Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER INTEGRALS AND FOURIER TRANSFORMS
Figure 4.16. (a) A chopped harmonic wave e i! 0t that lasts a ®nite time 2T. …b†
Fourier transform of e …i! 0t† ; jtj < T, and 0 otherwise.
the time T of the duration of the pulse. As T becomes larger the central peak
becomes higher and the width !…ˆ 2=T† becomes smaller. Considering only
the spread of frequencies in the central peak we have
! ˆ 2=T; or T ˆ 1:
Multiplying by the Planck constant h and replacing T by t, we have the relation
tE ˆ h:
…4:32†
A wave train that lasts a ®nite time also has a ®nite extension in space. Thus the
radiation emitted by an atom in 10 9 s has an extension equal to 3 10 8 10 9 ˆ
3 10 1 m. A Fourier analysis of this pulse in the space domain will yield a graph
identical to Fig. 4.11(b), with the wave numbers clustered around
k 0 …ˆ 2= 0 ˆ ! 0 =v†. If the wave train is of length 2a, the spread in wave number
will be given by ak ˆ 2, as shown below. This time we are chopping an in®nite
plane wave front with a shutter such that the length of the packet is 2a, where
2a ˆ 2vT, and 2T is the time interval that the shutter is open. Thus
Then
(
…x† ˆ eik 0x ; a x a
:
0; jxj > a
…k† ˆ…2† 1=2 Z 1
1
ˆ…2=† 1=2 a sin…k 0 k†a
…k 0 k†a :
…x†e ikx dx ˆ…2† 1=2 Z a
a
…x†e ikx dx
This function is plotted in Fig. 4.17: it is identical to Fig. 4.16(b), but here it is the
wave vector (or the momentum) that takes on a spread of values around k 0 . The
breadth of the central peak is k ˆ 2=a, orak ˆ 2.
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