By Adobe Creative Team, Adobe Press
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This fact will be exploited in Chapter 3 to solve the free Schr¨odinger equation with arbitrary initial conditions. While this chapter contains some material that is indispensable for a thorough description of quantum mechanics, there are some mathematically more elaborate sections that may be skipped at ﬁrst reading. ” 15 16 2. 1. Fourier Series of Complex-Valued Functions Fourier analysis is the art of writing arbitrary wave functions as superpositions of trigonometric functions. As a ﬁrst step we consider periodic functions and the associated Fourier series.
Linear structure The set L2 ([a, b]) of square-integrable functions has the structure of a linear space. For the following it is useful to bear in mind the analogy with the n-dimensional complex space Cn which is a more elementary example of a linear space. The vectors v ∈ Cn are n-tuples v = (v1 , . . , vn ), with vi ∈ C. 21) 22 2. FOURIER ANALYSIS These n-tuples can be added and multiplied by scalars. For a ∈ C, b ∈ C, v ∈ Cn and w ∈ Cn we can deﬁne the linear combination av + bw by (av + bw)i = a vi + b wi , for i = 1, .
92) ˆ As |k| → ∞, this function decreases so slowly that ik ψ(k) is not integrable. This, of course, reﬂects the fact that ψ is not diﬀerentiable at x = 0. But ˆ since ik ψ(k) is square-integrable, we can deﬁne its inverse Fourier transform in the L2 -sense as in Eq. 79). Thus, the inverse Fourier transform is the limit (with respect to the metric in L2 ) of the sequence n 1 ˆ dk. 93) 2π −n It can be shown that the L2 -limit of this sequence is given by x φ(x) = −e−|x| . 6. 4. The Fourier transform allows to deﬁne a generalized derivative.
Adobe(R) GoLive(R) 4.0 Classroom in a Book by Adobe Creative Team, Adobe Press