f 2 | H (I) |i f 1 | H (I) |i in .NET Creator Code 128 Code Set B in .NET f 2 | H (I) |i f 1 | H (I) |i

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f 2 . using .net toinclude code 128 for web,windows application ISSN H (I) . i f 1 H (I) . i. n+1 , n (4.54). a result to be use visual .net Code 128 d shortly. The perturbation method developed previously can still be used with appropriate modi cations to accommodate the fact that the eld is now quantized.

The Schr dinger equation still has the form of Eq. (4.19) but where now H 0 is given o (I) by Eq.

(4.48) and H by Eq. (4.

49). Ignoring all other atomic states except . a. Emission and absorption of radiation by atoms and b , the state vector can be written as (t) = Ci (t). a . n e i Ea t/ h e in t + C f1 (t). b . n 1 e i Eb t/ h e i(n 1) t + C f2 (t). b . n + 1 e i Eb t/ h e i(n+1) t (4.55). where, assuming that (0) = . a . n , Ci (0) = 1 and C f 1 (0) = C f 2 (0) = 0. Following the perturbative method used before, we obtain the rst-order correction for the amplitudes C f 1 and C f 2 associated with the atom being in state . b :. t (1) C f 1 (t). i = h dt f 1 H (I) . i ei(E f 1 Ei )t/ h , (4.56) dt f 2 H (I) . i ei(E f 2 Ei )t/ h ,. i (1) C f 2 (t) = h where the former i s associated with absorption and the latter with emission. The amplitude associated with the atom being in the state . b , regardless of .net vs 2010 Code-128 how it (1) (1) (1) got there, is just the sum of the amplitudes in Eqs. (4.

56), i.e. C f = C f 1 + C f 2 .

From Eqs. (4.50) we obtain.

(1) C f (t). ei( + ba )t .NET Code 128 Code Set C 1 i = (d E 0 )ab (n + 1)1/2 h ( + ba ). ei( ba )t 1 ( ba ) . (4.57). where ba = (E b E a )/ h and where the rst term is due to emission and the second to absorption. If the number of photons is large, n 1, then we can replace n + 1 by n in the rst term and then Eqs. (4.

57) and (4.33) are essentially the same, there being the correspondence between the classical and quantum eld amplitudes (E0 )cl (2iE0 n)quantum. This correspondence between quantum and classical elds has its limits, one being the case when n = 0, as already discussed.

If . b is the excited s Code 128 Code Set A for .NET tate then ba > 0 so if ba then the rst term of Eq. (4.

51) can be dropped, which is again the rotating wave approximation. Of course, if . a is the excited s tate, then ba < 0 and if ba , the second term of Eq. (4.57) can be dropped, and we notice that the remaining term does not vanish even when n = 0, the transition between .

a and b taking place by spontaneous emission. Thus the rotating wave approximation carries over into the case where the eld and the atom are quantized. It can be shown that Fermi s Golden Rule carries over in a similar fashion.

We conclude this section with a eld-theoretic derivation of the Planck distribution law. Suppose we have a collection of atoms interacting resonantly with a quantized eld of frequency = (E a E b )/ h,where . a and b are the atomic s tates with E a > E b . We let Na and Nb represent the populations of atoms in states . a and b respectively. Fu rther, we let Wemis represent the transition rate due to photon emission and Wabs the transition rate due to photon absorption..

4.3 Interaction of an atom with a quantized eld Because the atoms are constantly emitting and absorbing photons, the atomic populations change with time according to. d Na = Na Wemis + Nb Wabs dt d Nb = Nb Wabs + Na Wemis . dt (4.58).
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