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Introduction

In quantum mechanics textbooks [1,2] there are not much specific examples for the Heisenberg indeterminacy [3] relation

$\begin{equation} \Delta x \cdot \Delta k \ge \hbar / 2 .\end{equation}$

It is usually investigated what is the shape of the wave packet which minimises the indeterminacy, i.e. for which

$\begin{equation} \Delta x \cdot \Delta k = \hbar / 2 .\end{equation}$

It is found that this wave function is a Gaussian

$\begin{equation} \psi_{min}(x) = N \cdot \exp( - \frac{x^2}{ 4 { \left( \Delta x \right) } ^ 2} ) .\end{equation}$

The Gaussian minimises the indeterminacy because it is well localised both in coordinate (x) and momentum (p) space. Apart from this minimising wave packet usually only the ideal cases of complete localisation in x and k spaces ( $\psi(x) = \delta (x-x_0)$ and $\psi(x) = \exp ( i k_0 x )$ )are treated.

In this paper we analyse in detail how does the shape of the wave packet influence its indeterminacy and how does the wave function and the indeterminacy change during the time development of the wave packet.

For the sake of simplicity we use atomic units, i.e. $\hbar = m_{electron} = \vert q_{electron} \vert = 1$ . From this one gets the simple relations $p = k, \qquad E = \omega$ where p is the momentum, $k = {2 \pi} / {\lambda}$ the wave number, E is the energy, $\omega = {2 \pi} / {T}$ the angular frequency. In this paper we investigate only free time development. This means that there are no forces acting on our particles, hence the potential V(x) is constant. That constant can be chosen to be zero, so $V(x) \equiv 0$ is inserted into the Schrödinger equation. In classical limit the particle moves uniformly along a straight line in free space.

In Section(II) the properties of various initial wave packets are discussed. In Section(III) their time development is investigated. Section(IV) treats the indeterminacy as the function of time. Specific examples are analysed in Section(V). The summary of the results can be found in Section(VI).

Next: Initial states Up: Influence of the wave Previous: Influence of the wave