electrostatics manual v1.1

7/31/2019 Electrostatics Manual v1.1

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Electrostatics Manual v1.1

Salvatore Cardamone

21/07/2012


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1. The Laplace Equation1.1 Introduction

Laplaces Equation is a second orderpartial differential equation [PDE] of the form

where is a function which is twice-differentiable. Solutions to Laplaces Equation are termed harmonic

functions. In quantum mechanics [QM], is termed the wave function, and is a function of the positions of a

quantum particle in a system. Such a term features prominently in the Schrodinger equation

Here, denotes all coordinates for which is defined by. In our case, it will prove beneficial to adopt spherical

polar coordinates, i.e. . Note also that the above formulation of the Schrodinger equation is time-independent. The true Schrodinger equation accounts for the fact that the wave function is also a function of

time. However, the addition of this variable is of no use to us, and so we omit it.

In Cartesian coordinates, Laplaces equation may be written

The conversion of this to spherical polar coordinates is long and laborious, and so the result is presented. The

derivation requires simple trigonometry, and may be found in many introductory physics textbooks for the more

pedantic student

( ) ( )

Considering how this is a PDE, we must look for solutions of the form . Separatingthe variables in such a manner, however, requires that our coordinate system be orthogonal, which places certain

constraints upon our solutions, as we shall see later on. For now, we simply state that our coordinate system is

orthogonal, which allows us to write

( ) ( )

We may simplify this by separating the variables, which means multiplying both sides by the reciprocal of

, which leaves us with

Multiplying both sides by

The first term in the above expression contains both rand , meaning we have not fully separated the variables.

However, the final term is a function of alone, meaning that for the equation to be true, this term must be

equal to a constant, which we call m. Whether this constant is positive or negative is as of yet undetermined. We


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may, however, impose a few constraints, in that the potential must be single-valued, i.e. the points , so that .In other words, we require harmonic solutions, which impose a further constraint, in that m must be negative

Note the similarity of this ordinary differential equation [ODE] to that of classical simple harmonic motion

[SHM], whereby

which has solutions

Similarly

Additionally, we require periodicity, so that , which is true if and only ifm is an integer.We shall say that the solution to this part of the Laplace Equation may be represented by .We now replace the term withm2, and divide both sides by leading to

Now the first term is a function of only rwhilst the other two terms are both functions of, so that to be true forall rand , the first term must be equal to a constant,B

Additionally, the sum of the other two terms must equal this same constant, thereby giving

( )

We tackle the solutions to the radial and coltaitudal parts of Laplaces equation separately

1.2 Radial Solution

We have found that

In order to solve this, we implement the product rule


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A differential equation of this form represents anEuler-Cauchy Differential Equation, which is formally

represented as

We have here representedR(r) withy for simplicity in the following calculations. Now, remembering that if we may define , we say that

The second derivative then becomes

(

) [

(

) ]

*

+

We may utilise these results in our Euler-Cauchy Equation

( )

Which is a simple second order ODE, the characteristic equation for which is given in the expanded form

We assume 2 distinct roots to this equation, and , the general solution of which is given by

Remembering that

We shall return to the true identities of the and roots once we have discussed the solution to the coltaitudalpart of Laplaces Equation.

1.3 Colatitudinal Solution

We have found that

We may utilise the product rule once again to expand the term in brackets, and tidying up, we obtain


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Whilst not entirely recognisable in this form, if we define , where ||, the above becomes theSturm-Liouville Equation. Setting m = 0 results in theLegendre Differential Equation. The solutions of this

equation, , are theLegendre Polynomials, .We now use the fact that to derive the values of given as the roots to the radial part of theLaplace Equation. Primarily, we return to

, which reduces to

As such, we state that and , giving us the solution

We now have solutions to each of the variables which make up , meaning we can write as aproduct of all 3 solutions

It is possible to simplify this somewhat, by amalgamating the solutions for and into

These solutions, , are termed Spherical Harmonics of rankl and m, and are defined as the solution tothe angular portion of Laplaces Equation.

We must, however, remember that the values ofl and mare not fixed. The true solution to Laplaces Equation is

in fact a superposition of all allowed values ofl and m, i.e.

2. Legendre PolynomialsAn extremely important factor which we must mention is the orthonormalisation of the Legendre Polynomials.

We again state the function we used to generate a Sturm-Liouville equation

Reformulating this somewhat

* +

It is clear to see in this form that the Sturm-Liouville equation is in fact an eigenvalue equation. This imposes

certain constraints on available solutions. Most important of these is that the eigenfunctions form an

orthonormal basis, which shall be expanded upon now.

2.1 Azimuthally Symmetric

If we take the general Legendre Differential Equation which is azimuthally symmetric, i.e. ignore the solution to

theF()portion of Laplaces Equation


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[ ] And multiply through by

We subtract from this an equivalent expression, but with the roles ofl and n reversed

As opposed to integrating the entirety of this function, we integrate the first term by parts betweenx = -1 andx

= +1. If this term can be substantially simplified, the second term is essentially equivalent and we may do the

same to that

At the boundary conditions we impose, the term becomes zero, meaning that the entirety of theintegral is equal to zero. As such, we disregard the first two terms in the above expression, and are left with

Unless l(l+1) = n(n+1), this means that

Since l and m are non-negative integers, l(l+1) = n(n+1) can only be true ifl = n. As such, we can state the

orthogonality conditions of the Legendre Polynomials

We now need to solve for the constant terms, Cn, This may only be done once we have decided upon a

normalisation condition for the Legendre Polynomials. By convention, the normalisation condition is

In order to evaluate the above integral, we require an explicit way of expressing the Legendre Polynomials,

which may be done by use ofRodrigues Formula

We remind ourselves that we wish to calculate

By use of Rodrigues Formula and large amounts of rearranging, we find that


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It is not the objective of this document to perform long and laborious derivations, particularly those which focus

on extensive algebraic manipulations, but if you wish to see the full work through, the document entitled

Methods of Theoretical Physics: I gives a full account of the problem, and is listed in the references. As such,

we suffice in stating that the orthonormalisation condition of the Legendre Polynomials is given by

We note that this is written in Cartesian coordinates when our formulation of Laplaces equation was given in

spherical polar coordinates. Transformation between the two is simple, by use of the relationship x = cos .

Additionally, our boundary conditions must be altered, which is done by multiplying the integrand by sin and

changing the period over which we integrate to between 0 and

2.2 Associated Legendre Functions

This class of functions requires no constraints placed upon the Legendre functions being azimuthally symmetric.

As such, we state ourAssociated Legendre Equation

[ ] *

+ Where P(x) are the solutions to the Associated Legendre Equation. We begin by stating that ,and substitute this into the above

[ ] * + After substantial levels of supposedly simple algebra [which I cannot seem to derive], we obtain the following

relationship

Where is a Legendre Polynomial and is a solution to the Associated Legendre Equation.Substituting this back into the Associated Legendre Equation

* + * + It is clear that we must have otherwise the m-fold derivative of the lth Legendre Polynomial will givezero. Recall also Rodrigues Formula, which gives us an expression ofPl(x). Substituting this in


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Surprisingly enough, this formula now makes sense for negative values ofm as well as positive ones, provided

that || , as the derivative will still yield a result other than zero. Thus, we have constructed AssociatedLegendre Functions for all integers m in the interval .We also note that the equation itself is invariant under switching m tom, as the only term involving m is a

squared term in the Associated Legendre Equation. This means that if we take a solution with a given m, then

turning m intom gives another separate solution. Analysis of the formula for the Associated Legendre

Functions shows us that both and are regular, and so must be linearly dependent upon oneanother, i.e.

Use of the Associated Legendre Functions and the above relationship allows us to determine the value ofkquite

easily

If we look only at the highest power ofx

Using this result, we can quite easily determine the normalisation integral for the Associated Legendre

Functions. The relevant integral which we need to evaluate is

Using the same method which we already employed for the azimuthally symmetric case, it is easy to show that

this integral is equal to zero, unless l=n. For this, we can make use of the above case

**

We use Rodrigues Formula to evaluate the constants Clm


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Note that by making use of **, we have managed to cancel the terms as opposed to adding them.Integrating the above by parts l+m times, and noting that boundary terms give zero, we eventually obtain

In other words, we have shown that

Which is the orthonormalisation condition of the Associated Legendre Polynomials.

3. The Spherical Harmonics


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electrostatics manual v1.1

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