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Chapter 22: Problem 35

Use the operator separation technique to show that (a) the GF for the IIelmholtz operator \(\nabla^{2}+k^{2}\) in three dimensions is $$ G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-i k \sum_{l=0}^{\infty} \sum_{m=-l}^{l} j_{l}\left(k r_{<}\right) h_{l}\left(k r_{>}\right) Y_{l m}(\theta, \varphi) Y_{l m}^{*}\left(\theta^{\prime}, \varphi^{\prime}\right), $$ where \(r_{<}\left(r_{>}\right)\) is the smaller (larger) of \(r\) and \(r^{\prime}\) and \(j_{l}\) and \(h_{l}\) are the spherical Bessel and Hankel functions, respectively. No explicit BCs are assumed except that there is regularity at \(r=0\) and that \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right) \rightarrow\) 0 for \(|\mathbf{r}| \rightarrow \infty\) (b) Obtain the identity $$ \frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}=i k \sum_{l=0}^{\infty} \sum_{m=-l}^{l} j_{l}\left(k r_{<}\right) h_{l}\left(k r_{>}\right) Y_{l m}(\theta, \varphi) Y_{l m}^{*}\left(\theta^{\prime}, \varphi^{\prime}\right) . $$ (c) Derive the plane wave expansion [see Eq. (19.46)] $$ e^{i \mathbf{k} \cdot \mathbf{r}}=4 \pi \sum_{l=0}^{\infty} \sum_{m=-l}^{l} i^{l} j_{l}(k r) Y_{l m}^{*}\left(\theta^{\prime}, \varphi^{\prime}\right) Y_{l m}(\theta, \varphi), $$ where \(\theta^{\prime}\) and \(\varphi^{\prime}\) are assumed to be the angular coordinates of \(\mathbf{k}\). Hint: Let \(\left|\mathbf{r}^{\prime}\right| \rightarrow \infty\), and use $$ \left|\mathbf{r}-\mathbf{r}^{\prime}\right|=\left(r^{\prime 2}+r^{2}-2 \mathbf{r} \cdot \mathbf{r}^{\prime}\right)^{1 / 2} \rightarrow r^{\prime}-\frac{\mathbf{r}^{\prime} \cdot \mathbf{r}}{r^{\prime}} $$ and the asymptotic formula \(h_{l}^{(1)}(z) \rightarrow(1 / z) e^{i[z+(l+1)(\pi / 2)]}\), valid for large \(z\).

Short Answer

Expert verified
The exercise involves the manipulation of Green's Function, spherical Bessel and spherical Hankel functions through the separation of variables method to derive the given equations.

Step by step solution

01

Separation of Variables

Begin by separating the variables for the problem. Write the Laplacian in spherical coordinates and separate the angular and radial parts. This will give you two equations, one for the angular part, and one for the radial part. The solutions to the angular part are the spherical harmonics \(Y_{lm}\) and the radial part gives two types of solutions, regular at the origin (the spherical Bessel functions \(j_l\)) and regular at infinity (the Bessel functions of the third kind or the spherical Hankel functions \(h_l\)).
02

Formulation of Green's Function

For the Green's function, utilize the representation of the delta function in terms of spherical harmonics to integrate the product of two Green's functions over the angular coordinates. Then use the orthogonality of the spherical harmonics to simplify the expressions. The resulting integral over radius will then give the form of the Green's function \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) as stated in the exercise.
03

Derive the identity

The similarity of the given identity and the Green's function allows you to proceed by comparison. To make this comparison, expand the exponential term on the left hand side in spherical waves. This will give rise to Bessel functions. Matching the terms on both sides will then show the given relation.
04

Derive Plane Wave Expansion

For the plane wave expansion, you start with the equation from step 3. First, take the limit as \(|\mathbf{r'}| \rightarrow \infty\), then use the identity for \(|\mathbf{r} - \mathbf{r'}|\), and the asymptotic formula for \(h_l^{(1)}(z) \). This leads to the conclusion of the stated plane wave expansion equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Green's Function
The concept of Green's function is a critical tool in mathematical physics, particularly when dealing with differential equations. In essence, it serves as a fundamental solution to inhomogeneous differential equations. Think of it as a method to relate the source of a physical phenomenon, like a point charge in an electric field, to its impact on the surroundings.

In the context of the Helmholtz operator abla^{2}+k^{2}, Green's function specifically helps us determine how waves propagate in space when affected by a source. The operator here is a representation of the wave equation in a medium, where k is the wave number. Green's function technique cleverly uses the principle of superposition, summarizing the collective effect of each possible point source.
Spherical Harmonics
Spherical harmonics are the angular portion of the solutions to the Laplace equation in spherical coordinates. Think of them as the 3D analogue of Fourier series—a way to express functions on the surface of a sphere. They pop up everywhere in physics, from quantum mechanics to gravitation.

They are particularly important in problems with spherical symmetry, just like the one we're discussing. In our problem, they appear when separating the variables in the wave equation, representing the angular part of the solution. The indices l and m you see, often called the quantum numbers, denote the degree and order of the harmonic and come from the constraints of the problem being solved.
Helmholtz Equation
The Helmholtz equation, denoted by abla^{2}+k^{2}, is a fundamental equation in the study of wave phenomena. It comes up when you're investigating how waves, such as sound or light, behave under certain conditions. It's named after Hermann von Helmholtz, a towering figure in the study of acoustics.

This equation is what you get when you apply separation of variables to the wave equation, and it's at the heart of many problems in physics. In this problem, it's being solved in a very particular setup where the Green's function for the equation must be found without specific boundary conditions.
Bessel Functions
Bessel functions are a family of solutions to Bessel's differential equation that appear in a wide range of wave propagation and static potentials problems. You can think of them much like the sines and cosines in Fourier's world; they are particularly handy in cylindrical geometries.

There are several kinds of Bessel functions, but in our case, the Helmholtz equation leads to the radial part of the solution being expressed in terms of spherical Bessel functions—a variant that is designed for spherical symmetry and that behaves well at the origin.
Spherical Bessel Functions
Spherical Bessel functions are specific kinds of Bessel functions that have been adapted for problems with spherical symmetry. There are two types we're dealing with here: the first kind (denoted by j_l) and the Hankel functions, which are Bessel functions of the third kind (denoted by h_l).

The spherical Bessel functions of the first kind, j_l, appear in solutions to the radial equation that behave regularly at the origin, which is why they are used in the description of the Green's function in this problem for r < r'. They're critical for understanding how waves behave in a bounded spherical region.
Spherical Hankel Functions
Spherical Hankel functions, on the other hand, describe solutions that behave nicely at infinity—hence their use for r > r' in our Green's function. In physics, this is important for studying scattering problems where waves are emitted from a source and propagate to infinity. They exhibit oscillatory decay, making them suited for representing outgoing or incoming waves in spherical coordinates, important in studying radiative effects.
Plane Wave Expansion
The plane wave expansion is a powerful technique that expresses a plane wave, which moves in a straight line with a constant amplitude, as a sum over spherical waves, which spread out from a point source. This is pivotal in the analysis of wave scattering, antenna theory, and more. In the context of our problem, the plane wave is being expanded using spherical harmonics along with Bessel functions, revealing how a linear propagation of energy can be seen as a combination of spherical components.
Separation of Variables
The method of separation of variables is an approach to solve partial differential equations, such as the Helmholtz equation. It's like tackling a problem by dividing and conquering, assuming that the solution can be broken into parts, each depending only on a single coordinate. This method simplifies the complex three-dimensional problem into manageable one-dimensional parts, and it's essential for finding the Green's function and for expressing the wave equation solutions in terms of spherical harmonics and Bessel functions in our exercise.
Laplacian in Spherical Coordinates
The Laplacian is a mathematical operator that appears in the wave equation, representing the divergence of the gradient of a function. In a nutshell, it measures how much the function spreads out from a point. When we switch to spherical coordinates to exploit the symmetry of a problem, the Laplacian gets a new, more complex look but allows us to apply the separation of variables strategy effectively. Our understanding of the Laplacian in spherical coordinates is crucial for solving spherical wave problems as well as understanding the nature of potentials in spherical geometries.

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Most popular questions from this chapter

Consider GF for the Helmholtz operator \(\nabla^{2}+\mu^{2}\) in two dimensions. (a) Show that $$ G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-\frac{i}{4} H_{0}^{(1)}\left(\mu\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)+H\left(\mathbf{r}, \mathbf{r}^{\prime}\right) $$ where \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) satisfies the homogeneous Helmholtz equation. (b) Separate the variables and use the fact that \(H\) is regular at \(\mathbf{r}=\mathbf{r}^{\prime}\) to show that \(H\) can be written as $$ H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\sum_{n=0}^{\infty} J_{n}(\mu r)\left[a_{n}\left(\mathbf{r}^{\prime}\right) \cos n \theta+b_{n}\left(\mathbf{r}^{\prime}\right) \sin n \theta\right] . $$ (c) Now assume a circular boundary of radius \(a\) and the \(\mathrm{BC} G\left(\mathbf{a}, \mathbf{r}^{\prime}\right)=0\), in which a is a vector from the origin to the circular boundary. Using this \(\mathrm{BC}\), show that $$ \begin{aligned} a_{0}\left(\mathbf{r}^{\prime}\right)=& \frac{i}{8 \pi J_{0}(\mu a)} \int_{0}^{2 \pi} H_{0}^{(1)}\left(\mu \sqrt{a^{2}+r^{\prime 2}-2 a r^{\prime} \cos \left(\theta-\theta^{\prime}\right)}\right) d \theta, \\ a_{n}\left(\mathbf{r}^{\prime}\right)=& \frac{i}{4 \pi J_{n}(\mu a)} \\ & \times \int_{0}^{2 \pi} H_{0}^{(1)}\left(\mu \sqrt{a^{2}+r^{\prime 2}-2 a r^{\prime} \cos \left(\theta-\theta^{\prime}\right)}\right) \cos n \theta d \theta \\ b_{n}\left(\mathbf{r}^{\prime}\right)=& \frac{i}{4 \pi J_{n}(\mu a)} \\ & \times \int_{0}^{2 \pi} H_{0}^{(1)}\left(\mu \sqrt{a^{2}+r^{\prime 2}-2 a r^{\prime} \cos \left(\theta-\theta^{\prime}\right)}\right) \sin n \theta d \theta \end{aligned} $$ These equations completely determine \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) and therefore \(\bar{G}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\).

Use the operator separation technique to calculate the Dirichlet GF for the two-dimensional operator \(\nabla^{2}-k^{2}\) on the rectangle \(0 \leq x \leq a, 0 \leq\) \(y \leq b\). Also obtain an eigenfunction expansion for this GF.

Calculate the retarded GF for the wave operator in two dimensions and show that it is equal to $$ G_{s}^{(\mathrm{ret})}(\mathbf{r}, t)=\frac{\theta(t)}{2 \pi \sqrt{t^{2}-r^{2}}} $$ Now use this result to obtain the GF for any even number of dimensions: $$ G_{s}^{(\mathrm{ret})}(\mathbf{r}, t)=\frac{\theta(t)}{2 \pi}\left(-\frac{1}{2 \pi r} \frac{\partial}{\partial r}\right)^{n-1}\left[\frac{1}{\sqrt{t^{2}-r^{2}}}\right] \quad \text { for } n=m / 2 . $$

Use the operator separation technique to find the three-dimensional Dirichlet GF for the Laplacian in a circular cylinder of radius \(a\) and height \(h\).

Write the bound-state Schrödinger integral equation for a non-local potential, noting that \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=e^{-\kappa\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\), where \(\kappa^{2}=-2 \mu E / \hbar^{2}\) and \(\mu\) is the mass of the bound particle. The homogeneous solution is zero, as is always the case with bound states. (a) Assuming that the potential is of the form \(V\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-g^{2} U(\mathbf{r}) U\left(\mathbf{r}^{\prime}\right)\), show that a solution to the Schrödinger equation exists iff $$ \frac{\mu g^{2}}{2 \pi \hbar^{2}} \int_{\mathbb{R}^{3}} d^{3} r \int_{\mathbb{R}^{3}} d^{3} r^{\prime} \frac{e^{-\kappa\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} U(\mathbf{r}) U\left(\mathbf{r}^{\prime}\right)=1 $$ (b) Taking \(U(\mathbf{r})=e^{-\alpha r} / r\), show that the condition in (22.62) becomes $$ \frac{4 \pi \mu g^{2}}{\alpha \hbar^{2}}\left[\frac{1}{(\alpha+\kappa)^{2}}\right]=1 $$ (c) Since \(\kappa>0\), prove that the equation in (b) has a unique solution only if \(g^{2}>\hbar^{2} \alpha^{2} /(4 \pi \mu)\), in which case the bound- state energy is $$ E=-\frac{\hbar^{2}}{2 \mu}\left[\left(\frac{4 \pi \mu g^{2}}{\alpha \hbar^{2}}\right)^{1 / 2}-\alpha\right]^{2} . $$
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