Time-dependent wave equation

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

Munera and Guzman [56] obtained new explicit noncyclic solutions for the three-dimensional time-dependent wave equation in spherical coordinates. Their solutions constitute a new solution for the classical Maxwell equations. It is shown that the class of Lorenz-invariant inductive phenomena may have longitudinal fields as solution. But here, these solutions correspond to massless particles. Hence, in this framework a photon with zero rest mass may be compatible with a longitudinal field in contrast to that Lehnert, Evans, and Roscoe frameworks. But the extra degrees of freedom associated with this kind of longitudinal solution without nonzero photon mass is not clear, at least at the present state of development of the theory. More efforts are needed to clarify this situation. 

Solving the time-dependent wave equation for the electron with initial conditions I Ci(—o°) P = 1 and Cf(—°o) = 0, we obtain the expression for Cf(t) and hence the expression forP(6, u)... 

Equation (4.20) in tb, the so-called time-dependent wave equation, is often considered to be a diffusion equation, as it shows only a first time derivative. 

The Dirac equation was derived in several steps [5, 6], starting with the time-dependent wave equation for a free particle in the Schrddinger representation ... 

The effect of these forces on the nucleus essentially follows Newton s second law, which is re-expressed in quantum mechanics as the time-dependent wave equation [1]. They appear in this equation as a time-dependent potential V(t superimposed upon the much stronger, static potential from the main magnetic field Bq. The effect of this potential V(t), as will be seen in more detail in following sections, is to create a transition probability W between... 

Insertion of (4.9) into the time-dependent wave equation, including the time-dependent perturbation V t) described in the introduction of section 4.2, enables one to deduce the absorption rate... 

T) relates to the state of the relative motion (direction and size of the relative velocity), and k to the inner state of the atoms. One substitutes in the time-dependent wave equation, multiplies with and integrates over the and i -space. In... 

We must now investigate how such transition probabilities as W and W W are derived from first principles. The underlying law is expressed in the (time-dependent) wave equation ih dil//dt = which is the quantum-mechanical version of Newton s second law. Let us once again consider a simple two-level spin system, described by the spin wave function... 

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

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

We present state-to-state transition probabilities on the ground adiabatic state where calculations were performed by using the extended BO equation for the N = 3 case and a time-dependent wave-packet approach. We have already discussed this approach in the N = 2 case. Here, we have shown results at four energies and all of them are far below the point of Cl, that is, E = 3.0 eV. 

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. 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

The time evolution of the wavefunetion E is determined by solving the time-dependent Sehrodinger equation (see pp 23-25 of EWK for a rationalization of how the Sehrodinger equation arises from the elassieal equation governing waves, Einstein s E=hv, and deBroglie s postulate that )i=h/p)... 

Electrons are very light particles and cannot be described by classical mechanics. They display both wave and particle characteristics, and must be described in terms of a wave function, T. Tlie quantum mechanical equation corresponding to Newtons second law is the time-dependent Schrbdinger equation (h is Plancks constant divided by 27r). 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

In the time dependent wave packet analysis (21) the Schrodinger equation... 

We now illustrate the utility of Eq. (27) in relating the RWP dynamics based on the arccosine mapping, Eq. (16), to the usual time-dependent Schrodinger equation dynamics, Eq. (1). We carried out three-dimensional (total angular momentum 7 = 0) wave packet calculations for the... 

Equation (4.a) states that the wave function must obey the time-dependent Schrodinger equation with initial condition /(t = 0) = < ),. Equation (4.b) states that the undetermined Lagrange multiplier, x t), must obey the time-dependent Schrodinger equation with the boundary condition that x(T) = ( /(T))<1> at the end of the pulse, that is at f = T. As this boundary condition is given at the end of the pulse, we must integrate the Schrodinger equation backward in time to find X(f). The final of the three equations, Eq. (4.c), is really an equation for the time-dependent electric field, e(f). 

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. 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

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