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Time development of the different wave packets

The wave function of a quantum system evolves in time in (non-relativistic) quantum mechanics under the control of the (time dependent) Schödinger equation:

$\begin{eqnarray} i \frac { \partial \Psi (t) } { \partial t } & = & {\bf {H}} \Psi (t) .\end{eqnarray}$

Given the initial state $\Psi_0 = \Psi (t_0)$ the state $\Psi (t)$ at any subsequent time t is determined because (13) is of first order in t . The Schödinger equation in coordinate representation is a partial differential equation which can be solved generally only with approximate methods. The Schödinger equation in wave number representation, however, is only an ordinary differential equation in the special case of $V(x) \equiv 0$ :

$\begin{eqnarray} i \frac { d \varphi (k,t) } { d t } &=& \frac{1}{2} k^2 \varphi(k,t)\end{eqnarray}$

which can be solved easily:

$\begin{eqnarray} \varphi (k,t) &=& \varphi_0 (k) \cdot \exp{ -i \frac{k^2}{2} t } .\end{eqnarray}$

Hence if $\varphi_0 (k)$ is the initial (t=0 ) wave number wave function of the quantum particle than the wave number wave function at any time t is simply the product of the initial wave function with a k dependent phase factor. We can immediately see from this that the wave number space probability density remains constant (in free space) because the exponent in the k and t dependent phase factor in (15) is purely imaginary (until t remains real), i.e. $\varrho (k,t) = \varrho_0 (k)$ .

The coordinate space wave function is given by the inverse Fourier transform of $\varphi (k,t)$ . Hence the time development of the initial $\psi_0 (x)$ wave function is given by:

$\begin{eqnarray} \psi (x,t) &=& {\cal F}^{-1} \left[ \varphi_0 (k) \exp{ -i ... ...(x) \\ \varphi_0 (k) &=& {\cal F} \left[ \psi_0 (x) \right] (k)\end{eqnarray}$

The product under the Fourier transform in (16) can be expressed directly in coordinate space as a convolution. One must convolute the initial $\psi_0 (x)$ wave function with the so called free propagator function:

$\begin{eqnarray} \psi(x,t) &=& \psi_0(x) * P_{Free}(x,t) \\ P_{Free}(x,t) &=... ...{ 2 \pi t } } \exp{ -i \frac{\pi}{4} } \exp{ i \frac{x^2}{2 t} }\end{eqnarray}$

The explicit time development of the wave function can be calculated analytically for all of our initial wave packets studied in Section V except for the ''Good function'' (VD) case where it has to be calculated numerically.

Next: Indeterminacy as the function Up: Influence of the wave Previous: Initial states