Wave packet probability density

Figure 3 The time evolution of the wave packet prepared by the pump process. Contours of the wave packet probability density are displayed for different times.

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... 

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

To obtain a first impression of the nonadiabatic wave-packet dynamics of the three-mode two-state model. Fig. 34 shows the quantum-mechanical probability density P (cp, f) = ( (f) / ) (p)(cp ( / (f)) of the system, plotted as a function of time t and the isomerization coordinate cp. To clearly show the... 

We turn now to study the properties of the metastable state in more detail. We, therefore, concentrate on the long-time behavior, i.e., t > f0, and defer the discussion of the short-time dynamics to a later section. Figure 1.3 shows snapshots of the probability density of the evolving wave packet at different... 

Figure 1.5 Snapshots of the probability density (solid line, left y-axis) at different times. The phase S(x,f), see Eq. (3), of the wave packet is also displayed (dashed line, right y-axis). Note that at times different than zero, the phase is x dependent, meaning that we are not in a stationary state. Furthermore, note that in the interaction region, i.e., between the barriers, the phase is approximately constant.

Figure 1.7 The probability density of the wave packet at time t = 200 on a logarithmic scale. Note that outside the interaction region, we have an exponentially diverging function.

At this point we will, briefly, describe some of the fundamental qualitative differences between a quantum mechanical and a classical mechanical description. First of all, a trajectory R(t) is replaced by a wave packet, which implies that a deterministic description is replaced by a probabilistic description. x(R,t) 2 is a probability density, giving the probability of observing the nuclei at the position R at time t. In... 

Fig. 4.2.1 The probability density associated with the Gaussian wave packet. The most probable position is at x = xt, which also coincides with the expectation (average) value of the time-dependent position. The width is related to the time-dependent uncertainty (Ax)t, i.e., the standard deviation of the position.

Since we have scattering from wave packets uniformly distributed over 6, the total scattering probability into the solid angle element dCl is / dbP(dil <— b). If this number is multiplied by the (relative) flux density of molecules in the beam, we get the flux of molecules that show up in dfl. Thus the cross-section is simply... 

Fig. 6.12. Probability density of the wavepacket after 30 fs of field-free propagation of the wave packet shown in Fig. 6.1 as a function of a the coordinates of the proton b the coordinates of the N=C bond vector c the momentum of the proton (d) the momentum of the N=C bond vector...

The magnitude of the wave function and therefore also the probability density at any point do not change with time. To discuss electron dynamics we must consider linear combinations of energy eigenstates of different energy. The convenient choice is a trave packet. In particular, we construct a packet, using states with wave numbers near ko and parallel to it in the Brillouin Zone ... 

One has to emphasize that Eqs. (82) and (83) do not involve the Born-Oppenheimer approximation although the nuclear motion is treated classically. This is an important advantage over the quantum molecular dynamics approach [47-54] where the nuclear Newton equations (82) are solved simultaneously with a set of ground-state KS equations at the instantaneous nuclear positions. In spite of the obvious numerical advantages one has to keep in mind that the classical treatment of nuclear motion is justified only if the probability densities (R, t) remain narrow distributions during the whole process considered. The splitting of the nuclear wave packet found, e.g., in pump-probe experiments [55-58] cannot be properly accounted for by treating the nuclear motion classically. In this case, one has to face the complete system (67-72) of coupled TDKS equations for electrons and nuclei. 

We note that the choice of a Hilbert space of square-integrable functions as the state space of the evolution equation is perfectly natural for the Schrodinger equation. The solutions of the Schrodinger equation are in the Hilbert space L (R ) (they have only one component), and the expression tj x,t) is interpreted as a density for the position probability at time t. Hence the norm of a Schrodinger wave packet,... 

Figure 1 shows four snapshots from the (numerically calculated) time-evolution -0(t) of the initial function -tpoix) = iVexp(—x /2)(l,l). This is a spinor with Gaussian initial functions in both components. More precisely, the pictures show (according to our interpretation) the position probability density 4> x)f + 1 2 x) p. We see that the shape of the wave packet at later times shows strange distortions, similar to distortions caused by interference phenomena. Moreover, consider the expectation value of the position x in the state which is (ac-... 

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