next up previous
Next: Minimising wave packet Up: Influence of the wave Previous: Indeterminacy as the function

Examples (with figures)

  In this section specific examples of wave packets are given. For each wave packet its real space and spectral properties are analysed. We then calculate the $\Delta x$ and $\Delta k$indeterminacies for each initial state and examine how does $\Delta x \cdot \Delta k$ differ from 1/2 . Finally the time dependent wave functions and probability densities are worked out and investigated.

For simplicity the wave number and position expectation values of all but the Gaussian wave packet are chosen to be zero. The effect of the non zero $x_0=<{\bf {x}}\gt$ and $k_0=<{\bf {k}}\gt$is demonstrated in Section(VA).


Table 1. summarises the important formulae for each example treated in this section.

Graphs of the initial states in coordinate and wave number representation are plotted in Fig. 1. while their decay forms are demonstrated in Fig. 2. The time dependence of the wave functions is shown in Fig. 3. and the evolution of their indeterminacies can be seen in Fig. 4.

(Click to enlarge image)
Figure 1: Various initial wave packets. The real part of the coordinate (left) and momentum (right) wave functions are shown. For all the four wave packets the $\Delta x$ values and the scale of the figures are the same. The x and y axes are plotted as thin lines.

(Click to enlarge image)
Figure 2: Asymptotic behaviour of the various wave packets in k space. The four thick lines are the envelopes of the spectral density $\varrho (k)$. (The envelope is defined as a smooth curve fitted to the maxima of the spectral density.) For the square and triangular wave packets the straight lines indicate polynom decay ($1/k^2$ and $1/k^4$, respectively). For the good and Gaussian packets the envelopes are not straight lines. This verifies the faster than polynomial decay of their spectral densities.

(Click to enlarge image)
Figure 3: Time evolution of the various wave packets. The four columns of figures show the time development of the real part of the coordinate wave function for the four initial states (whose names are given in the figure). Time is measured in units of $t_2 / 8$ where $t_2$ is the doubling time of the Gaussian (Cf. Fig. 4.) defined by $a_{Gauss} (t=t_2) = 2 a_{Gauss} (t=0)$. The x, y scale is the same for all figures. The x and y axes are plotted as thin lines.