NOTES ON FOURIER SERIES and FOURIER TRANSFORM

Transcription

1 NOTES ON FOURIER SERIES and FOURIER TRANSFORM Physics 141 (2003) These are supplemental math notes for our course. The first part, (Part-A), provides a quick review on Fourier series. I assume that you have seen some of these materials in various equivalent forms, ( e.g., in treating either wave motions or driven osccillators.) For quantum mechanics, we prefer using a complex represntation. The second part, (Part- B), discusses how one moves from Fourier series to Fourier transform when extending previous discussion to functions defined the entire real line. These notes spell out more fully than discussions provide by Griffiths, Sec. 2.4, as well as Problem However, you shoould be aware of the fact that some notations used here are different from that used by Griffiths. However, the differences are minor. 1

2 PART-A: FOURIER SERIES I. STANDARD REPRESENTATION Consider a piecewise continuous and differentiable function, f(θ), 0 θ. Itcan be expanded in a Fourier series as follows: f(θ) = a n cos nθ + b n sin nθ, n=0 n=1 (I.1) where a 0 = 1 0 dθ f(θ); a n = 1 dθ cos nθ f(θ), π 0 b n = 1 dθ sin nθ f(θ), n 1. (I.2) π 0 COMMENTS: (1) In terms of Eq. (I.1), f(θ) can be extended to <θ< ; itbecomes a periodic function, with a period, i.e., f(θ) =f(θ +). For instance, instead of [0, ], we can use [, π] for limits in Eq. (I.2). (2) We will in general be interested in functions which are complex. It follows that the Fourier coefficients, {a n }, and {b m }, are complex numbers. It is often more convenient to re-write (I.1) in a complex form. (3) We will also be interested in periodic functions of period, i.e., F(x) =F(x + ), where can be large. This can easily be done by a variable change: θ x θ. We will first discuss the complex representation for a Fourier series. We next discuss the variable change θ x θ, which permits us to make an easy transition from a Fourier series to a Fourier transform by taking the limit. 2

3 II. COMPEX REPRESENTATION Using the identity e ±inθ = cos nθ ± i sin nθ, wecan re-express Eq. (I.1) as a sum over e inθ and e inθ, i.e., f(θ) = 1 n= c n e inθ. (II.1) After regrouping terms and comparing Eq. (II.1) with Eq. (I.1), one finds that the new Fourier coefficients are related to the old ones by: c n = (a n ib n )/2, n 0; c n = (a n + ib n )/2, n < 0; or, more directly, one has c n = 1 π dθ e inθ f(θ), (II.2) for all integers n. COMMENTS (1) Fourier coefficients {c n } are generally complex, even if the original function, f(θ), is real. (If f(θ) isreal, one has a relation between c n and c n, i.e., c n = c n.) (2) et us consider the family of functions, {u n (θ)}, <n<, where u n (θ) 1 e inθ. (II.3) One readily verifies by direct integrations: π Orthonormality : u n u m dθ u n(θ)u m (θ) = 1 π dθ e i(m n)θ = δ n,m (II.4) where δ n,m =1ifn = m and δ n,m =0ifn m. That is, δ n,m is the Kronecker delta symbol. 3

4 (3) In Eq. (II.4), we have introduced an inner product : For any pair of periodic functions g(θ) and f(θ), we associate a complex number g f π dθ g (θ)f(θ). (II.5) (4) The fact that f(θ) can be expanded as a sum over the family of functions, {u n (θ)}, i.e., Eq. (II.1), implies this set of functions constitutes a complete set of basis functions. Substituting (II.2) into (II.1), one has f(θ) = π dθ [ 1 n= e in(θ θ ) ]f(θ ). It follows that the expression between the square brackets is nothing but a Dirac delta function, i.e., for θ, θ π, Completeness Relation : n= u n(θ) u n (θ )= 1 n= e in(θ θ) = δ(θ θ ). (II.6) (5) The notion of Dirac delta function reuqires us to generalize the notion of a function. Indeed, in mathematics, it is often referred to as a generalized fubction or a distribution. For us, it is adequate to define it informally, asisdone in Sec. 2.5 of Griffiths. (6) When Eqs. (II.4) and (II.6) are satisfied, one says that {u n (θ)} form a complete orthonormal set of basis functions. Note that this is an infinite set, but labelled by a discrete index, n. Interms of these basis functions, (II.1) and (II.2) can be re-written as f(θ) = c n u n (θ), n= (II.7) c n = u n f = π dθ u n(θ)f(θ). (II.8) 4

5 III. FOURIER SERIES FOR FUNCTIONS WITH PERIOD Given a periodic function F(x), where F(x) = F(x + ), it can be converted to a periodic function of θ with a period by the following transformation: f(θ) F(x = θ). Conversely, one can easily adopt our analysis of Section II for F(x) byreplacing θ by x. Therefore, (using capital letters), let us define a complete set of orthonormal basis functions, {U n (x)}, <n< U n (x) 1 e inx. (III.1) It is clear that these functions are periodic in x with a period. With our normalization, (III.1), it follows from Eqs. (II.4) and (II.6) that Orthonormality : U n U m /2 /2 dx U n(x)u m (x) = 1 /2 /2 dx e i(m n)x = δ n,m (III.2) Completeness Relation : n= U n(x) U n (x )= 1 n= e in(x x) = δ(x x ), (III.3) where /2 x and x /2. Instead of Eqs. (II.1), (II.2), (II.7), and (II.8), we have F(x) = C n U n (x), n= (III.4) C n = U n F = /2 /2 dx U n(x)f(x). (III.5) That is, we have defined a new inner product: G F /2 /2 dx G (x)f(x). (III.6) 5

6 COMMENTS (1) It is clear that the space of functions of type {f(θ) =f(θ +)} is not the same as the space of functions where {F(x) =F(x + )},. However, these two spaces are in one-to-one correspondence. (2) The inner product g f, (II.5), used in Eqs. (II.4) and (II.8), is different from the corresponding inner product G F, (III.6), used in Eqs. (III.5) and (III.2). Although we have employed the same notation, each takes on a different integration range and no confusion should arise. 6

7 PART-B: FOURIER SERIES AND FOURIER TRANSFORM IV. FOURIER TRANSFORM et us now consider functions, {f(x)}, which are piecewise continuous and differentiable over the entire real line, i.e., xɛ(, ). It is intuitively clear that, physically, one must be able to first consider periodic functions, {F(x) =F(x + )}, defined over [ /2,/2], and then to take the limit. COMMENTS (1) et us first examine the basis functions, U n (x) 1 e inx, < n <. The normalization is certainly unfortunate, since U n (x) 0as.Ofcourse, this difficulty can easily be overcome by working with a new set, {φ n (x)}, where φ n (x) U n(x) = 1 e inx. (2) If one takes the limit with x and n fixed, φ n (x) a nontrivial result is obtained if the limit is taken with k n n to grow with. Inthis limit, φ n (x) φ k (x), where 1. Since <n<, kept fixed, i.e., one allows φ k (x) 1 e ikx, (IV.1) with kɛ(, ). (3) With large and n =1,itfollows that k = n = 1. Therefore, the sums over n in Eqs. (III.3) and (III.4) can be replaced by an integral, ( ) dk. For 7

8 instance, instead of (III.4), one arrives at f(x) lim F(x) = lim n= C n U n (x) = 1 dk g(k) e ikx = dk g(k) φ k (x), (IV.2) where /2 g(k) lim,n C n = lim dx Un(x)F(x),n /2 = 1 dx e ikx f(x) = dx φ k(x) f(x), (IV.3) with k = n fixed. We shall refer to g(k) asthe Fourier transform of f(x), given by (IV.3), and f(x) asthe inverse Fourier transform of g(k), given by (IV.2). (4) Substituting (IV.3) into (IV.2), one has g(k) = dk [ dx φ k (x) φ k (x)] g(k ). Since this must hold for all g(k), one arrives at Orthonormality : dx φ k(x) φ k (x) = 1 dx e ix(k k) = δ(k k ). (IV.4) This is analogous to Eqs. (II.6) and (III.3). The only difference is the fact that the label, k, for the complete basis functions, {φ k (x)}, takes on continuous values. (5) Substituting (IV.2) into (IV.3), one has f(x) = dx [ dk φ k (x) φ k(x )] f(x ). It follows that Completeness : dk φ k(x) φ k (x )= 1 dk e ik(x x) = δ(x x ). (IV.5) Unlike (II.4) and (III.2), our complete basis functions satisfy Dirac delta function orthonormal condition. 8

9 (6) Finally, we can also introduce an inner product: g f dx g (x) f(x). (IV.6) Using this notation, the orthonormality condition, Eq. (IV.4), can be expressed as and the Fourier transform, (IV.3), becomes φ k φ k = δ(k k ), (IV.7) g(k) = φ k f. (IV.8) V. ORTHOGONAITY AND COMPETENESS To repeat, any reasonable functions we need to work with, f(x), defined over the real line, <x<, can always be represented it by a Fourier integral, f(x) = 1 dk g(k) e ikx = dk g(k) φ k (x), (V.1) where g(k) = 1 dx e ikx f(x) = dx φ k(x) f(x), (V.2) is the Fourier transform of f(x). The set of functions, φ k (x), labelled by a continuous index, <k<, again forms a complete ON set. That is: Orthonormality : dx φ k(x) φ k (x) = 1 dx e ix(k k) = δ(k k ). (V.3) Completeness : dk φ k(x) φ k (x )= 1 dk e ik(x x) = δ(x x ). (V.4) These complete basis functions satisfy Dirac delta function orthonormal condition. 9