Transition probability per unit time

The absorption spectmm, 0(01), is the ratio of transition probability per unit time/incident photon flux. The incident photon flux is the number of photons per unit area per unit time passing a particular location, and is... 

Actually, we are always interested only in the transition probability per unit time to a group of final states with density pf = dnfldEf. This transition rate is given by... 

The expression for the transition probability per unit time W taking into account the process of activation has the form... 

The problem of the physical meaning of the quantity Hx and of the reorganization energy of the medium Es has been analyzed in Ref. 11. Following Ref. 11, we write the expression for the transition probability per unit time in the form3... 

To formulate the basic model, we consider the transfer of a proton from a donor AHZ,+1 to an acceptor B 2 in the bulk of the solution. For reactions in the condensed phase, at any fixed distance R between the reactants, the transition probability per unit time W(R) may be introduced. Therefore, we will consider first the transition of the proton at a fixed distance R and then we will discuss the dependence of the transition probability on the distance between the reactants. 

Let us assume that all the nuclear subsystems may be separated into several subsystems (R, q9 Q, s,...) characterized by different times of motion, for example, low-frequency vibrations of the polarization or the density of the medium (q), intramolecular vibrations, etc. Let (r) be the fastest classical subsystem, for which the concept of the transition probability per unit time Wlf(q, Q,s) at fixed values of the coordinates of slower subsystems q, Q, s) may be introduced. 

We may also introduce the transition probability per unit time at fixed values of the coordinates of slower subsystems, Wlf(q9 Q) and WfXq, < ), and consider the master equations for the corresponding probability densities RXq, Q) and Rf(q, Q), etc. 

Thus, in this limit, P(t) increases linearly with t and the concept of the transition probability per unit time w may be introduced. The calculation for long times leads to w decreasing with an increase... 

Thus unlike the previous case where the transition probability per unit time exists at some small time and is determined by the frequency characteristics of the reactive oscillator, here the concept of the transition probability per unit time exists only at some sufficiently long time. Note two more differences between the formulas (161)-(162) and (171)-(172). In the first case the frequency factor transition probability (i.e., preexponential factor) is determined mainly by the frequency of the reactive oscillator co. In the second case it depends on the inverse relaxation time r l = 2T determined by the interaction of the reactive oscillator with the thermal bath. 

Fig. 8.24. Schematic behaviour of gas mass, total mass and metallicity in the Simple model (left), the extreme inflow model of Larson (1972) (middle) and a model with time-decaying inflow (right). The abscissa is u = /( co(t )dt where o> is the (constant or otherwise) transition probability per unit time for gas to change into stars. The initial mass has been taken as unity in each case.

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

The coefficient Cb ,(t) is then obtained by a standard first-order perturbation calculation which takes into account the initial conditions defined by Eq. (7). This gives the transition probability per unit time from the initial state Xav l a to the isoenergetic continuum of states Xbw l b in the form ... 

In the example discussed above, the transition X- X sta s simply for a single spin flip at a randomly chosen lattice site, and W(X - ) = 1 if 5< 0 while W(X- X ) = Q — 3 lkgT) for >0 should be interpreted as transition probability per unit time. Note that other choices for W would also be possible provided they satisfy the principle of detailed balance ... 

The photoelectric cross-section o is defined as the one-electron transition probability per unit-time, with a unit incident photon flux per area and time unit from the state to the state T en of Eq. (2). If the direction of electron emission relative to the direction of photon propagation and polarization are specified, then the differential cross-section do/dQ can be defined, given the emission probability within a solid angle element dQ into which the electron emission occurs. Emission is dependent on the angular properties of T in and Wfin therefore, in photoelectron spectrometers for which an experimental set-up exists by which the angular distribution of emission can be scanned (ARPES, see Fig. 2), important information may be collected on the angular properties of the two states. In this case, recorded emission spectra show intensities which are determined by the differential cross-section do/dQ. The total cross-section a (which is important when most of the emission in all direction is collected), is... 

The starting point for all calculations of transition probabilities is the well-known formula (22) sometimes called the Golden Rule. It expresses the transition probability per unit time A in terms of the density of final states... 

An elementary treatment of a three-level system under pulsed excitation was given in Sec. II.C. Pollack treats the steady-state condition and the turn-off condition as well. These cases are quite interesting and deserve further discussion. Figure 51 shows the three-level system he analyzed. The quantities Ni(f), N2(f), and N3(f)are the electron-distribution functions (populations) and a9 b, and c represent the total transition probabilities per unit time. The boundary condition Nx + N2 + N = JV =constant is assumed. 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Here W(y2 yi) is the transition probability per unit time from yt to y2 and hence... 

Exercise. The transition probability per unit time for the velocity V of a Rayleigh piston in a gas is... 

Fig. 3.8. Energy levels, transition frequencies and transition probabilities per unit time (Wo. W and W2) in a magnetically coupled I-S system ((A) dipolar coupling (B) contact coupling). A is the contact hyperfine coupling constant The order of the levels is for g, < 0.

The element p 7 e = p 71 cos (0), where 0 is the angle between these two vectors. However this angle is ji/2 different from the coordinate angle evaluated in d3r = r drd sin(0) 0 + n/2. This means that the transition probability per unit time assumes the form... 

For a quantitative treatment of establishing connections between vibronic coupling and vibrational progressions in electronic spectra, band profiles from vibronic wavefunctions must be calculated using common procedures of time-dependent perturbation theory and Fermi s golden rule [57], For emission, e.g., the transition rate which is the transition probability per unit time summed over... 

The probability rises linearly with time and therefore it leads to a constant transition rate, i.e., transition probability per unit time interval,... 

When a(f) = 1, the field E t) in Eq. (7.24) describes a continuous wave with amplitude Eq. The transition probability to the excited state is given by (X2(t) x2(t), and in this case a constant transition probability per unit time is found (after a few oscillations of the electromagnetic field). For a direct reaction, this is equal to the rate constant of Eq. (7.5), kn(hv). Using Eq. (7.28), it is found [3,4] that... 

The transition probabilities per unit time and per scattering center in Eqs (B.ll) and (B.12) are for a collision process, often written in a more explicit form to emphasize the particular transition considered ... 

In order to calculate the IR and Raman transition probabilities per unit time between the vibration state, i, to the vibration state, j, it is necessary to express in quantum mechanical terms the operator describing the interaction between the molecule and the electromagnetic radiation, which is given by... 

The microscopic processes occurring in a system, along with their corresponding transition probabilities per unit time, are an input to a KMC simulation. This information can be obtained via the multiscale ladder using DFT,... 

The net-event KMC (NE-KMQ or lumping approach has been introduced by our group. The essence of the technique is that fast reversible events are lumped into an event with a rate equal to the net, i.e., the difference between forward and backward transition probabilities per unit time (Vlachos, 1998). The NE-KMC technique has recently been extended to spatially distributed systems (Snyder et al., 2005), and it was shown that savings are proportional to the separation of time scales between slow and fast events. The method is applicable to complex systems, and is robust and easy to implement. Furthermore, the method is self-adjusted, i.e., it behaves like a conventional KMC when there is no separation of time scales or at short times, and gradually switches to using the net-event construct, resulting in acceleration, only as PE is approached. A disadvantage of the method is that the noise is reduced. 

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