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Download Visual quantum mechanics : selected topics with computer by Bernd Thaller PDF

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By Bernd Thaller

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Extra resources for Visual quantum mechanics : selected topics with computer generated animations of quantum mechanical phenomena ; CD-ROM included

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137) The Fourier transform of a distribution is defined as ˆ tˆ(ψ) = t(ψ). 138) A distribution is said to be regular if there exists a (locally integrable) function t(·) such that t(ψ) = t(x) ψ(x) dnx. 139) If a distribution is regular, then the linear mapping t determines the function t(x) uniquely (almost everywhere), and vice versa. Hence one usually does not distinguish between the regular distribution and the function describing it. 9. SPECIAL TOPIC: DIRAC DELTA DISTRIBUTION 47 Any square-integrable function ψ can also be interpreted as a regular distribution.

Several cate the phase of the complex coefficients ψ(k examples are given in Color Plate 8. A square-integrable function need not be continuous. 16) has discontinuous jumps at x = ±1, but it is certainly square-integrable on any interval [−L, L]. 3. 16). If you take only finitely many summands of the Fourier series of a squareintegrable function, you get an approximation by a finite Fourier sum. Any finite Fourier sum is a smooth function with the property ψN (−L) = ψN (L). You can see in Fig. 2 how such an approximation of a discontinuous function looks like.

Each ψn is in the domain of F, and the sequence (ψn ) converges in H. The continuity of F implies the convergence of the sequence (Fψn ) and thus it makes sense to define the Fourier transform of ψ to be the limit of the sequence Fψn . 79) where ψn is any sequence of integrable functions converging to ψ, that is, ψ = lim ψn . 80) 34 2. 6. Further Results About the Fourier Transformation The following discussion quotes some elementary properties that make the Fourier transform a very important tool in analysis.

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