Puzzled about Section 3.12 of Jackson's E&M book

[ForgetfulPhysicist]

Hi wizards,

I'm working through Jackson's book on E&M (3rd edition) and got stuck in section 3.12 on expansions of Green functions. I have three questions regarding section 3.12:

First, why is Jackson trying to find a Green function that satisfies equation 3.156? To my beginner mind, it seems incorrect to pursue 3.156 because I thought Green functions in electrostatics must always follow equation 1.39, not eqn 3.156. Equation 3.156 seems drastically different that 1.39 due to the additional (f(x) + gamma)G term. What am I not understanding? What electrostatics geometry would lead to equation 3.156? In all previous derivations of Green functions for electrostatics, the process started with 1/|x-x'| and creating a series expansion of 1/|x-x'| in the compatible coordinate system, wherein the series expansion converges to 1/|x-x'| in the domain and simultaneously produces the desired boundary conditions on the boundaries. Equation 3.156 doesn't start with 1/|x-x'| and also doesn't start with equation 1.39. What am I not understanding?

Second, why is "the z coordinate is singled out for special treatment" in equation 3.168? In that paragraph Jackson considers a rectangular box with sides at x = 0, y = 0, z = 0, x = a, y = b, z = c, and ALL of these planes need a boundary condition correct? So why single out just the z-dimension?

Third, can anyone give me some references that would actually USE the final product (aka equation 3.168) of section 3.12? It seems useless, but I'm a rookie. It would be nice if someone had a reference that created a Green function for a box with all sides V=0... can that be accomplished with equation 3.168? It doesn't seem so, since equation 3.168 wasn't derived with equation 1.39 as the starting point.

Thank you all who can help sort out my puzzled mind.

 

[Orodruin]

Could you please write out the equations you are referring to? Everybody does not have Jackson easily accessible.

 

[ForgetfulPhysicist]

Orodruin and all, attached are pdfs of the relevant sections of the book, but there may be context or information missing, so please let me know if you want other sections uploaded.

 

[Orodruin]

I believe you are mixing up the specific Green's function of the Laplace equation with the general concept of Green's functions.

A Green's function $G(x,x')$ to a linear differential operator $L_x$ acting on the $x$ variable is a function that satisfies
$$
L_x G(x,x') = \delta(x-x')
$$
(sign conventions may differ). For the Laplace operator, the Green's function is proportional to $1/|x-x'|$ (in three dimensions), but this is not generally true for any differential operator. The Green's function mentioned in Chapter 3 is not the Green's function of the Laplace operator, it is the Green's function of $\nabla^2 + f(x) + \lambda$.Edit: If you have access to my book, Chapter 7 covers Green's functions in general and their application to several situations in physics.

 

[vanhees71]

Fortunately there's much more freedom. A Green's function for the (negative) Laplace operator is defined by
$$-\Delta G(\vec{x},\vec{x}')=\delta^{(3)}(\vec{x}-\vec{x}').$$
The most simple solution is
$$G_0(\vec{x},\vec{x}')=\frac{1}{4 \pi |\vec{x}-\vec{x}'|},$$
but of course you can add any harmonic function to it, i.e.,
$$G(\vec{x},\vec{x}')=G_0(\vec{x},\vec{x}') + h(\vec{x},\vec{x}')$$
with
$$\Delta_{\vec{x}} h(\vec{x},\vec{x}')=0$$
is a Green's function of $-\Delta$ too, and that's indeed needed, if you want a Green's function that satisfies certain boundary conditions, as discussed in the said long chapter in Jackson. E.g., if you want a Green's function for the solution of the Poisson equation,
$$-\Delta \Phi(\vec{x})=\frac{1}{\epsilon} \rho(\vec{x})$$
with a given charge density $\rho(\vec{x})$ and otherwise "vacuum", $G_0$ is the right Green's function to use, i.e., the solution in this case is
$$\Phi(\vec{x})=\frac{1}{\epsilon} \int_{\mathbb{R}^3} \mathrm{d}^3 x' G_0(\vec{x},\vec{x}') \rho(\vec{x}').$$
But if you have in addition some matter around, you need other Green's functions to fulfill boundary conditions. E.g., you might have a conducting grounded sphere with radius $a$ around the origin. Then the sphere must be an equipotential surface with $\Phi(\vec{x})|_{\vec{x} \in S}=0$, and you can use $h(\vec{x},\vec{x}')$ to fulfill this boundary condition (a Cauchy boundary condition). In other situations you may have other constraints. Another viable boundary condition is the Neumann boundary condition, where you give the values of $\vec{n} \cdot \vec{\nabla} \Phi$, where $\vec{n}$ is a unit-normal vector to the surface, etc. For all these situations, if mathematically consistent (e.g., you can in general not give both Cauchy and Neumann condistions for a given surface) you can look for the corresponding Green's function.

 

[Orodruin]

but Green's functions relevant to electrostatic problems should always utilize the Laplacian as the linear operator, right?

That sounds a bit too summary. It will be very common that the linear operator is the Laplace operator, but not really that it will always be the case.

 

[vanhees71]

I think it's all about Green's functions of the Laplace operator in these chapters. At one place Jackson, however, uses a complete set of orthonormal eigenfunctions to derive series expansions for Green's functions in terms of these orthonormal sets. Among the most important examples are the (solid) spherical harmonics ("multipole expansion").

 

[ForgetfulPhysicist]

Fortunately there's much more freedom. A Green's function for the (negative) Laplace operator is defined by
$$-\Delta G(\vec{x},\vec{x}')=\delta^{(3)}(\vec{x}-\vec{x}').$$
The most simple solution is
$$G_0(\vec{x},\vec{x}')=\frac{1}{4 \pi |\vec{x}-\vec{x}'|},$$
but of course you can add any harmonic function to it, i.e.,
$$G(\vec{x},\vec{x}')=G_0(\vec{x},\vec{x}') + h(\vec{x},\vec{x}')$$
with
$$\Delta_{\vec{x}} h(\vec{x},\vec{x}')=0$$
is a Green's function of $-\Delta$ too, and that's indeed needed, if you want a Green's function that satisfies certain boundary conditions, as discussed in the said long chapter in Jackson. E.g., if you want a Green's function for the solution of the Poisson equation,
$$-\Delta \Phi(\vec{x})=\frac{1}{\epsilon} \rho(\vec{x})$$
with a given charge density $\rho(\vec{x})$ and otherwise "vacuum", $G_0$ is the right Green's function to use, i.e., the solution in this case is
$$\Phi(\vec{x})=\frac{1}{\epsilon} \int_{\mathbb{R}^3} \mathrm{d}^3 x' G_0(\vec{x},\vec{x}') \rho(\vec{x}').$$
But if you have in addition some matter around, you need other Green's functions to fulfill boundary conditions. E.g., you might have a conducting grounded sphere with radius $a$ around the origin. Then the sphere must be an equipotential surface with $\Phi(\vec{x})|_{\vec{x} \in S}=0$, and you can use $h(\vec{x},\vec{x}')$ to fulfill this boundary condition (a Cauchy boundary condition). In other situations you may have other constraints. Another viable boundary condition is the Neumann boundary condition, where you give the values of $\vec{n} \cdot \vec{\nabla} \Phi$, where $\vec{n}$ is a unit-normal vector to the surface, etc. For all these situations, if mathematically consistent (e.g., you can in general not give both Cauchy and Neumann condistions for a given surface) you can look for the corresponding Green's function.

Thank you Vanhees71,

Yes I understood your point, thanks, but let me drill down directly at my confusion:

I thought the methodological approach for using a Green's function to solve an electrostatics problem went via the following steps, in order of operation:
1) create a representation of 1/|x-x'| via a series expansion of 1/|x-x'| in the relevant coordinate system
2) create an "add on" function (h(x,x') as you call it (wherein Laplacian of h(x,x') = 0) that causes the Green's function to behave as desired at the boundaries.

But Jackson's equation 3.156 doesn't follow this order of operations. Rather, it starts with a Laplacian differential equation (equation 3.156) with added terms (namely, (f(x) + gamma)G ) that don't appear to have any relevance to any electrostatics problem in the entire book. I can't imagine why adding (f(x) + gamma)G over the entire domain would help G satisfy an electrostatics boundary condition. This causes the beginning of section 3.12 to seem to be irrelevant and useless for electrostatics understandings ... sigh...

 

[ForgetfulPhysicist]

Moving onto the second part of my questions about section 3.12:
Why is "the z coordinate is singled out for special treatment" in equation 3.168? A rectangular box needs ALL of the bounding planes to have a boundary condition correct? So why would Jackson single out just the z-direction for "special" treatment? The decision to single out the z-direction seems puzzling and lacking direction or motivation or reason.

It's sad that EVERY paragraph of section 3.12 seems to have directionless logic/reason. And no application to a real world problem is given. Can anyone point to a Green's function used in electrostatics that is similar to the discussion of section 3.12?

 

[ForgetfulPhysicist]

I believe you are mixing up the specific Green's function of the Laplace equation with the general concept of Green's functions.

A Green's function $G(x,x')$ to a linear differential operator $L_x$ acting on the $x$ variable is a function that satisfies
$$
L_x G(x,x') = \delta(x-x')
$$
(sign conventions may differ). For the Laplace operator, the Green's function is proportional to $1/|x-x'|$ (in three dimensions), but this is not generally true for any differential operator. The Green's function mentioned in Chapter 3 is not the Green's function of the Laplace operator, it is the Green's function of $\nabla^2 + f(x) + \lambda$.Edit: If you have access to my book, Chapter 7 covers Green's functions in general and their application to several situations in physics.

Insofar as Jackson's explanation of Green's functions goes, Green's functions only need to satisfy L_x G(x,x') = \delta(x-x') -- see Chapter 1, especially equation 1.31. Jackson's explanations of Green's functions give no reason to pursue arbitrary operators like $\nabla^2 + f(x) + \lambda$.

What would ever motivate an electrostatics Green's function with arbitrary operator $\nabla^2 + f(x) + \lambda$?

I understand why a f(x) "add on" term might be needed to force the Green's function to satisfy boundary condition requirements, but why would it be written like $\nabla^2 + f(x) + \lambda$ instead of just a generic F(x,x') that satisfies L_x F(x,x') = 0 as was done by Jackson in eq 1.41?

 

[Orodruin]

Insofar as Jackson's explanation of Green's functions goes, Green's functions only need to satisfy L_x G(x,x') = \delta(x-x') -- see Chapter 1, especially equation 1.31. Jackson's explanations of Green's functions give no reason to pursue arbitrary operators like $\nabla^2 + f(x) + \lambda$.

What would ever motivate an electrostatics Green's function with arbitrary operator $\nabla^2 + f(x) + \lambda$?

I understand why a f(x) "add on" term might be needed to force the Green's function to satisfy boundary condition requirements, but why would it be written like $\nabla^2 + f(x) + \lambda$ instead of just a generic F(x,x') that satisfies L_x F(x,x') = 0 as was done by Jackson in eq 1.41?

The additional terms are not related to the boundary conditions. They form part of the linear differential operator $L_x$.

 

[ForgetfulPhysicist]

The additional terms are not related to the boundary conditions. They form part of the linear differential operator $L_x$.

How do you know they aren't related to the boundary conditions?

The fact that they are "part of the linear differential operator" is irrelevant to whether they are related to the boundary conditions or not.

Jackson explains that the core fundamental operator for Green's functions for electrostatics problems is the Laplacian. Thus, it seems that any "additional" terms would only be added because of an effort to satisfy the boundary conditions.

 

[Orodruin]

How do you know they aren't related to the boundary conditions?

Because boundary conditions show up in the boundary conditions of the Green’s function, not in the linear operator. The linear operator related to a Green’s function by definition is the linear operator that results in a delta inhomogeneity when acting on the Green’s function.

The fact that they are "part of the linear differential operator" is irrelevant to whether they are related to the boundary conditions or not.

No, it most certainly is not.

Thus, it seems that any "additional" terms would only be added because of an effort to satisfy the boundary conditions.

That would be an erroneous conclusion. Boundary conditions are handled through boundary conditions on the Green’s function itself and its resulting values and derivatives on the boundary.

 

[ForgetfulPhysicist]

Because boundary conditions show up in the boundary conditions of the Green’s function, not in the linear operator. The linear operator related to a Green’s function by definition is the linear operator that results in a delta inhomogeneity when acting on the Green’s function.No, it most certainly is not.That would be an erroneous conclusion. Boundary conditions are handled through boundary conditions on the Green’s function itself and its resulting values and derivatives on the boundary.

Ok, then what other purpose do those extra $f(x) + \lambda$ terms serve? Green's theorem in equation 1.35 has only a Laplacian operator. It seems that it doesn't allow other linear operators to enter in -- only a Laplacian. So can you explain why the extra $f(x) + \lambda$ would ever be needed in the operator? There seems to be no way for those extra terms to enter into Green's Theorem for an electrostatics problem. Can you explain?

 

[Orodruin]

It seems that it doesn't allow other linear operators to enter in -- only a Laplacian.

Why would you think so? If you are looking only on the Green’s function of the Laplace operator, yes, but the method of Green’s functions is so much more than that.

Either way, the section that you are reading is not really related to that, but more related to finding an expansion of the Green’s function in terms of eigenfunctions of the Laplace operator or, more generally to a linear differential operator of the given form. It is explicitly stated later that there is a specialisation to f(x)=0.

 

[ForgetfulPhysicist]

Why would you think so? If you are looking only on the Green’s function of the Laplace operator, yes, but the method of Green’s functions is so much more than that.

Either way, the section that you are reading is not really related to that, but more related to finding an expansion of the Green’s function in terms of eigenfunctions of the Laplace operator or, more generally to a linear differential operator of the given form. It is explicitly stated later that there is a specialisation to f(x)=0.

Yes this post is looking only at electrostatics; the conversation was always clarified to be been limited to electrostatics; and it was mentioned a couple times that the scope of Green's functions was limited accordingly. So yes, it seems we are limited to Green's theorem's that have a lone Laplacian operator.

 

[vanhees71]

I also find the treatment of the method with eigenfunctions not very illuminating in Jackson. In my opinion one should anyway flip the order in the standard theoretical-physical curriculum and teach quantum mechanics 1 (non-relativistic quantum mechanics in the 1st-quantization formalism) first. Then the very important technique for solving linear partial equations using complete sets of orthonormal function systems on the Hilbert space $\mathrm{L}^2(\mathbb{R}^3)$ becomes much more natural.

E.g., the multipole expansion is nothing else than using $r=|\vec{x}^2|$, $\vec{L}^2=-(\vec{x} \times \vec{\nabla})^2$, and $\hat{L}_z=-\mathrm{i} \partial_{\varphi}$ (the latter in spherical coordinates $(r,\vartheta,\varphi)$ with the azimuthal angle, $\varphi$) as a complete compatible set of self-adjoint operators to build the corresponding complete set of orthonormal functions.

If you have a boundary-value problem, where the surfaces are easily formulated in terms of such a choice of coordinates, it's usually possible to find the appropriate Green's function for these boundary conditions. That's why these "generalize Fourier transformations" are so important.

 

[ForgetfulPhysicist]

I also find the treatment of the method with eigenfunctions not very illuminating in Jackson. In my opinion one should anyway flip the order in the standard theoretical-physical curriculum and teach quantum mechanics 1 (non-relativistic quantum mechanics in the 1st-quantization formalism) first. Then the very important technique for solving linear partial equations using complete sets of orthonormal function systems on the Hilbert space $\mathrm{L}^2(\mathbb{R}^3)$ becomes much more natural.

E.g., the multipole expansion is nothing else than using $r=|\vec{x}^2|$, $\vec{L}^2=-(\vec{x} \times \vec{\nabla})^2$, and $\hat{L}_z=-\mathrm{i} \partial_{\varphi}$ (the latter in spherical coordinates $(r,\vartheta,\varphi)$ with the azimuthal angle, $\varphi$) as a complete compatible set of self-adjoint operators to build the corresponding complete set of orthonormal functions.

If you have a boundary-value problem, where the surfaces are easily formulated in terms of such a choice of coordinates, it's usually possible to find the appropriate Green's function for these boundary conditions. That's why these "generalize Fourier transformations" are so important.

Agreed! Also, Jackson's previous sections (e.g., 3.9) had ALREADY performed an expansion via eigenfunctions when he derived the Legendre equation from the Laplacian in spherical coordinates and then connected the eigenfunction expansion to 1/|x-x'| in terms of the spherical harmonics (eigenfunctions). So it's puzzling that Jackson would create section 3.12 and pretend it's new and different -- simultaneously not giving any practical applications for it.

 

[TSny]

Also, Jackson's previous sections (e.g., 3.9) had ALREADY performed an expansion via eigenfunctions when he derived the Legendre equation from the Laplacian in spherical coordinates and then connected the eigenfunction expansion to 1/|x-x'| in terms of the spherical harmonics (eigenfunctions). So it's puzzling that Jackson would create section 3.12 and pretend it's new and different -- simultaneously not giving any practical applications for it.

I believe that there are important differences between the methods in sections 3.9 and 3.12. I found it very helpful to work through the details of Jackson's rectangular box problem which starts at the bottom of page 120 in section 3.12 of the 2nd edition. Using the method in this section, the Green function shown as (3.167) is obtained very quickly from the general result (3.160) with $\lambda = 0$.

Equation (3.168) shows a different form of the Green function for the same problem. This result is obtained by the different method illustrated in sections 3.9 and 3.11. I found this method to be much more tedious than the method in section 3.12 (at least for this example).

The method of section 3.9 leads to (3.168) which is a double-sum expansion for the Green function. The individual terms are functions of the primed and unprimed variables, $f_{lm}(x, y, z, x', y', z')$. You can check that each term individually satisfies Laplace's equation. That is, $\nabla^2 f_{lm}(x, y, z, x', y', z') = 0$ (except at $z = z'$ where the second derivative with respect to $z$ of the term does not exist). The derivatives in the Laplacian are taken with respect to the unprimed variables. In contrast, the method of section 3.12 yields (3.167) which is a triple-sum for the Green function. The individual terms $F_{lmn}(x, y, z, x', y', z')$ of this sum do not satisfy Laplace's equation. Instead, they are eigenfunctions of the Laplacian operator with nonzero eigenvalues: $\nabla^2 F_{lmn}+ k_{lmn}^2 F_{lmn}= 0$. The $k_{lmn}$ are given explicitly in (3.166).

So, section 3.9 expands the Green function in terms of solutions of Laplace's equation; whereas, section 3.12 expands in terms of eigenfunctions of the Laplacian.

Problem 3.21 at the end of the chapter shows three different expressions for the Green function for a grounded cylindrical box. I believe the first two expressions are obtained using the method of section 3.9 (or 3.11). In the first expression, the variable $z$ was "singled out for special treatment". In the second expression, the variable $\rho$ was singled out. The third expression is found using the method of 3.12.