3 THE INITIAL WAVEPACKET


FUNCTIONAL FORM

The initial wavepacket is the product of a plane harmonic wave with a gaussian-shaped envelope function:
 
 

 Eqtn 3.1

For numerical reasons, we have chosen the wavevector of the plane wave to be 100*pi=314.2 inverse distance units, corresponding to a wavelength of .02 distance units. Thus one can fit 500 wavelengths into the width of one movie frame, which extends over one unit of distance from x= -0.5 to x=0.5. The gaussian envelope function has a width parameter "sigma" chosen to be .05 distance units. It is straightforward to show that this implies that the envelope drops to one-half its peak value at a distance of ln4 * sigma = (1.39 sigma) distance units from the peak.

 PROBABILITY AMPLITUDE

In the movies, we follow not the wavefunction psi itself, but the probability amplitude, which is psi psi*, and usually the wave character is not readily apparent. For example, taking psi psi* for the initial wavepacket (Eq. 3.1), the result is a simple gaussian function (whose amplitude drops to 50% of its peak value at a distance of ln2 * sigma = 0.69 sigma distance units from the peak centre). Hence psi psi* does not have any apparent wavelike character in this case, (and incidentally is a more strongly localised function than is psi itself.)

It is therefore useful to keep in mind throughout this resource the primary and essential character of the wavefunction, that of a wave, and to reinforce this caveat we show below just once the wavefunction psi itself. Since psi is complex, we can only graph the real or imaginary components, and in the graph below we show the real part of the initial wavepacket:-
 
 

 Real part of Wavefunction




NORMALISATION

The wavefunction 3.1 is not normalised in the usual sense that,
 
 

 Eqtn 3.2

because it is handy when interpreting the movies to have probability amplitudes whose initial peak value is just unity; this makes it a little easier to determine quantities such as the reflection coefficient direct from the movies. If required, the value of the integral in Eq 3.2 for a wavefunction of the form given in Eq 3.1 can be calculated: the result is sigma * square root(pi).

SPECTRAL COMPOSITION

The wavepacket Eq. 3.1 has been specifically chosen so that it shows the localisation expected of particle-like solutions to Schroedingers equation. However, much of the familiar theory of one dimensional scattering is formulated in terms of incident plane waves. It is therefore interesting to ponder how pertinent these formulae might be to predicting the scattering of the particle-like wavepacket. To answer this question, we need to write Eq 3.1 not as a product of a single plane wave with a function, but rather as an equivalent sum over simple plane waves. This is the subject of Fourier analysis. The result of taking the Fourier transform of Eq. 3.1 is that the wavefunction may also be written as the integral ( = "infinite sum") of plane waves whose amplitudes follow a distribution which is again gaussian in shape, with a half-width in wavevector space of 0.69/sigma:
 
 

Eqtn 3.3

where the term in brackets represents the amplitude of each plane wave component. The first exponential in the bracket is just a phase factor; we can plot the second term for the values of wavevector k0 and width sigma use here to get an impression of the "spectral purity" of the wavepacket:-
 
 

 Spectral Content of Wavepacket

The amplitude function is quite strongly peaked around k0, and so we may expect that the behaviour of the wavepacket will be similar to that of a plane wave of wavevector k0. However, in situations where the spectral composition of the packet may change, this simple approximate equivalence may well be in serious error.
Comments etc to : jo@nat.vu.nl