Hamiltonian time-dependent Schrodinger equation

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

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

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 nonstationary state evolves, governed by the time-dependent Schrodinger equation... 

If the PES are known, the time-dependent Schrodinger 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. 

Since the Hamiltonian is symmetric in space coordinates the time-dependent Schrodinger equation prevents a system of identical particles in a symmetric state from passing into an anti-symmetric state. The symmetry character of the eigenfunctions therefore is a property of the particles themselves. Only one eigenfunction corresponds to each eigenfunction and hence there is no exchange degeneracy. 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

Tully has discussed how the classical-path method, used originally for gas-phase collisions, can be applied to the study of atom-surface collisions. It is assumed that the motion of the atomic nucleus is associated with an effective potential energy surface and can be treated classically, thus leading to a classical trajectory R(t). The total Hamiltonian for the system can then be reduced to one for electronic motion only, associated with an electronic Hamiltonian Jf(R) = Jf t) which, as indicated, depends parametrically on the nuclear position and through that on time. Therefore, the problem becomes one of solving a time-dependent Schrodinger equation ... 

In complete analogy to the diabatic case, the equations of motion in the adiabatic representation are then obtained by inserting the ansatz (29) into the time-dependent Schrodinger equation for the adiabatic Hamiltonian (7)... 

Our main concern in this section is with the actual propagation forward in time of the wavepacket. The standard ways of solving the time-dependent Schrodinger equation are the Chebyshev expansion method proposed and popularised by Kossloff [16,18,20,37 0] and the split-operator method of Feit and Fleck [19,163,164]. I will not discuss these methods here as they have been amply reviewed in the references just quoted. Comparative studies [17-19] show conclusively that the Chebyshev expansion method is the most accurate and stable but the split-operator method allows for explicit time dependence in the Hamiltonian operator and is often faster when ultimate accuracy is not required. All methods for solving the time propagation of the wavepacket require the repeated operation of the Hamiltonian operator on the wavepacket. It is this aspect of the propagation that I will discuss in this section. 

Then any state vector of a system, (f)) that satisfies the time-dependent Schrodinger equation with Hamiltonian H(t) and is represented in terms of the basis states S can also be represented in terms of the basis states D t) via the transformation... 

A formal description of an experiment can be given briefly as follows. At time to a state H i of the full Hamiltonian H is prepared this state then evolves in time according to the time-dependent Schrodinger equation... 

To describe a chemical reaction from a physical standpoint at the nonrelati-vistic level, one must first construct the Hilbert space associated with all quantum states related to the system defined by its molecular hamiltonian, Hm. For the isolated system the time-dependent Schrodinger equation... 

For a time-independent Hamiltonian, the wavefunction satisfying the time-dependent Schrodinger equation contains a multiplicative factor exp(-iEt/h). For positronium this time-dependent factor leads to a probability density proportional to... 

The result for a free electron in an electromagnetic field can be transferred to the Hamiltonian H of an atom by using the same approach. Because the electromagnetic field depends on time, one starts with the time-dependent Schrodinger equation... 

Let us again consider the photodissociation of the linear triatomic molecule with coordinates R and r (Figure 2.1). We want to solve the time-dependent Schrodinger equation (4.1) with the Hamiltonian given in (2.39) and the initial condition (4.4). 

To simplify the notation we have assumed that the light pulse has prepared the system in a single bound state. The probabilities for finding the system in states I J/ n) and 1 2(E,0)) at time t are aij(i) 2 and a2(t) E,/3) 2, respectively. Inserting (7.3) into the time-dependent Schrodinger equation with the full Hamiltonian which also includes the coupling W and utilizing (7.1) and (7.2) yields the coupled equations... 

The detectable signal in the time intervals between the moments of exchanges can be determined by solving the time-dependent Schrodinger equation for the specific conformer however, in this case, the Hamiltonian is independent of time.18 The advantage of this method is its smaller memory requirement its disadvantage is the longer computational time because of the Monte Carlo simulation and that it was not possible to apply it to coupled spin systems so far. 

Here Hq is the molecular Hamiltonian, and fi e(t) is the interaction between the molecule and the laser field in the dipole approximation, where (i is the transition dipole moment of the molecule. Time evolution of the system is determined by the time-dependent Schrodinger equation,... 

Given the Hamiltonian of Eq. (1.42), the dynamics of the particles in the presence of the field are obtained by solving for the wave function VF(R, t) via the time-dependent Schrodinger equation ... 

Let the evolution of the nuclear subsystem be given by a trajectory Q = Q(t). Consequently, the electronic Hamiltonian f/el becomes time dependent [through g(t)], and the state (j>(Q(t), q) of the electronic subsystem is, in general, nonstationary (j)(Q(t), q) obeys the time-dependent Schrodinger equation (20)... 

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