Time-dependent Hamiltonians

Much of the formal discussion below, as well as the two-level example discussed above deal with systems and processes characterized by time-independent Hamiltonians. Many processes of interest, however, are described using time-dependent Hamiltonians a familiar example is the semi-classical description of a system interacting with the radiation field. In this latter case the system-field interaction can be described by a time-dependent potential, for example. 

We can formalize this type of approximation in the following way. Consider a Hamiltonian that describes two interacting systems, 1 and 2. In what follows we use M, and M2, R2 as shorthand notations for the masses and coordinates of systems 1 and 2, which are generally many-body systems 

In writing (2.35) we have deviated from our standard loose notation that does not usually mark the difference between a coordinate and the corresponding operator. For reasons that become clear below we emphasize that Ri,R2 are operators, on equal footings with other operators such as H or V. 

Next we assume that the solution of the time-dependent Schrodinger equation can be written as a simple product of normalized wavefunctions that describe the 

64 Quantum dynamics using the time-dependent Schrodinger equation 

The time-dependent Schrodinger equation for the overall system is 

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It remains to investigate the zeros of Cg t) arising from having divided out by. The position and number of these zeros depend only weakly on G, but depends markedly on the fomi that the time-dependent Hamiltonian H(x, () has. It can be shown that (again due to the smallness of ci,C2,...) these zeros are near the real axis. If the Hamiltonian can be represented by a small number of sinusoidal terms, then the number of fundamental roots will be small. However, in the t plane these will recur with a period characteristic of the periodicity of the Hamiltonian. These are relatively long periods compared to the recurrence period of the roots of the previous kind, which is characteristically shorter by a factor of... 

One may attempt to approximate to such an experimental situation by considering a subsystem with small dimensions in the direction of the flow, so that a single temperature may be sufficiently precise in describing it. In this model one would have to provide a time-dependent hamiltonian operating in such a way as to feed energy into the system at one boundary and to remove energy from the other boundary. We would therefore be obliged to discuss systems with hamiltonians that are explicitly functions of time, and also located on the boundaries of the macrosystem. 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

A physically acceptable theory of electrical resistance, or of heat conductivity, must contain a discussion of the explicitly time-dependent hamiltonian needed to supply the current at one boundary and remove it at another boundary of the macrosystem. Lacking this feature, recent theories of such transport phenomena contain no mechanism for irreversible entropy increase, and can be of little more than heuristic value. 

In the stochastic theory of lineshape developed by Blume [31], the spectral lines are calculated under the influence of a time-dependent Hamiltonian. The method has been successfully applied to a variety of dynamic effects in Mossbauer spectra. We consider here an adaptation due to Blume and Tjon [32, 33] for a Hamiltonian fluctuating between two states with axially symmetric electric field gradients (efg s), the orientation of which is parallel or perpendicular to each other. The present formulation is applicable for states with the same... 

An important class of nonequilibrium systems are those in which mechanical work, either steady or varying, is performed on the subsystem while it is in contact with a heat reservoir. Such work is represented by a time-dependent Hamiltonian, t), where p(f) is the work parameter. (For example, this... 

To this end, we expand the time-dependent Hamiltonian (32), written in diagonal coordinates P and Q, around the saddle point of the autonomous potential to obtain... 

For time-dependent Hamiltonian systems we chose in Section IVB to use a normal form that decouples the reactive mode from the bath modes, but does not attempt a decoupling of the bath modes. This procedure is always safe, but in many cases it will be overly cautious. If it is relaxed, the dynamics within the center manifold is also transformed into a (suitably defined) normal form. This opens the possibility to study the dynamics within the TS itself, as has been done in the autonomous case, for example in Ref. 107. One can then try to identify structures in the TS that promote or inhibit the transport from the reactant to the product side. 

Fet us consider a system described by an explicitly time-dependent Hamiltonian p, q.t) where (p, q) = z is a point in phase space. Hamilton s equation of motion are... 

For simplicity, let us consider a molecular system with a Hamiltonian J o(z) that is coupled to a harmonic spring with spring constant k and a time-dependent minimum r(t). The explicitly time-dependent Hamiltonian of the complete system is then... 

However, in the presence of a strong quadrupolar interaction, the second-order cross-terms between the quadrupolar interaction and the CSA [27, 28] or the heteronuclear dipolar interaction [29, 30] also have to be taken into account. For instance, upon including the CSA in the expression of the time-dependent Hamiltonian in (9), the second-order term becomes... 

Another situation of interest concerns a time-dependent Hamiltonian, i.e. which transforms from (go,Po) at to (q,p) at t. 

For time-dependent Hamiltonians, one can reformulate the problem with the (t, l ) scheme,227 in which the time is treated as an extra degree of freedom. Thus, the techniques developed for stationary problems can be applied in a straightforward manner. Applications of recursive methods to laser-driven dynamics have been reported by several authors.99,228-230... 

If the time-dependent Hamiltonian under MAS commutes with itself at all points in time, i. e., TLih) = 0 for all values of and t2, magic-angle spinning will lead to... 

If the time-dependent Hamiltonian does not commute with itself at all times, then one does not necessarily observe a sharp side-band spectrum under MAS. Only for spinning... 

We now wish to write the ME explicitly for time-dependent Hamiltonians of the following form [11] ... 

The time-evolution generated by the time-dependent hamiltonian is given by a time-ordered exponential form,... 

For systems with time-dependent Hamiltonians one cannot use the eigenstate representation to easily solve the problem. Even an instantaneous diagonaliza-tion at every moment in time does not cure this problem because one would... 

One of the main assets of the time-dependent theory is the possibility of treating some degrees of freedom quantum mechanically and others classically. Such composite methods necessarily lead to time-dependent Hamiltonians which obviously exclude time-independent approaches. We briefly outline three approximations that are frequently used in molecular dynamics studies. To be consistent with the previous sections we consider the collinear triatomic molecule ABC with Jacobi coordinates R and r. 

To describe the motion or the position of the two spins (such as 13C and H in Fig. 1) with respect to the external field (or the laboratory-fixed coordinate system), it is customary and useful in treating DD relaxation to introduce relevant space functions. These functions, F(q)(t), appear in the time-dependent Hamiltonian describing DD interactions, which can be written as a product of two second-rank tensors25 A and F ... 

We assume further that this system is subjected to a weak external perturbation. This perturbation may be as weak as one wishes it will be treated semiclassically by a time-dependent hamiltonian (an incident electromagnetic field, for example). 

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