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Initial states

Quantum mechanics is deterministic in the sense that the initial state determines the state at any subsequent time. It is important to learn how do the different properties of the initial state influence the properties of the time evolution.

We investigate the initial states both in the one dimensional coordinate and momentum representation. Let x be the position variable. Its operator $\bf {x}$ is the multiplication with x , i.e.

$\begin{equation} {\bf {x}} \psi(x) = x \cdot \psi(x) .\end{equation}$

Let p be the momentum variable. In atomic units p = k and this allows us to use the k wave number instead of the p momentum throughout this paper. The operator of the wave number k is

$\begin{equation} {\bf {k}} \psi(x) = \frac{1}{i} \frac{\partial \psi(x)} {\partial x} .\end{equation}$

Moreover, x and k are conjugated variables, i.e.

$\begin{equation}[ {\bf {x}}, {\bf {k}} ] = i .\end{equation}$

The wave number space wave function is the Fourier transform of the coordinate space wave function:

$\begin{eqnarray} \varphi(k) = {\cal F} [ \psi(x) ] (k) &=& \frac{1}{ \sqrt{2 \... ... \sqrt{2 \pi} } \int_{-\infty}^{\infty} \varphi(k) e^{i k x} dk .\end{eqnarray}$

The form of the position and wave number operator in wave number space is the following:

$\begin{eqnarray} {\bf {x}} \varphi(k) &=& \frac{1}{i} \frac{\partial \varphi(k)} {\partial k} \\ {\bf {k}} \varphi(k) &=& k \cdot \varphi(k) .\end{eqnarray}$

For a free particle the energy and wave number operators commute, $[ {\bf {H}}, {\bf {k}} ] = 0$ , hence the eigenfunctions of the $\bf {H}$ Hamilton operator which are the solutions of the free space stationary Schrödinger equation

$\begin{eqnarray} {\bf {H}} \Psi &=& E \Psi \\ - \frac{1}{2} \frac {\partial^2 \psi(x)} {\partial x^2} &=& E \psi(x)\end{eqnarray}$

are eigenfunctions of $\bf {k}$ as well. Hence k is a ''good quantum number'', the solutions of the free space stationary Schrödinger equation can be uniquely labelled with k values.

For the sake of comparison it is good to choose the parameters of all our initial wave packets so that their initial position indeterminacy values are all equal with a common $\Delta x$ .(This is not possible for the two completely localised cases VE and VF because in these cases one can not choose $\Delta x$ freely, its value is for x localisation and $\infty$ for k localisation.)

Some of our initial wave functions have compact support (bounded support).

Definition. A function has compact support if it is zero outside some finite interval.

If a coordinate wave function has compact support this means that it is localised to the given interval, i.e. there is no chance to find the quantum particle outside this interval. This kind of localisation can be achieved in principle by enclosing the particle into an infinite potential box. A finite potential box is not enough because the quantum particle always has a non-zero probability density in the forbidden region, i.e. there is a non-zero chance to find it at any far distance. In practice the localisation to a finite x interval can be achieved with a high enough finite potential box. The core electrons are e.g. well localised in a neutral atom or the valence electrons are well localised in a piece of metal.

If a wave function has compact support in k it means that its spectrum is limited. As we will see in Section(III) the dynamics in the x and k spaces are different because the $\bf {H}$ Hamilton operator is not symmetric in x and k in free space.

If a function does not have compact support than it is said to have infinite support. If a wave function has infinite support in x this means that the quantum particle has a non-zero probability of being found at any x value.

There is an important theorem for wave functions with compact support:

Theorem 1. If a wave function has compact support in the x coordinate (k wave number) space than it has infinite support in the conjugated k (x ) space.

Proof: $\varphi(k)$ is the Fourier transform of $\psi(x)$ .The Fourier transform of a compact supported function has always infinite support.

The wave number space wave function $\varphi(k)$ corresponding to a compact supported $\psi(x)$ can however decrease fast if $\psi(x)$ is chosen adequately. This can be formulated as the following theorem [4]:

Theorem 2. If a function f(x) has compact support in x and it is infinitely smooth then its Fourier transform g(k) and all of its derivatives $g^{(n)}(k)$ decrease faster than any polynomial of 1 / |k| .

A function is infinitely smooth if it has continuous derivatives of all orders on all points, i.e. it is infinitely differentiable at all points. At first sight it is not obvious that there exist infinitely differentiable compact supported functions at all because the simple compact supported functions (e.g. square or triangle) are not infinitely differentiable at the edges of the support interval. But there is a whole class of such functions: these are the so called ''good functions'' or ''test functions'' of distribution theory [4]. These functions are infinitely differentiable also at the edges of the support interval because they join smoothly the $f(x) \equiv 0$ function. An example of a ''good function'' is the one we use at VD.

Next: Time development of the Up: Influence of the wave Previous: Introduction