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## 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 . The construction is basically the same for all these sets and it is illustrated by the following example.

Subdivide the real line, , of the Fourier domain into equal subintervals of length and consider a function, , whose Fourier transform is zero everywhere except in one of these subintervals,

 (267)

We demand that , is an integer so that can be pictured as a finite complex amplitude in the th frequency window . See Figure 2.14.

Note that

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 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

 (268)

We see that the transformation

which is represented by the matrix'',

is a unitary transformation because it preserves inner products between elements in , or equivalently, because it preserves the inner products between the square integrable basis elements

Next: Orthonormal Wave Packets: Definition Up: Orthonormal Wave Packet Representation Previous: Orthonormal Wave Packet Representation   Contents   Index
Ulrich Gerlach 2010-12-09