Time-dependent Hamiltonian

Ho is the unperturbed Hamiltonian, describing the Zeeman interaction and Hx(t) is the dipole-dipole Hamiltonian, time-dependent through variation in the orientation of the r/s vector. The DD Hamiltonian can conveniently be expressed using scalar contraction of irreducible tensors (16). 

Dynamic processes on the molecular level influence the nuclear spin system by rendering the spin Hamiltonian time dependent. Depending on the relation between the characteristic times of the molecular motions, T, and the strength of the modulated interaction, coj, one can identify different regimes. 

In systems where the Hamiltonian has time dependence (such as in the absorption of electromagnetic radiation), the separation of the TDSE into spatial and time differential equations is generally not possible or very difficult. One approach to solving these problems is to treat the time dependence in the Hamiltonian as a perturbation of the time independent Hamiltonian time-dependent perturbation theory). The Hamiltonian for the system is separated into a time dependent part, H, (the perturbation), and a time independent part. Ho. The TDSE is written as follows ... 

After second quantization, a general basis is then created. For the characterization of QBs, we need to make the Hamiltonian time-dependent. Let us take the help of temporal evolution of number of bosons at each site of the system ... 

The time-dependent Sclirodinger equation allows the precise detemiination of the wavefimctioii at any time t from knowledge of the wavefimctioii at some initial time, provided that the forces acting witiiin the system are known (these are required to construct the Hamiltonian). While this suggests that quaiitum mechanics has a detemihiistic component, it must be emphasized that it is not the observable system properties that evolve in a precisely specified way, but rather the probabilities associated with values that might be found for them in a measurement. 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

We start from the time-dependent Sclirodinger equation for the state fiinction (wave fiinction (t)) of the reactive molecular system with Hamiltonian operator // ... 

There are numerous methods for solving the time dependent Scln-ddinger equation (A3.13.43). and some of them were reviewed by Kosloff [118] (see also [119. 120]). Wlienever projections of the evolving wave fiinction on the spectroscopic states are usellil for the detailed analysis of the quanPun dynamics (and this is certainly the case for tlie detailed analysis of IVR), it is convenient to express the Hamiltonian based on spectroscopic states I... 

In the language of quanPim meehanies, the time-dependent B -field provides a perturbation with a nonvanishing matrix element joining the stationary states a) and P). If the rotating field is written in temis of an amplitude a perturbing temi in tlie Hamiltonian is obtained... 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

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... 

They unfold a connection between parts of time-dependent wave functions that arises from the structure of the defining equation (2) and some simple properties of the Hamiltonian. 

Interaction with light changes the quantum state a molecule is in, and in photochemistry this is an electronic excitation. As a result, the system will no longer be in an eigenstate of the Hamiltonian and this nonstationaiy state evolves, governed by the time-dependent Schrddinger equation... 

If the PES are known, the time-dependent Schrbdinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrodinger equation... 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

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 ... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

Schrddinger s equations are usually written in a more succinct manner by invoking the Hamiltonian operator H, so for example the time-dependent equation for a single particle... 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

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