random perturbation
   1. D-1 Statistical properties of the perturbation
   2. D-2 Perturbative computation of the transition probability
   3. D-3 Validity of the perturbation treatment
5. E Long-time behavior for a two-level atom
   1. E-1 Sinusoidal perturbation
   2. E-2 Random perturbation
   3. E-3 Broadband optical excitation of an atom

A. Statement of the problem

Consider a physical system with Hamiltonian H₀. The eigenvalues and eigenvectors of H₀ will be denoted by E_n and |φ_n〉:

(A-1)

For the sake of simplicity, we shall consider the spectrum of H₀ to be discrete and non-degenerate; the formulas obtained can easily be generalized (see, for example, § C-3). We assume that H₀ is not explicitly time-dependent, so that its eigenstates are stationary states.

At t = 0, a perturbation is applied to the system. Its Hamiltonian then becomes:

(A-2)

with:

(A-3)

where λ is a real dimensionless parameter much smaller than 1 and Ŵ(t) is an observable (which can be explicitly time-dependent) of the same order of magnitude as H₀, and zero for t<0.

The system is assumed to be initially in the stationary state |φ_i〉, an eigenstate of H₀ of eigenvalue E_i. Starting at t = 0 when the perturbation is applied, the system evolves: the state |φ_i〉 is no longer, in general, an eigenstate of the perturbed Hamiltonian. We propose, in this chapter, to calculate the probability P_if(t) of finding the system in another eigenstate |φ_f〉 of H₀ at time t. In other words, we want to study the transitions that can be induced by the perturbation W(t) between the...