An Introduction to Modern Astrophysics · Problem 5.18

Telescopes — Carroll & Ostlie

Physics — Astrophysics · fourier-series-wave-packets

Problem

One of the most important ideas of the physics of waves is that any complex waveform can be expressed as the sum of the harmonics of simple cosine and sine waves. That is, any wave function

f(x)

can be written as

\begin{aligned} f(x) &= a_0 + a_1 \cos x + a_2 \cos 2x + a_3 \cos 3x + a_4 \cos 4x + \cdots \\ &\quad + b_1 \sin x + b_2 \sin 2x + b_3 \sin 3x + b_4 \sin 4x + \cdots. \end{aligned}

The coefficients

a_n

and

b_n

tell how much of each harmonic goes into the recipe for

f(x)

. This series of cosine and sine terms is called the Fourier series for

f(x)

. In general, both cosine and sine terms are needed, but in this problem you will use only the sine terms; all of the

a_n \equiv 0

. On page 130, the process of constructing a wave pulse by adding a series of sine waves was described. The Fourier sine series that you will use to construct your wave employs only the odd harmonics and is given by

\Psi = \frac{2}{N+1}\left(\sin x - \sin 3x + \sin 5x - \sin 7x + \cdots \pm \sin Nx\right) = \frac{2}{N+1}\sum_{\substack{n=1 \\ n\text{ odd}}}^{N} (-1)^{(n-1)/2} \sin nx,

where

N

is an odd integer. The leading factor of

2/(N+1)

does not change the shape of

\Psi

but scales the wave for convenience so that its maximum value is equal to 1 for any choice of

N

. (a) Graph

\Psi

for

N = 5

, using values of

x

(in radians) between 0 and

\pi

. What is the width,

\Delta x

, of the wave pulse? (b) Repeat part (a) for

N = 11

. (c) Repeat part (a) for

N = 21

. (d) Repeat part (a) for

N = 41

. (e) If

\Psi

represents the probability wave of a particle, for which value of

N

is the position of the particle known with the least uncertainty? For which value of

N

is the momentum of the particle known with the least uncertainty?

Related theory

Fourier Series Of A Square Wave

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