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Additional resources for An introduction to generalized coordinates in mechanics and physics
90) ψ = F −1 (ik ψ) is called the “generalized derivative” of ψ. Let me explain this concept with an example. The function (see Fig. 91) is integrable and its Fourier transform is ˆ ψ(k) = 2 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.
Here “close together” means that ψ − φ is small. The Fourier–Plancherel theorem implies that ψˆ and φˆ are close together in the same sense because ψˆ − φˆ = ψ − φ . 5. 1. Basic deﬁnitions The Fourier transformation F is a mapping from a set of integrable functions ψ in a Hilbert space to functions ψˆ belonging to the same Hilbert space. The mapping F is an example of a linear operator. You will learn in Chapter 4 that in the quantum-mechanical formalism the linear operators play an essential role (all physical observables are represented by linear operators).
4) Because of the periodicity, it is suﬃcient to restrict the consideration to an interval of length 2L, say, the interval [−L, L]. 5) n=−N (with arbitrary complex numbers cn ) is a smooth function on [−L, L] with the property ψ(−L) = ψ(L). It can, of course, also be considered a periodic function on R. 5) is called a trigonometric sum or Fourier sum. For n ≥ 0 we call (L) cn un(L) (x) + c−n u−n (x) the summand of order n. 1. 1 is an interactive demonstration showing how new functions can be built by adding trigonometric functions.