Heisenberg's Quantum Mechanics

Heisenberg's Quantum Mechanics

Mohsen Razavy

Language: English

Pages: 680

ISBN: 9814304115

Format: PDF / Kindle (mobi) / ePub

This book provides a detailed account of quantum theory with a much greater emphasis on the Heisenberg equations of motion and the matrix method. No other texts have come close to discuss quantum theory in terms of depth of coverage. The book features a deeper treatment of the fundamental concepts such as the rules of constructing quantum mechanical operators and the classical-quantal correspondence; the exact and approximate methods based on the Heisenberg equations; the determinantal approach to the scattering theory and the LSZ reduction formalism where the latter method is used to obtain the transition matrix. The uncertainty relations for a number of different observables are derived and discussed. A comprehensive chapter on the quantization of systems with nonlocalized interaction is included. Exact solvable models, and approximate techniques for solution of realistic many-body problems are also considered. The book takes a unified look in the final chapter, examining the question of measurement in quantum theory, with an introduction to the Bell's inequalities.

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with the Hamiltonian H= 1 2 1 p + mω 2 q 2 , 2m 2 (1.138) Infinitesimal Canonical Transformation 21 and determine the coordinate q(t) in terms of q(0) and p(0) we first find the Poisson brackets p(0) , (1.139) {q, H}t=0 = m {{q, H} , H}t=0 = −ω 2 q(0), (1.140) {{{q, H} , H} , H} = − ω2 p(0), m (1.141) and so on. Then from the expansion q(t) = q(0) + (∆t)2 ∆t {q, H}t=0 + {{q, H} , H}t=0 + · · · , 1! 2! (1.142) we get q(t) ω 2 (∆t)2 ω 4 (∆t)4 + − ··· 2! 4! p(0) ω 2 (∆t)3 + + ··· ∆t − m

coordinates, Eq. (3.216) is important in the formulation of the quantum theory of collective phenomena, such as the surface oscillations of a nucleus [36]. 3.11 Summation Over Normal Modes In a number of problems, for example the quantization of free electromagnetic field, it is convenient to formulate the problem in a large cubic box of volume L3 , and then take the limit of L going to infinity. Here by imposing the boundary conditions such as the condition of perfect reflectivity at the

walls of the box we obtain a set of normal modes kx = 2nx π , L ky = 2ny π , L and kz = 2nz π , L (3.230) where nx , ny and nz are integers. The vector n = (nx , ny , nz ) represents a normal mode in the lattice space. Let us consider a small volume located between nx and nx + ∆nx , ny and ny + ∆ny and nz and nz + ∆nz . The 80 Heisenberg’s Quantum Mechanics number of lattice points which is the number of normal modes in this volume is ∆nx ∆ny ∆nz . From (3.230) we find L∆kx = 2π∆nx ,

mechanics the time-reversal invariance changes the sign of momentum p, but leaves the position of the particle r unchanged; r = θ r θ−1 = r, (6.51) p = θ p θ−1 = −p. (6.52) Under this transformation the orbital angular momentum L of the particle as well as its spin angular momentum will also change sign [4],[5]; L = θ L θ−1 = −L, or σ = θσ θ−1 = −σ (6.53) Time-Reversal Invariance 147 S = θ S θ−1 = −S. (6.54) For example Eq. (6.53) follows from the definition of L L = θ (r ∧ p) θ−1 = −r

(p, L, S) = (p · L)VpL (r) + (p · S)VpS (r) + (L · S)VLS (r) + [p · (L ∧ S)] VpLS (r) + [S · (p ∧ L)] VSpL (r) + [L · (p ∧ S)] VLpS (r). (6.82) All these terms commute with the rotation operator with the generator L + S. The requirement of the time reversal transformation means that the last three terms must be set equal to zero. Also the condition that the Hamiltonian should be invariant under parity transformation requires that VpL = VpS = 0. Thus the only remaining term will be the

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