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Appendix C Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Fourier coefficients (a_n) and (b_n) of a function (constant function) defined on an interval, using equations (C.6) and (C.7)

Find all the Fourier coefficients \(a_n\) (\(n = 0, 1, 2, \ldots\)) and \(b_n\) (\(n = 1, 2, 3, \ldots\)) of the function \(f(x) = 1\) (constant function) defined on the interval \([0, L]\), using equations (C.6) and (C.7).

Hint

When integrating \(\cos\!\left(\frac{2\pi n}{L}x\right)\) from \(0\) to \(L\), for \(n \geq 1\) the result corresponds to one full period of \(\sin\). Note that when \(n = 0\), the integrand becomes \(1\).

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B-2. Using the orthogonality of complex exponentials (Eq. (C.12)), evaluate the following integral

Using the orthogonality of complex exponentials (Eq. (C.12)), evaluate the following integral.

\[ \int_0^L e^{i \frac{2\pi \cdot 3}{L} x}\, e^{-i \frac{2\pi \cdot 5}{L} x}\, dx \]
Hint

Combine the integrand into the form \(e^{i\frac{2\pi(m-n)}{L}x}\). Confirm that when \(m = 3\) and \(n = 5\), we have \(m \neq n\).

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B-3. Using Euler's formula, show the following

Using Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\), show the following.

\[ \cos\!\left(\frac{2\pi n}{L}x\right) = \frac{1}{2}\left(e^{i\frac{2\pi n}{L}x} + e^{-i\frac{2\pi n}{L}x}\right) \]

Furthermore, derive the relations between the real Fourier coefficients \(a_n, b_n\) and the complex Fourier coefficients \(c_n\) of a real function \(f(x)\):

\[ c_n = \frac{a_n - i b_n}{2} \quad (n \geq 1), \qquad c_{-n} = \frac{a_n + i b_n}{2} \quad (n \geq 1), \qquad c_0 = \frac{a_0}{2} \]

by comparing Eq. (C.5) and Eq. (C.10).

Hint

Rewrite the \(\cos\) and \(\sin\) in Eq. (C.5) as complex exponentials using Eq. (C.9), then compare the coefficients of \(e^{ik_n x}\) and \(e^{-ik_n x}\) with \(c_n\) and \(c_{-n}\) in Eq. (C.10).

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B-4. Find all Fourier coefficients and on the interval

Find all the Fourier coefficients \(a_n\) and \(b_n\) of \(f(x) = \sin\!\left(\frac{2\pi}{L}x\right)\) on the interval \([0, L]\).

Hint

For the calculation of \(a_n\), the orthogonality relation in equation (C.4) can be used directly. For the calculation of \(b_n\), use equation (C.3).

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B-5. Gaussian Integral Formula

Using the Gaussian integral formula

\[ \int_{-\infty}^{\infty} e^{-\alpha t^2}\,dt = \sqrt{\frac{\pi}{\alpha}} \qquad (\alpha > 0) \]

compute the Fourier transform \(\tilde{f}(k)\) of the function \(f(x) = e^{-3x^2}\) using convention (b) (Eq. (C.16)).

Hint

Complete the square in the exponent of \(\tilde{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-3x^2} e^{-ikx}\,dx\) as \(-3\!\left(x + \frac{ik}{6}\right)^2 - \frac{k^2}{12}\).

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B-6. Following the definition of convolution (Eq. (C.21)), compute the convolution of and ( is a constant). Here is D

Following the definition of convolution (Eq. (C.21)), compute the convolution \((f * g)(x)\) of \(f(x) = e^{-|x|}\) and \(g(x) = \delta(x - a)\) (where \(a\) is a constant). Here \(\delta\) is the Dirac delta function, which satisfies \(\int_{-\infty}^{\infty} h(x')\,\delta(x' - a)\,dx' = h(a)\).

Hint

Using the property of \(\delta(x' - a)\) (the sifting property), the integration over \(x'\) can be carried out immediately.

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B-7. Evaluate the following integral using the Fourier integral representation of the δ function (Eq. (C.19))

Evaluate the following integral using the Fourier integral representation of the δ function (Eq. (C.19)):

\[ \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i(k - k')x}\,dx = \delta(k - k') \]
\[ \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i \cdot 7 x}\,dx \]
Hint

Simply set \(k = 7\) and \(k' = 0\) in Eq. (C.19).

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B-8. Verify the following using Parseval's equality (Eq. (C.18)) for the case

Using Parseval's equality (Eq. (C.18)), verify the following. For \(f(x) = e^{-|x|}\),

\[ \int_{-\infty}^{\infty} |f(x)|^2\,dx = \int_{-\infty}^{\infty} |\tilde{f}(k)|^2\,dk \]

holds. Show this by directly computing the left-hand side, computing the right-hand side using \(\tilde{f}(k) = \sqrt{\frac{2}{\pi}}\,\frac{1}{1+k^2}\) (convention (b)), and demonstrating that the two results agree.

Hint

Left-hand side: The integral of \(|f(x)|^2 = e^{-2|x|}\) is twice \(\int_0^{\infty} e^{-2x}\,dx\). Right-hand side: Use the fact that \(\int_{-\infty}^{\infty}\frac{dk}{(1+k^2)^2} = \frac{\pi}{2}\) (you may use partial fraction decomposition, or simply apply \(\int_{-\infty}^{\infty}\frac{dk}{(1+k^2)^2} = \frac{\pi}{2}\) as a known formula).

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Medium

M-1. Find the Fourier coefficients and of the function defined on the interval , and write down the Fourier series (Equation (C.5

Find the Fourier coefficients \(a_n\) and \(b_n\) of the function \(f(x) = x\) defined on the interval \([0, L]\), and write down the Fourier series (Equation (C.5)). Furthermore, verify that the value of the series obtained by substituting \(x = L/2\) agrees with \(f(L/2) = L/2\).

Hint

\(a_n\): Compute \(\int_0^L x\cos\!\left(\frac{2\pi n}{L}x\right)dx\) by integration by parts. \(a_0\) is twice the mean value. \(b_n\): \(\int_0^L x\sin\!\left(\frac{2\pi n}{L}x\right)dx\) is also computed by integration by parts. Check the values of \(\sin\) and \(\cos\) at \(x = L/2\).

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M-2. Fourier Transform of a Gaussian Function and Parseval's Theorem

Find the Fourier transform of the Gaussian function \(f(x) = e^{-ax^2}\) (\(a > 0\)) using convention (b), and show that the result is again a Gaussian function. Furthermore, using Parseval's theorem (Eq. (C.18)), verify that

\[ \int_{-\infty}^{\infty} e^{-2ax^2}\,dx = \int_{-\infty}^{\infty} \frac{1}{2a}\,e^{-k^2/(2a)}\,dk \]

holds.

Hint

Complete the square in the exponent as \(-a\!\left(x + \frac{ik}{2a}\right)^2 - \frac{k^2}{4a}\). Use the Gaussian integral formula \(\int_{-\infty}^{\infty} e^{-\alpha t^2}\,dt = \sqrt{\pi/\alpha}\).

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M-3. Solve the following problem using the convolution theorem (Eq. (C.22))

Solve the following problem using the convolution theorem (Eq. (C.22)).

For \(f(x) = e^{-x^2}\) and \(g(x) = e^{-x^2}\), find the convolution \((f * g)(x)\) not by direct calculation, but by computing the product in Fourier transform space and then taking the inverse transform.

Hint

Use \(\tilde{f}(k) = \tilde{g}(k) = \frac{1}{\sqrt{2}} e^{-k^2/4}\) (the case \(a = 1\) in D5). Calculate \(\widetilde{(f*g)}(k) = \sqrt{2\pi}\,\tilde{f}(k)\,\tilde{g}(k)\) from Eq. (C.22), then take the inverse Fourier transform.

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M-4. Fourier Integral Representation of the δ Function

Starting from the Fourier integral representation of the δ function

\[ \delta(x - x') = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ik(x - x')}\,dk \]

derive the following.

(a) \(\delta(x)\) is an even function: \(\delta(-x) = \delta(x)\)

(b) Scaling rule: \(\delta(\alpha x) = \frac{1}{|\alpha|}\,\delta(x)\) (\(\alpha \neq 0\))

(c) \(x\,\delta(x) = 0\)

Hint

(a) In the Fourier integral representation, substitute \(x \to -x\) and change the integration variable \(k \to -k\). (b) In the Fourier representation of \(\delta(\alpha x)\), make the variable substitution \(k \to k/\alpha\). (c) Compute \(\int x\,\delta(x)\,\phi(x)\,dx\) for an arbitrary test function \(\phi(x)\).

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M-5. Differentiation Property of the Fourier Transform

Derive the differentiation property of the Fourier transform. In convention (b), given that the Fourier transform of \(f(x)\) is \(\tilde{f}(k)\):

(a) Show that the Fourier transform of \(f'(x) \equiv \frac{df}{dx}\) is \(ik\,\tilde{f}(k)\).

(b) Using this result, solve the differential equation \(f'(x) + \beta f(x) = 0\) (\(\beta > 0\)) in Fourier transform space, and reproduce \(f(x) = Ce^{-\beta x}\) (\(x > 0\)).

Hint

(a) Integrate \(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f'(x)\,e^{-ikx}\,dx\) by parts, and use the fact that \(f(x)\) vanishes as \(x \to \pm\infty\). (b) Note that the equation after Fourier transformation is not \((ik + \beta)\tilde{f}(k) = 0\), but rather an algebraic equation whose right-hand side depends on the initial conditions. Alternatively, one may directly compute the Fourier transform by writing \(f(x) = C e^{-\beta x}\theta(x)\) (where \(\theta\) is the Heaviside step function).

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Advanced

A-1. Fourier-Analytic Proof of the Uncertainty Relation

Fourier-Analytic Proof of the Uncertainty Relation

Suppose a function \(f(x)\) and its Fourier transform \(\tilde{f}(k)\) (convention (b)) are both normalized (\(\int |f(x)|^2\,dx = 1\)). Define the "width" in position and the "width" in wave number respectively as

\[ (\Delta x)^2 \equiv \int_{-\infty}^{\infty} x^2\,|f(x)|^2\,dx, \qquad (\Delta k)^2 \equiv \int_{-\infty}^{\infty} k^2\,|\tilde{f}(k)|^2\,dk \]

(for simplicity, assume \(\langle x \rangle = 0\), \(\langle k \rangle = 0\)).

(a) In the Cauchy–Schwarz inequality

\[ \left|\int_{-\infty}^{\infty} u(x)^*\,v(x)\,dx\right|^2 \leq \int_{-\infty}^{\infty}|u(x)|^2\,dx \cdot \int_{-\infty}^{\infty}|v(x)|^2\,dx \]

set \(u(x) = x\,f(x)\), \(v(x) = f'(x)\) and derive \(\Delta x \cdot \Delta k \geq \frac{1}{2}\).

(b) Find the condition for equality to hold, and show that it corresponds to the Gaussian function \(f(x) = \left(\frac{1}{2\pi\sigma^2}\right)^{1/4} e^{-x^2/(4\sigma^2)}\).

(c) Confirm that by setting \(p = \hbar k\), one obtains the quantum mechanical uncertainty relation \(\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\).

Hint

(a) Integrate \(\int x\,f(x)^*\,f'(x)\,dx\) by parts and use \(\int |f(x)|^2\,dx = 1\). Also, use the result from D5, S5(a) (the Fourier transform of \(f'\) is \(ik\tilde{f}\)) and Parseval's theorem to show \(\int |f'(x)|^2\,dx = \int k^2|\tilde{f}(k)|^2\,dk = (\Delta k)^2\). (b) The equality condition in Cauchy–Schwarz is \(v(x) = \lambda\,u(x)\) (\(\lambda\) is a constant). This becomes the differential equation \(f' = \lambda\,x\,f\). (c) Simply substitute \(\Delta p = \hbar\,\Delta k\).

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A-2. From Fourier Series to Parseval's Identity: Derivation of \(\zeta(2) = \pi^2/6\)

From Fourier Series to Parseval's Identity: Derivation of \(\zeta(2) = \pi^2/6\)

(a) Using the Fourier series of \(f(x) = x\) on the interval \([0, L]\) (the result from S1), derive the Fourier series version of Parseval's identity:

\[ \frac{1}{L}\int_0^L |f(x)|^2\,dx = \frac{|a_0|^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty}\left(|a_n|^2 + |b_n|^2\right) \]

(Hint: Integrate \(|f(x)|^2\) from Eq. (C.5) and use orthogonality.)

(b) By substituting the Fourier coefficients of \(f(x) = x\) into the result from (a), prove that

\[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \]

This is the value of \(\zeta(2)\) for the Riemann zeta function \(\zeta(s) = \sum_{n=1}^{\infty} n^{-s}\) (the Basel problem).

Hint

(a) Substitute Eq. (C.5) twice into the left-hand side \(\frac{1}{L}\int_0^L |f(x)|^2\,dx\), and use the orthogonality relations (C.2)–(C.4) to eliminate the cross terms. (b) For \(f(x) = x\), we have \(\frac{1}{L}\int_0^L x^2\,dx = \frac{L^2}{3}\), and substituting the \(a_n, b_n\) found in S1 and canceling \(L\) determines the value of \(\sum 1/n^2\).

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