Orthonormal Wave Packets: General Construction

There are many different complete sets of orthonormal wave packets. Each set is a countable basis for the Hilbert space $L^2(-\infty ,\infty )$ . The construction is basically the same for all these sets and it is illustrated by the following example.

Subdivide the real line, $-\infty <\omega <\infty$ , of the Fourier domain into equal subintervals of length $\varepsilon$ and consider a function, $P_{j\ell }(t)$ , whose Fourier transform is zero everywhere except in one of these subintervals,

Figure 2.14: The imaginary part of the localized Fourier amplitude of the wave packet $P_{j\ell }(t)$ . In this graph $\ell =4$ and the mean frequency of the wave packet is $(j+\frac {1}{2})\varepsilon$ .

$$F_{j\ell}(\omega ) = \left\{\begin{array}{lr} 0 &\omega~\textrm{n... ...varepsilon}} &j\varepsilon \le\omega\le (j+1)\varepsilon \end{array}\right. ~~.$$ (267)

We demand that $\ell =0,\pm 1,\pm 2, \cdots$ , is an integer so that $F_{j\ell}( \omega )$ can be pictured as a finite complex amplitude in the $j$ th frequency window $j\varepsilon\le\omega\le (j+1)\varepsilon$ . See Figure 2.14.

Note that

$$\int^\infty_{-\infty} \overline{F}_{j\ell }(\omega )F_{j'\ell '} (\omega ) d\omega =\delta_{jj'}\delta_{\ell\ell '}\,,$$

which can be easily verified. Such an orthonormality relation is the key to constructing complete sets of orthonormal wave packets. Simply invent a set of functions $F_{j\ell}( \omega )$ which satisfy such an orthonormality property. The example in Figure 2.14 illustrates this idea. Then use these functions to construct your own set of wave packets

$$P_{j\ell}(t)=\int^\infty_{-\infty} F_{j\ell} (\omega ) \frac{e^{\displaystyle i\omega t}}{\sqrt{2\pi}} d\omega\,.$$ (268)

We see that the transformation

$${\cal F}^{-1}\colon ~~ L^2(-\infty ,\infty ) \longrightarrow L^2(-\infty,\infty)$$

$$F_{j\ell} (\omega ) \sim\!\!\rightarrow {\cal F}^{-1}[F_{j\ell}] \equiv \int^\infty_{-\infty} F_{j\ell}(\omega )\frac{e^{i\omega t}}{\sqrt{2\pi}} d\omega$$

$$~~\qquad \equiv P_{j\ell}(t)~,$$

which is represented by the ``matrix'',

$$\frac{e^{i\omega t}}{\sqrt{2\pi}}~,$$

is a unitary transformation because it preserves inner products between elements in $L^2(-\infty ,\infty )$ , or equivalently, because it preserves the inner products between the square integrable basis elements

$$\delta_{jj'}\delta_{\ell\ell '}=\langle F_{j\ell},F_{j'\ell '}\ra... ...cal F}^{-1}[F_{j'\ell '}]\rangle = \langle P_{j\ell }, P_{j'\ell '}\rangle\,.$$