Theory of transition probabilities

Spectroscopy is concerned with the observation of transitions between stationary states of a system, with the accompanying absorption or emission of electromagnetic radiation. In this section we consider the theory of transition probabilities, using time-dependent perturbation theory, and the selection rules for transitions, particularly those relevant for rotational spectroscopy. 

We will consider a two-level system with states f-a and /// which have energies Ea and Eb. The system is acted upon by a time-dependent perturbation, 

We look for a solution of the time-dependent Schrodinger equation 

If we substitute (6.266) into (6.265) and (6.264) we find that the coefficients satisfy the 

We now suppose that at time t = 0 the system starts in // in other words Ca(0) = 1, 0,(0) = 0. Integration of equation (6.268) with Ca = 1 gives the first-order correction to Cb, and after a time T we find 

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In these cases the critical surface can be localized relatively easily at some intersection line or surface of the electronic states involved. In addition, however, the transition probabilities must be included in (see equation (1.90)) these are different at different points of the surface. For this, the theory of transition probabilities of Landau and Zener and extended theories must be used. For details see refs. 6, 66f. 

This is the one dimensional version of what is usually called the Bom approximation in scattering theory. The transition probability obtained from equation A3.11.43() is... 

All ab initio applications of multiple scattering theory in dilute substitutional alloys rely on the one-to-one correspondence configuration. This holds both for the calculation of transition probabilities [7], represented by Eq. (7), and the electronic structure [8], represented by the Green s function equation [9]... 

In Sect. 2.1, the electron transfer rate was defined as the Boltzmann average of transition probabilities, which were calculated through time-dependent perturbation theory by using the Born-Oppenheimer and Frank-Condon approx-... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

In case the collision takes place according to Newtonian mechanics, the relation (1.3) can be proved by means of Liouville s theorem. In quantum mechanics, Eq. (1.3) is practically one of the postulates of the theory, following directly from quantum mechanical calculations of transition probabilities from one state to another. For our present purpose, considering that this is an elementary discussion, we shall simply assume the correctness of relation (1.3). This relation is sometimes called the principle of microscopic reversibility. 

This is the famous result of Landau and Teller. In many of the more detailed theories, to be discussed below, this result appears explicitly as the principal contribution to the dependence of transition probability on temperature. In practical units, equation (6) is frequently written in the form ... 

In first order of the perturbation theory the transition probability per unit time is described by the well-known golden rule... 

Transition probabilities in fact provide a particularly severe test of atomic calculations, because they are rather sensitive to the wavefunctions of both levels involved in the transition and to the approximations used, especially when electron correlations and relativistic effects are considered. On a more fundamental level, measurements of transition probabilities are also being used to explore the non-conservation of parity predicted by the unified theories of the weak and electromagnetic interaction here, highly forbidden transitions must be tested for an admixture of allowed transition probabilities [8,9]. 

The calculation of transition probabilities on the new quantum theory is thus essentially an evaluation of integrals of the... 

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