By J. J. Sakurai

ISBN-10: 0201067102

ISBN-13: 9780201067101

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Extra info for Advanced Quantum Mechanics Sakurai (1967)

Example text

O)e-i't, a[,," - a[,,(0)ei"t. 78) Finally, we have A(x, t) I ,V4 tL 4 , ^ , 1 ! - ^ l a u . * ( 0 ) e ' " ' e i k ' x - i a t4, 1 . ( 2 . 7 9 ) Note that the quantized field operator A is Hermitian while the classical field A is real. In particular, x and I should not be regarded as the spacetime coordinatesof the photon. Fluctuations and the uncertainty relations. We are now in a position to discuss some of the peculiarities arising from the quantum nature of the radiation field. AIp, nor the total number operator defined by nr:>> N r .

The radiative transition due to this term is known as an electricquadtupole(E2) transition since xp*px:(imlh)lHo,xxl, {Bethe and SalPeter(1957),P. 266. r4s) Becauseof the transversality condition k'e(o) :0, it is legitimate to replace xx by its tracelesspart whose individual componentsare Tii - xixi - (8ir/3)lx l'. r46) Note that it has five independent components which can be written as linear combinations of Yfl. It then follows from the Wigner-Eckhart theorem that the angular momentum selection rule for an E2 ttansition is lJ"-J"l<2

Eq. Meanwhile, according to classical electrodynamics,the time averageof E2 can be equated with the energy density of the electromagneticwave. e2) (E')*u""u*"- nh(clL), where n stands for the number of photons per unit volume. For this we must have n> llL3. e3) In other words, the description of physical phenomena based on classicalelectrodynamicsis reliable when the number of photons per volume /u3is muchgreater than one. Thus the classical approximation is an extremely good one. The historical development of quantum mechanicswas guided by an analogy betweenthe electron and the photon both of which were recognizedto exhibit the famous wave-particle duality.

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Advanced Quantum Mechanics Sakurai (1967) by J. J. Sakurai

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