Rabi formula

TIME-DEPENDENT PERTURBATION THEORY THE RABI FORMULA... 

If at time f = 0 we start in states Un for the radiation field and ua for the system, then the probability that at time t the radiation field will be in state Un 11 and the system will be in state uh is, in analogy to the Rabi formula of Eq. (3.34.19),... 

The electron hopping frequency may be estimated from time-dependent perturbation theory. If Hab is treated as a constant perturbation, the system will start to oscillate between the two diabatic states once the perturbation is turned on. In a bimolecular reaction, for example, the perturbation is turned on upon formation of the precursor complex, while in a covalently attached (bridged) binuclear system it can be turned on upon reduction (oxidation) of one end of the fully oxidized (reduced) system by an external reagent or by photoexcitation. If the system is in the diabatic reactant state at / = 0, then the probability of it being in the product state at some later time t is given by the Rabi formula [27]. 

In intermediate fields the calculation is more complicated. For I or J = 1/2, the Breit-Rabi formula can be expressed in analogy with the fine structure case, see (2.31). For J = 1/2 we have... 

From the measured level-crossing positions the fine-structure splitting can be calculated using the Breit-Rabi formula for the fine structure, (2.31). 

In Chap. 7 we have discussed how hyperfine structure can be determined by level-crossing spectroscopy. Clearly, alkali atom states can readily be studied using this technique after stepwise excitation. We will here instead choose an example illustrating fine-structure measurements. In Fig. 9.17 the example of the inverted sodium 4d 5/2,3/2 state is given. From the measured level-crossing positions the fine-structiue splitting can be calculated using the Breit-Rabi formula for the fine structure, (2.31). 

We have used here the resonance detect D = C02 — u> — (o = X22-X11 for real w, and a real V-= 1 121" From these simple expressions one can calculate all desired quantities, in particular also the "Rabi formula in equation (90) for the population p2 of the upper level, when at f = 0 the lower level is populated (pi(0) = 1) ... 

The full width (FWHM) is 2V, sometimes called the power broadening width . A few other simple one and multiphoton excitation problems can be solved by equally simple back-of-the-envelope analytical expressions in the quasiresonant approximation. An interesting generalization of the Rabi formula has been given by Shirley for higher odd order resonances with n = 2p + 1 (integer p > 0)... 

The quadrupole interaction vanishes for J = - so that the Breit-Rabi formula gives an exact description of the hyperfine structure, in this case. If the small term gjVjjBM is... 

Fig.18.3. A plot of the Breit-Rabi formula for the case I = The abscissa is given by Cgj + gjin/M) PgB/hvjjpg where hvp pg is the energy difference between the levels F=2 and F=1 in zero field.

Finally, by solving the secular equation, determine the energies of the hyperfine sub-levels in fields of arbitrary strength. Verify the correctness of the results using the Breit-Rabi formula and Fig.18.16. 

Expand the Breit-Rabi formula up to terms of order x and thus obtain a precise expression for the frequency of the Zeeman magnetic resonance transitions AF=0,... 

By expanding the Breit-Rabi formula in the weak field approximation prove that the frequency of the AF= 1, Mp=0 Mp=0 transition is given by... 

AMj=0, AMj= 1 transitions at fields which were not quite large enough for the complete Paschen-Back formula, equation (18.37), to be applicable. By expanding the Breit-Rabi formula up to terras of order... 

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