Schrodinger equation initial conditions

In addition to initial conditions, solutions to the Schrodinger equation must obey eertain other eonstraints in form. They must be eontinuous funetions of all of their spatial eoordinates and must be single valued these properties allow T T to be interpreted as a probability density (i.e., the probability of finding a partiele at some position ean not be multivalued nor ean it be jerky or diseontinuous). The derivative of the wavefunetion must also be eontinuous exeept at points where the potential funetion undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This eondition relates to the faet that the momentum must be eontinuous exeept at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

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). 

We have investigated another procedure for reducing the computational expense of the AIMS method, which capitalizes on the temporal nonlocality of the Schrodinger equation and the deterministic aspect of the AIMS method. Recall that apart from the Monte Carlo procedure that we employ for selecting initial conditions, the prescription for basis set propagation and expansion is deterministic. We emphasize the deterministic aspect because the time-displaced procedure relies on this property. 

Gaussian wavepacket propagation, 377-381 initial condition selection, 373-377 nuclear Schrodinger equation, 363-373 Adiabatic-to-diabatic transformation (ADT). 

The KG equation is Lorentz invariant, as required, but presents some other problems. Unlike Schrodinger s equation the KG equation is a second order differential equation with respect to time. This means that its solutions are specified only after an initial condition on bothand d /dt has been given. However, in contrast to k, d /dt has no direct physical interpretation [61]. Should the KG equation be used to define an equation of continuity, as was done with Schrodinger s equation (4), it is found to be satisfied by... 

Suppose they approach each other, interact, then move away. This scattering process is described by a joint Schrodinger equation for lP(ql, q2, t) with initial condition (1.17). The point is that after the interaction has taken place the wave function (4i,42,0 no longer factorizes as in (1.17). The separated molecules cannot be described by separate wave functions, but only by the joint wave function, or else by their density matrices pj and p2. This is considered a paradox by those who have not learned to live with quantum mechanics. ... 

Equation (4.9) is the equation of motion of the Lagrange multiplier that restricts the solution to satisfy the Schrodinger equation it is to be solved subject to the final-state condition (4.10). Equation (4.11) is the Schrodinger equation for our system it is to be solved subject to the initial condition (4.12). The field that results from these calculations is given by... 

For the matrix Uy defined at a time t, we have the solution to the Schrodinger equation with this initial condition ... 

The proton exchange is regulated by the time-dependent Schrodinger equation, which tells us the distribution of the wave packet and the associated probability for finding the proton. At the time of a DNA replication, the proton has to choose sides , and this leads to a new initial condition with the proton fully on one side. 

The decay of the complex is described by exactly the same equations of motion independent of its creation, namely Hamilton s equations in classical mechanics and Schrodinger s equations in quantum mechanics. Only the initial conditions for the ultimate fragmentation step are different because they reflect how the intermediate complex was formed. However, these differences lead to several important implications ... 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

The first term on the right-hand side of this equation, ( W(tf)) = (f(tf) W f(tf)), is the expectation value of the target operator W at the final time tf. The second term represents the cost penalty function for the laser pulses with a time-dependent weighting factor Ait). The third term represents the constraint that the wave function fit) should satisfy the time-dependent Schrodinger equation with a given initial condition. Here i=(t) is the time-dependent Lagrange multiplier. 

In a time-dependent picture, one can imagine an electron being annihilated from a specific valence bond orbital at time t=0. This is obviously not a stationary state of the ion and hence one must follow the time dependent evolution of the system from this initial condition. This requires solving the time-dependent Schrodinger equation which may be written as the following set of coupled equations for the time dependent probabihty amplitudes, aj (t) ... 

The solution of the time-dependent Schrodinger equation (2) with initial condition (3) corresponds to a stationary point of the quantum mechanical action integral... 

Although this equation looks formally like the semiclassical Schrodinger equation (2), we emphasize that it is still different because it is defined in the enlarged Hilbert space X and the phase 0 does not have a definite value, since it is a dynamical variable on the same footing as x. In order to recover the semiclassical equation from Eq. (39), we have to reduce it to an equation defined in the Hilbert space Ji. From a mathematical point of view, this can be done by fixing a particular value of 0, as we did in Section II.A. Physically, this can be achieved, as we show in the following, by choosing the initial condition of the photon field as a coherent state. 

This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with a number state as the initial condition for the photon field 11 0,0) It shows that the solution dressed state vector v /(t) (the transfer state, which in the bare basis is given by / /(0,0) mainly projects on the transfer eigenvector during the process. Additional data of the dressed solution during time are shown in Fig. 21a and 21c. Figure 21a displays the probabilities of being in the bare states 1, 2, and 3 ... 

Comparing Figs. 20a and 21a, we notice that, as expected, the solution of the dressed Schrodinger equation, with a number state as initial condition for the... 

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