By Bernd Thaller
ISBN-10: 0387989293
ISBN-13: 9780387989297
Read Online or Download Visual quantum mechanics : selected topics with computer generated animations of quantum mechanical phenomena ; CD-ROM included PDF
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Additional info for Visual quantum mechanics : selected topics with computer generated animations of quantum mechanical phenomena ; CD-ROM included
Example text
8. Inequalities In this section you will read about some inequalities related to Fourier analysis. Here you will find a proof of the famous uncertainty relation that has been of great importance in the development of quantum mechanics and its interpretation. But there are other inequalities sometimes even more useful in the mathematical theory of quantum mechanics. 1. The uncertainty relation The uncertainty relation is an inequality that relates the width of a function ψ in position space with the width of the Fourier transform ψˆ in momentum space.
It is a very important observation that the original function and the function describing the continuous spectrum of wave numbers depend on each other in a very symmetrical way. This relationship—the Fourier transformation—can be described as a linear operator acting in the Hilbert space of square integrable functions. The properties of the Fourier transformation make it a very useful tool in quantum mechanics. For example, the derivative of a function corresponds via the Fourier transformation to a simple multiplication by k in momentum space.
102) √ √ Hence we find I = 2π. Hence fˆ(0) = I/ 2π = 1. Therefore, fˆ satisfies the same differential equation as f , with the same initial condition. We conclude that fˆ(k) = f (k) = exp(−k 2 /2). This proves Eq. 97) for α = 1. The general case now follows easily with a scaling transformation according to Eq. 86). 40 2. 9. 17 show various other examples of the Fourier transformation. These examples include oscillating and discontinuous functions and functions with a nonintegrable Fourier transform.
Visual quantum mechanics : selected topics with computer generated animations of quantum mechanical phenomena ; CD-ROM included by Bernd Thaller
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