Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 17 Eigenfunction methods for differential equations - all with Video Answers

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Chapter Questions

Problem 1

By considering $\langle h \mid h\rangle$, where $h=f+\lambda g$ with $\lambda$ real, prove that, for two functions $f$ and $g$
$$
\langle f \mid f\rangle\langle g \mid g\rangle \geq \frac{1}{4}[\langle f \mid g\rangle+\langle g \mid f\rangle]^{2}
$$
The function $y(x)$ is real and positive for all $x$. Its Fourier cosine transform $\tilde{y}_{\mathrm{c}}(k)$ is defined by
$$
\tilde{y}_{\mathrm{c}}(k)=\int_{-\infty}^{\infty} y(x) \cos (k x) d x
$$
and it is given that $\tilde{y}_{\mathrm{c}}(0)=1$. Prove that
$$
\tilde{y}_{\mathrm{c}}(2 k) \geq 2\left[\tilde{y}_{\mathrm{c}}(k)\right]^{2}-1
$$.

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Problem 2

(a) Write the homogeneous Sturm-Liouville eigenvalue equation for which $y(a)=y(b)=0$ as
$$
\mathscr{L}(y ; \lambda) \equiv\left(p y^{\prime}\right)^{\prime}+q y+\lambda \rho y=0
$$
where $p(x), q(x)$ and $\rho(x)$ are continuously differentiable functions. Show that if $z(x)$ and $F(x)$ satisfy $\mathcal{L}(z ; \lambda)=F(x)$ with $z(a)=z(b)=0$ then
$$
\int_{a}^{b} y(x) F(x) d x=0
$$
(b) Demonstrate the validity of result (a) by direct calculation for the case in which $p(x)=\rho(x)=1, q(x)=0, a=-1, b=1$ and $z(x)=1-x^{2}$.

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Problem 3

Consider the real eigenfunctions $y_{n}(x)$ of a Sturm-Liouville equation
$$
\left(p y^{\prime}\right)^{\prime}+q y+\lambda \rho y=0, \quad a \leq x \leq b
$$
in which $p(x), q(x)$ and $\rho(x)$ are continuously differentiable real functions and $p(x)$ does not change sign in $a \leq x \leq b$. Take $p(x)$ as positive throughout the interval, if necessary by changing the signs of all eigenvalues. For $a \leq x_{1} \leq x_{2} \leq b$, establish the identity
$$
\left(\lambda_{n}-\lambda_{m}\right) \int_{x_{1}}^{x_{2}} \rho y_{n} y_{m} d x=\left[y_{n} p y_{m}^{\prime}-y_{m} p y_{n}^{\prime}\right]_{x_{1}}^{x_{2}}
$$
Deduce that if $\lambda_{n}>\lambda_{m}$ then $y_{n}(x)$ must change sign between two successive zeroes of $y_{m}(x)$. (The reader may find it helpful to illustrate this result by sketching the first few eigenfunctions of the system $y^{\prime \prime}+\lambda y=0$, with $y(0)=y(\pi)=0$, and the Legendre polynomials $P_{n}(z)$ given in subsection 16.6.1 for $n=2,3,4,5$.)

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Problem 4

(a) Show that the equation
$$
y^{\prime \prime}+a \delta(x) y+\lambda y=0
$$
with $y(\pm \pi)=0$ and $a$ real, has a set of eigenvalues $\lambda$ satisfying
$$
\tan (\pi \sqrt{\lambda})=\frac{2 \sqrt{\lambda}}{a}
$$
(b) Investigate the conditions under which negative eigenvalues, $\lambda=-\mu^{2}$ with $\mu$ real, are possible.

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Problem 5

Express the hypergeometric equation
$$
\left(x^{2}-x\right) y^{\prime \prime}+[(1+\alpha+\beta) x-\gamma] y^{\prime}+\alpha \beta y=0
$$
in Sturm-Liouville form, determining the conditions imposed on $x$ and on the parameters $\alpha, \beta$ and $\gamma$ by the boundary conditions and the allowed forms of weight function.

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Problem 6

(a) Find the solution of $\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+b y=f(x)$ valid in the range $-1 \leq x \leq 1$ and finite at $x=0$, in terms of Legendre polynomials.
(b) If $b=14$ and $f(x)=5 x^{3}$, find the explicit solution and verify it by direct substitution.

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Problem 7

Use the generating function for the Legendre polynomials $P_{n}(x)$ to show that
$$
\int_{0}^{1} P_{2 n+1}(x) d x=(-1)^{n} \frac{(2 n) !}{2^{2 n+1} n !(n+1) !}
$$
and that, except for the case $n=0$,
$$
\int_{0}^{1} P_{2 n}(x) d x=0
$$.

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Problem 8

The quantum mechanical wavefunction for a one-dimensional simple harmonic oscillator in its $n$th energy level is of the form
$$
\psi(x)=\exp \left(-x^{2} / 2\right) H_{n}(x)
$$
where $H_{n}(x)$ is the $n$th Hermite polynomial. The generating function for the polynomials (17.53) is
$$
G(x, h)=e^{2 h x-h^{2}}=\sum_{n=0}^{\infty} \frac{H_{n}(x)}{n !} h^{n}
$$.
(a) Find $H_{i}(x)$ for $i=1,2,3,4$.
(b) Evaluate by direct calculation.
$$
\int_{-\infty}^{\infty} e^{-x^{2}} H_{p}(x) H_{q}(x) d x
$$
(i) for $p=2, q=3$; (ii) for $p=2, q=4$; (iii) for $p=q=3$. Check your answers against equation (17.52). (You will find it convenient to use
$$
\int_{-\infty}^{\infty} x^{2 n} e^{-x^{2}} d x=\frac{(2 n) ! \sqrt{\pi}}{2^{2 n} n !}
$$
for integer $n \geq 0$.)

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Problem 9

The Laguerre polynomials, which are required for the quantum mechanical description of the hydrogen atom, can be defined by the generating function (equation (17.58))
$$
G(x, h)=\frac{e^{-h x /(1-h)}}{1-h}=\sum_{n=0}^{\infty} \frac{L_{n}(x)}{n !} h^{n}
$$
By differentiating the equation separately with respect to $x$ and $h$, and resubstituting for $G(x, h)$, prove that $L_{n}$ and $L_{n}^{\prime}\left(=d L_{n}(x) / d x\right)$ satisfy the recurrence relations
$$
\begin{aligned}
L_{n}^{\prime}-n L_{n-1}^{\prime}+n L_{n-1} &=0 \\
L_{n+1}-(2 n+1-x) L_{n}+n^{2} L_{n-1} &=0
\end{aligned}
$$
From these two equations and others derived from them, show that $L_{n}(x)$ satisfies the Laguerre equation
$$
x L_{n}^{\prime \prime}+(1-x) L_{n}^{\prime}+n L_{n}=0
$$.

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Problem 10

Starting from the linearly independent functions $1, x, x^{2}, x^{3}, \ldots$, in the range $0 \leq x<\infty$, find the first three orthogonal functions $\phi_{0}, \phi_{1}$ and $\phi_{2}$, with respect to the weight function $\rho(x)=e^{-x} .$ By comparing your answers with the Laguerre polynomials generated by the recurrence relation derived in exercise $17.9$, deduce the form of $\phi_{3}(x)$.

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Problem 11

Consider the set of functions $\{f(x)\}$ of the real variable $x$, defined in the interval $-\infty<x<\infty$, that $\rightarrow 0$ at least as quickly as $x^{-1}$ as $x \rightarrow \pm \infty .$ For unit weight function, determine whether each of the following linear operators is Hermitian when acting upon $\{f(x)\}:$
(a) $\frac{d}{d x}+x$;
(b) $-i \frac{d}{d x}+x^{2}$;
(c) $i x \frac{d}{d x} ;$
(d) $i \frac{d^{3}}{d x^{3}}$

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Problem 12

The Chebyshev polynomials $T_{n}(x)$ can be written as
$$
T_{n}(x)=\cos \left(n \cos ^{-1} x\right)
$$. (a) Verify that these functions do satisfy the Chebyshev equation.
(b) Use de Moivre's theorem to show that an alternative expression is
$$
T_{n}(x)=\sum_{r \mathrm{even}}^{n}(-1)^{r / 2} \frac{n !}{(n-r) ! r !} x^{n-r}\left(1-x^{2}\right)^{r / 2}
$$.

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Problem 13

A particle moves in a parabolic potential in which its natural angular frequency of oscillation is $1 / 2 .$ At time $t=0$ it passes through the origin with velocity $v$ and is suddenly subjected to an additional acceleration of $+1$ for $0 \leq t \leq \pi / 2$, and then $-1$ for $\pi / 2<t \leq \pi .$ At the end of this period it is at the origin again. Apply the results of the worked example in section $17.6$ to show that
$$
v=-\frac{8}{\pi} \sum_{m=0}^{\infty} \frac{1}{(4 m+2)^{2}-\frac{1}{4}} \approx-0.81
$$.

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Problem 14

Find an eigenfunction expansion for the solution with boundary conditions $y(0)=y(\pi)=0$ of the inhomogeneous equation
$$
\frac{d^{2} y}{d x^{2}}+\kappa y=f(x)
$$
where $\kappa$ is a constant and
$$
f(x)= \begin{cases}x, & 0 \leq x \leq \pi / 2 \\ \pi-x, & \pi / 2<x \leq \pi\end{cases}
$$.

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Problem 15

(a) Find those eigenfunctions $y_{n}(x)$ of the self-adjoint linear differential operator $d^{2} / d x^{2}$ that satisfy the boundary conditions $y_{n}(0)=y_{n}(\pi)=0$, and hence construct its Green's function $G(x, z)$.
(b) Construct the same Green's function using the methods of subsection $15.2 .5$, showing that it is
$$
G(x, z)= \begin{cases}x(z-\pi) / \pi, & 0 \leq x \leq z \\ z(x-\pi) / \pi, & z \leq x \leq \pi\end{cases}
$$
(c) By expanding the function given in (b) in terms of the eigenfunctions $y_{n}(x)$, verify that it is the same function as that derived in (a).

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Problem 16

(a) The differential operator $\mathscr{L}$ is defined by
$$
\mathcal{L} y=-\frac{d}{d x}\left(e^{x} \frac{d y}{d x}\right)-\frac{e^{x} y}{4}
$$
Determine the eigenvalues $\lambda_{n}$ of the problem
$$
\mathcal{L} y_{n}=\lambda_{n} e^{x} y_{n} \quad 0<x<1
$$
with boundary conditions
$$
y(0)=0, \quad \frac{d y}{d x}+\frac{y}{2}=0 \quad \text { at } \quad x=1
$$
(b) Find the corresponding unnormalised $y_{n}$, and also a weight function $\rho(x)$ with respect to which the $y_{n}$ are orthogonal. Hence, select a suitable normalisation for the $y_{n}$.
(c) By making an eigenfunction expansion, solve the equation
$$
\mathscr{L} y=-e^{x / 2}, \quad 0<x<1
$$
subject to the same boundary conditions as previously.

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Problem 17

Show that the linear operator
$$
\mathscr{L} \equiv \frac{1}{4}\left(1+x^{2}\right)^{2} \frac{d^{2}}{d x^{2}}+\frac{1}{2} x\left(1+x^{2}\right) \frac{d}{d x}+a
$$
acting upon functions defined in $-1 \leq x \leq 1$ and vanishing at the endpoints of the interval, is Hermitian with respect to the weight function $\left(1+x^{2}\right)^{-1}$

By making the change of variable $x=\tan (\theta / 2)$, find two even eigenfunctions, $f_{1}(x)$ and $f_{2}(x)$, of the differential equation
$$
\mathscr{L} u=\lambda u
$$.

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Problem 18

By substituting $x=\exp t$ find the normalized eigenfunctions $y_{n}(x)$ and the eigenvalues $\lambda_{n}$ of the operator $\mathcal{L}$ defined by
$$
\mathcal{L} y=x^{2} y^{\prime \prime}+2 x y^{\prime}+\frac{1}{4} y, \quad 1 \leq x \leq e
$$
with $y(1)=y(e)=0 .$ Find, as a series $\sum a_{n} y_{n}(x)$, the solution of $\mathcal{L} y=x^{-1 / 2}$.

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Problem 19

with $y(1)=y(e)=0$. Find, as a series $\sum a_{n} y_{n}(x)$, the solution of $\mathcal{L} y=x^{-1 / 2}$. Express the solution of Poisson's equation in electrostatics,
$$
\nabla^{2} \phi(\mathbf{r})=-\rho(\mathbf{r}) / \epsilon_{0}
$$
where $\rho$ is the non-zero charge density over a finite part of space, in the form of an integral and hence identify the Green's function for the $\nabla^{2}$ operator.

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Problem 20

In the quantum mechanical study of the scattering of a particle by a potential, a Born-approximation solution can be obtained in terms of a function $y(\mathbf{r})$ that satisfies an equation of the form
$$
\left(-\nabla^{2}-K^{2}\right) y(\mathbf{r})=F(\mathbf{r})
$$
Assuming that $y_{\mathbf{k}}(\mathbf{r})=(2 \pi)^{-3 / 2} \exp (i \mathbf{k} \cdot \mathbf{r})$ is a suitably normalised eigenfunction of $-\nabla^{2}$ corresponding to eigenvalue $-k^{2}$, find a suitable Green's function $G_{K}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ By taking the direction of the vector $\mathbf{r}-\mathbf{r}^{\prime}$ as the polar axis for a $\mathbf{k}$-space integration, show that $G_{K}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ can be reduced to
$$
\frac{1}{4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \int_{-\infty}^{\infty} \frac{w \sin w}{w^{2}-w_{0}^{2}} d w
$$
where $w_{0}=K\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$.
(This integral can be evaluated using a contour integration (chapter 20 ) to give $\left.\left(4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{-1} \exp \left(i K\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right) .\right)$

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