Time dependent perturbation theory

When dealing with time-dependent fields one has to find solutions to the time-dependent electronic Schrodinger equation 

In the rest of this chapter we will not consider time-dependent magnetic perturbations and have therefore neglected the second-order perturbation Hamiltonian Gen- 

In the length gauge, Eq. (2.122), the operator could be the electric dipole or quadrupole operator, defined in Appendix A. It depends on coordinates and momenta of the electrons but it is independent of time, whereas we assume that the time-dependent field. F. (t) does not depend on any electronic variables. The subscript p - again denotes components of a tensor of appropriate rank. On the other hand, in the velocity gauge, Eq. (2.125), the operator is equal to the total canonical 

For monochromatic linear polarized radiation in the dipole approximation the time dependence of a component of the field vector can be expressed as 

For a general pulse of coherent but polychromatic radiation this then becomes 1 1 

In time-dependent perturbation theory we begin with a hamiltonian Ho whose eigenfunctions and eigenvalues 4° are known and form a complete orthonormal set. We then turn on a time-dependent perturbation Hi(t) and express the new time-dependent 

Introducing the expression of Eq. (B.57) on the left-hand side of this equation we obtain 

multiplying both sides of this last equation from the left by (pj and using the completeness and orthonormality properties of the unperturbed wavefunctions, we obtain 

This is very useful for finding the transition probability between states of the system induced by the perturbation Suppose that at r = 0 the system is in the eigenstate i of Ho, in which case c,(0) = 1 and Cj 0) = 0 for Then, writing the above expression in 

If we assume that the perturbation Hi(t) has the simple time dependence 

3 TIME DEPENDENCE AS A PERTURBATION 7.3.1 Time-Dependent Perturbation Theory 

We introduce a set of time-independent state wave functions which are eigenfunctions of a time-independent Hamiltonian  

In the absence of time-dependent perturbations. Equation 7.5 holds for a solution of the time-dependent SE. If time dependence is introduced as a perturbation V, the time-independent state wave functions /n may still be used, but the coefQcients Q(t) will be time dependent. P(t) satisfies the time-dependent SE  

Equation 7.28 is a systan of simultaneous linear homogeneous differential equations called the equation of motion for C. Note that the summation includes n = m. In matrix form. Equation 7.4 may be written as 

One may now derive expressions for the coefficients C j for different situations. Assume, for instance, that the system is originally (t = -oo) in a stationary state n of the time-independent Hamiltonian, so that 

Equation (2.73) is particularly useful in cases where the time evolution carried by Hq can be easily evaluated, and the effect of Eis to be determined perturbatively. Equation (2.73) is a direct route to such a perturbation expansion. We start by integrating it to get 

Note that the order of the operators r](Z) inside the integrand is important These operators do not in general commute with each other because they are associated with different times. It is seen from Eq. (2.76) that the order is such that operators associated with later times appear more to the left. 

Problem 2.9. Show that Eq. (2.74) is equivalent to the operator identity 

Quantum dynamics using the time-dependent Schrodinger equation 

The polarizabilities a, and y are often calculated by the methods of time-dependent perturbation theory, which I shall now describe. 

Strictly speaking, all perturbations must be time-dependent we cannot arrange for them to have been in existence since t = —oo and we must instead switch them on. As in an electrical circuit, such switching-on causes initial transient behaviour that eventually dies away to leave a steady state. 

The corresponding zero-order time-dependent states are (from Chapter 0) 

In this section I will write h for the time-dependent states and ijr for the time-independent ones. The jri may themselves depend on the space and spin variables of all the particles present. 

If I write the state of the perturbed system l v(t) then it must satisfy the time-dependent Schrodinger equation 

Our derivation of Equations (2.1) and (2.2) follows very closely the presentation of Loudon (1983 ch.2). The basic concept is to describe the molecule quantum mechanically, the photon field classically, and to treat the interaction between them in first-order perturbation theory. 

In spectroscopy, we start with a system in some stationary state, expose it to electromagnetic radiation (light), and then observe whether the system has made a transition to another stationary state. The radiation produces a time-dependent potential-energy term in the Hamiltonian, so we must use the time-dependent Schrbdinger equation. The most convenient approach here is an approximate one called time-dependent perturbation theory. 

Let the system (atom or molecule) have the time-independent Hamiltonian in the absence of the radiation (or other time-dependent perturbation), and let H t) be the time-dependent perturbation. The time-independent Schrbdinger equation for the unperturbed problem is 

First suppose that H t) is absent. The unperturbed time-dependent Schrodinger equation is 

The system s possible stationary-state functions are given by (7.98) as T = where the functions are the eigenfunctions of H [Eq. 

Now suppose that H t) is present.The function (9.117) is no longer a solution of the time-dependent Schrodinger equation. However, because the unperturbed functions T form a complete set, the true state function can at any instant of time be expanded as a linear combination of the functions according to T = S t i k-Because H is time dependent, will change with time and the expansion coefficients bk will change with time. Therefore, 

The system s possible stationary-state state functions are given by (7.99) as = exp —iElt/h)tpl, where the functions are the eigenfunctions of H° [Eq. (9.112)]. Each is a solution of (9.114). Moreover, the linear combination 

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The selection niles are derived tlnough time-dependent perturbation theory [1, 2]. Two points will be made in the following material. First, the Bolu frequency condition states that the photon energy of absorption or emission is equal... 

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

The Time Dependent Processes Seetion uses time-dependent perturbation theory, eombined with the elassieal eleetrie and magnetie fields that arise due to the interaetion of photons with the nuelei and eleetrons of a moleeule, to derive expressions for the rates of transitions among atomie or moleeular eleetronie, vibrational, and rotational states indueed by photon absorption or emission. Sourees of line broadening and time eorrelation funetion treatments of absorption lineshapes are briefly introdueed. Finally, transitions indueed by eollisions rather than by eleetromagnetie fields are briefly treated to provide an introduetion to the subjeet of theoretieal ehemieal dynamies. 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

The mathematical machinery needed to compute the rates of transitions among molecular states induced by such a time-dependent perturbation is contained in time-dependent perturbation theory (TDPT). The development of this theory proceeds as follows. One first assumes that one has in-hand all of the eigenfunctions k and eigenvalues Ek that characterize the Hamiltonian H of the molecule in the absence of the external perturbation ... 

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

The tools of time-dependent perturbation theory can be applied to transitions among electronic, vibrational, and rotational states of molecules. 

Time dependent perturbation theory provides an expression for the radiative lifetime of an excited electronic state, given by Tr ... 

In the time-dependent perturbation theory [Landau and Lifshitz 1981] the transition probability from the state 1 to 2 is related with the perturbation by the golden rule,... 

R. Dickman, I. Jensen. Time-dependent perturbation theory for nonequilibrium lattice models. Phys Rev Lett 67 2391-2394, 1991. 

The time-dependent Hartree-Fock theory was first discussed by Dirac (1930b) and subsequently in perturbative form by Dalgamo and Victor (1966). Its relationship to time-dependent perturbation theory has been discussed by Langhoff, Epstein and Karplus (1972). 

The statistical matrix is then computed via Eq. (7-78). When the expectation value of the energy, Tr (pH), is then calculated in different orders of V, the successive orders of time-dependent perturbation theory emerge. 

Its poles are determined to any order of by expansion of M. However, even in the lowest order in the inverse Laplace transformation, which restores the time kinetics of Kemni, keeps all powers to Jf (t/xj. This is why the theory expounded in the preceding section described the long-time kinetics of the process, while the conventional time-dependent perturbation theory of Dirac [121] holds only in a short time interval after interaction has been switched on. By keeping terms of higher order in i, we describe the whole time evolution to a better accuracy. 

Analysis of dynamics in terms of eigenstates, both for discrete and continuous spectra. Time dependent perturbation theory. 

Time-dependent perturbation theory, electron nuclear dynamics (END), molecular systems, 340-342... 

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. 

Time-dependent perturbation theory and transition probabilities... 

The power dissipation is computed by standard time-dependent perturbation theory (7.2.1). 

Time-dependent perturbation theory shows that the linewidth of an n-quantum transition, generated by a single pumping frequency, should be 1/n of the corresponding... 

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