Compact supported triangular shape wave packet

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Compact supported triangular shape wave packet

The coordinate wave function $\psi(x)$ of this packet has its maximum value at the middle of the support interval (of length L ) and it is linearly decreasing to zero towards the edges of the interval.

$\begin{eqnarray} \psi_{Triangular} ( x; L ) &=& \cases{ 0 , & if $x \in ( -\i... ... \Delta x \cdot \Delta k &=& \sqrt{ \frac{3}{10} } \gt \frac{1}{2}\end{eqnarray}$

The properties of the triangular shaped packet:

It has compact support in x , the particle is confined to the [ - L / 2, L / 2 ] interval.
In k it has infinite support and it is poorly localised because $\varrho (k)$ decreases only slowly as .
The wave function is continuous but its derivative has jumps in the origin and at the edges of the support interval.

It is interesting to note that the self convolution of a square shaped wave packet (52) gives a triangular shaped packet only the support of the resulting triangular packet will become twice as wide and the normalisation will change. This makes it possible to write the triangular shaped packet in the following way:

$\begin{eqnarray} \psi_{Triangular} ( x; L ) = \sqrt{\frac{3}{L}} \cdot \psi_{S... ...\frac{L}{2} \right) * \psi_{Square} \left( x; \frac{L}{2} \right)\end{eqnarray}$

With this convolution formula the time evolution of the triangular initial state is:

$\begin{eqnarray} \psi_{Triangular}(x,t;L) &=& \psi_{Triangular} (x,t=0;L) * P_{... ...{L}{2} \right) * \psi_{Square} \left( x, t; \frac{L}{2} \right) .\end{eqnarray}$

In the last equation the convolution identity (A*B)*C = A*(B*C) was utilised. This result means that the time evolution of the triangular initial wave function is given as a time dependent wave function of a square initial state (with support L/2 ) convoluted with a square function. As can be seen in Fig. 3. $\psi_{Triangular} (x,t)$ oscillates with much smaller amplitudes than $\psi_{Square} (x,t)$ because the big oscillations of the latter are ''smeared out'' by the convolution.

Next: A ''good'' initial state Up: Examples (with figures) Previous: Compact supported square shape