A ''good'' initial state

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A ''good'' initial state

(Click to enlarge animation)

As it was mentioned in Section(II) if a coordinate wave function has compact support and it is infinitely smooth then the corresponding wave number wave function is well localised. An example of such a ''good function'' is the following:

$\begin{equation} \psi_{Good} ( x; a ) = \cases{ N(a) \, \exp { - \frac{a^2}{... ...-a, a )$\space ; \cr 0, & if $x \not\in ( -a, a )$\space . \cr }\end{equation}$

where N(a) is the normalisation factor.

The integrals of such a function can not be calculated analytically but we can calculate the wave number space wave function and the indeterminacies numerically. The behaviour of $\varphi_{Good} ( k; a )$ can be compared with that of the other initial packets in Figs. 1. and 2. From the log-log scale plot Fig. 2. one can verify that $\varphi_{Good} ( k; a )$ really decreases faster than any polynomial of 1/k for increasing k .

From numerical integrations one gets the normalisation factor $N(a) = 2.741 / \sqrt {a}$ and the values for the indeterminacies which are summarised in Table 1.

$\psi_{Good}$ has the following - good - properties:

It has compact support in x , the particle is confined to the [ -a, a ] interval.
$\psi_{Good} ( x; a )$ is infinitely differentiable.
In k it has infinite support but it is well localised because $\varrho (k)$ decreases fast ( faster than any polynomial of 1 / |k| ).
The wave function and all of its derivatives are continuous.

$\psi_{Good} ( x,t; a )$ was calculated numerically with the Fourier transform formula (16). The results are shown in Fig. 3.

Next: Perfect localisation in coordinate Up: Examples (with figures) Previous: Compact supported triangular shape