Transitions Between States Time-Dependent Perturbation Theory

5 Transitions Between States Time-Dependent Perturbation Theory 

If the change in H is relatively small, we can write the total Hamiltonian of the perturbed system as a sum of the time-independent Ho and a smaller, time-dependent term, H (t)  

To find the wavefunction of the perturbed system, let s express it as a linear combination of the eigenfunctions of the unperturbed system  

Suppose we know that the molecule is in state Pa before we introduce the perturbation H. How rapidly does the wavefunction begin to resemble that of some other basis state, say Pb The answer should lie in the time-dependent Schrodinger equation (Eq. 2.9). Using Eqs. (2.54) and (2.55), we can expand the left-hand side of the Schrodinger equation to  

The right-hand side of the Schrodinger equation can be expanded similarly to ih PadCaldt + PbdCbldt + + CadPa/dt + CbdPbldt + ]. (2.57) 

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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. 

In the framework of the Born-Oppenheimer approximation, radiationless transitions from one surface to another are impossible. (See, e.g., Michl and BonaCit -Koutecky, 1990.) It is therefore necessary to go beyond the Born-Oppenheimer approximation and to include the interaction between different electronic molecular states through the nuclear motion in order to be able to describe such transitions. Using the time-dependent perturbation theory for the rate constant of a transition between a pair of states one arrives at... 

In the quantum-mechanical theories the intersection of the potential energy surfaces is deemphasized and the electron transfer is treated as a radiationless transition between the reactant and product state. Time dependent perturbation theory is used and the restrictions on the nuclear configurations for electron transfer are measured by the square of the overlap of the vibrational wave functions of the reactants and products, i.e. by the Franck-Condon factors for the transition. Classical and quantum mechanical description converge at higher temperature96. At lower temperature the latter theory predicts higher rates than the former as nuclear tunneling is taken into account. 

This is the final result of the first-order time-dependent perturbation theory treatment of light-indueed transitions between states i and f. 

The presence of the electron acceptor site adjacent to the donor site creates an electronic perturbation. Application of time dependent perturbation theory to the system in Figure 1 gives a general result for the transition rate between the states D,A and D+,A. The rate constant is the product of three terms 1) 27rv2/fi where V is the electronic resonance energy arising from the perturbation. 2) The vibrational overlap term. 3) The density of states in the product vibrational energy manifold. 

The interest aroused by the field of radiationless transitions in recent years has been enormous, and several reviews have been published 72-74) Basically, the ideas of Robinson and Frosch 75) who used the concepts on non-stationary molecular states and time-dependent perturbation theory to calculate the rate of transitions between Born-Oppenheimer states, are still valid, although they have been extended and refined. The nuclear kinetic energy leads to an interaction between different Born-Oppenheimer states and the rate of radiationless transitions is given by... 

The simplest approach to understanding the radiation- (light-) induced transition between electronic states is to invoke time-dependent perturbation theory. Thus, one starts from the time-dependent Schrodinger equation... 

Equations 2.85 and 2.86 may be considered the Schrodinger representation of the absorption of radiation by quantum systems in terms of spectroscopic transitions between states i) and /). In the Schrodinger picture, the time evolution of a system is described as a change of the state of the system, as implemented here in the form of the time-dependent perturbation theory. The results hardly resemble the classical relationships outlined above, compare Eqs. 2.68 and 2.86, even if we rewrite Eq. 2.86 in terms of an emission profile. Alternatively, one may choose to describe the time evolution in terms of time-dependent observables, the Heisenberg picture . In that case, expressions result that have great similarity with the classical expressions quoted above as we will see next. 

Spectroscopy involves transitions between states. To discuss transitions, we must use the time-dependent Schrodinger equation, which is harder to deal with than the time-independent equation. Rather than attempt an exact solution, we will use an approximate treatment called time-dependent perturbation theory. 

For the specific case of differential circularly polarized emission, starting with time-dependent perturbation theory, one obtains the following relationship between the time-dependent experimental observable, AI(X, ), the excited state population, Nn, and the differential transition probability, AW = WLeft - Wpjght, following an excitation pulse of polarization, k, at time t = 0... 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

To find the relative intensities of the four AB lines we use the results of time-dependent perturbation theory, as described in Section 2.3. For N spins, the probability Pmn of a transition between states m and n is given by a matrix element of the x component of magnetization or spin ... 

The off-diagonal elements depend on the x and y components of the local field, which contain many fluctuating components oscillating at different frequencies. The parts which oscillate at the ESR frequency ( ) induce transition between the a- and P-states. By making a Fourier analysis of V(f) and using time-dependent perturbation theory [22], the transition probability between the a- and p-states (P< ) is given by... 

Time-dependent perturbation theory shows that, when an atom or molecule in a stationary state is exposed to electromagnetic radiation of frequency v, the molecule may make a transition between two stationary states m and n whose energy difference is hv, provided the transition dipole moment is nonzero for states m... 

The e and the neutrino are not present in the nucleus they are created only in the P-decay process, just as the y quantum in the y decay. In contrast with the strong interaction, the weak interaction between nucleons can be treated as a very weak perturbation. Then, according to the time-dependent perturbation theory of quantum mechanics, the transition probability in unit time Pa) between the initial (i) and final (f) states of the system is proportional to the final state density (pf = dn/dEo) and Ha, where Ha is the matrix element of the Hamiltonian H of the weak interaction ... 

Several quantum mechanical calculations have been made for electron transfer processes between metals and atoms or ions " in gaseous medium. In all the cases, the considerations concern the transition of electrons from a metal state to a bound atomic state or to a free continuum state or vice versa. The calculations of transition probabilities in the cited works have been based on Fermi s golden rule of time-dependent perturbation theory. However, it was pointed out by Gadzuk that the use of the golden rule usually presents a difficult problem if an estimate of the transition probability is desired, because it requires evaluation of a matrix element one must specify initial and final state (wave functions) and an interaction. This is not as straightforward as it seems. In a transition, e.g., between an atomic and a conduction band metal state, the initial and final states are eigenfunctions of different Hamiltonians. It seems meaningless to evaluate matrix elements, if the initial and final states are solutions of different Hamiltonians. 

From time-dependent perturbation theory of quantum-mechanics, it can be stated that a transition between two states ir) and ) is allowed provided that (Vf 77p ) 0. This takes place if v vq (ie, the resonance condition) and the alternative magnetic field Bi(t) is polarized perpendicularly to the static magnetic field Bo. Concerning a spin 7 = 1 (Fig. lb), similar calculations show that only the single-quantum transitions 0) 1> and -1) 0> (and those in the opposite directions) are allowed in the first approximation and occur at the same frequency, given by equation 3. 

The time-dependent perturbation theory in the form of the Fermi s Golden Rule may now be used to obtain Wthe transition probability between the two spin states, assuming = If) for simplicity, so that X = 2 IT. 

In its broadest sense, spectroscopy is concerned with interactions between light and matter. Since light consists of electromagnetic waves, this chapter begins with classical and quantum mechanical treatments of molecules subjected to static (time-independent) electric fields. Our discussion identifies the molecular properties that control interactions with electric fields the electric multipole moments and the electric polarizability. Time-dependent electromagnetic waves are then described classically using vector and scalar potentials for the associated electric and magnetic fields E and B, and the classical Hamiltonian is obtained for a molecule in the presence of these potentials. Quantum mechanical time-dependent perturbation theory is finally used to extract probabilities of transitions between molecular states. This powerful formalism not only covers the full array of multipole interactions that can cause spectroscopic transitions, but also reveals the hierarchies of multiphoton transitions that can occur. This chapter thus establishes a framework for multiphoton spectroscopies (e.g., Raman spectroscopy and coherent anti-Stokes Raman spectroscopy, which are discussed in Chapters 10 and 11) as well as for the one-photon spectroscopies that are described in most of this book. 

The time-dependent perturbation theory that we have used to treat resonance energy transfer and absorption of light assumes that we know that a system is in a given state (state 1), so that the coefficient associated with this state (Ci) is 1, while the coefficient for finding the system in a different state (C2) is zero. The resulting expression for the rate of transitions to state 2 (Eq. 2.61 or 7.8) neglects the possibility of a return to state 1. It can continue to hold at later times only if the transition to state 2 is followed by a relaxation that takes the two states out of resonance. Without such relaxations, the system would oscillate between the two states as described by the coupled equations... 

From a quantum mechanical standpoint, the calculation of the double-differential cross-section is based on time-dependent perturbation theory, which provides the general framework for calculating transition rates between quantum states. 

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