NORMALIZATION OF THE LEGENDRE POLYNOMIALS

8. NORMALIZATION OF THE LEGENDRE POLYNOMIALS

If we take the scalar product of a vector with itself, A · A = A 2 , we get the square of the length (or norm) of the vector. If we divide A by its length, we get a unit vector. In Chapter 3, Section 14 we showed that we can think of functions as the vectors of a vector space and we defined the norm N of a function A(x) on (a, b) by [see Chapter 3, equation (14.2)]

A ∗ (x)A(x) dx =

|A(x)| 2 dx = N 2 .

We also say that the function N −1 A(x) is normalized ; like a unit vector, a normal- ized function has norm = 1. The factor N −1 is called the normalization factor. For

Section 8 Normalization of the Legendre Polynomials 579 example, π

0 sin 2 nx dx = π/2. Then the norm of sin nx on (0, π) is

functions normalized orthogonal functions is called orthonormal. For example,

is an orthonormal set on (0, π). Such a set of orthonormal functions may remind us of i, j, k; like these unit vectors, the functions are orthogonal and have norm = 1. If the elements of a vector space are functions, we can then use a (complete) orthonormal subset of the functions as the basis vectors of the space. We think of expanding other functions in terms of them (by analogy with writing a three-dimensional vector in terms of

i, j, k). For example, suppose we have expanded a given function f (x) on (0, π) in

a Fourier sine series:

f (x) =

sin nx.

We call f (x) a vector with components B n in terms of the basis vectors Thus, in quantum mechanics, we often refer to a function which describes the state of a physical system as either a state function or a state vector. Just as we can write a three-dimensional vector in terms of i, j, k, or in terms of another basis, say

e r ,e θ ,e φ , so we can expand a given f (x) in terms of another orthonormal set of functions and find its components relative to this new basis. In Section 9, we shall see how to expand functions in Legendre series.

Just as we needed the norm of sin nx in Fourier series, so we shall need the norm of P l (x) in expanding functions in Legendre series. We shall prove that

[P l (x)] 2 dx = 2 .

Then the functions l (x) are an orthonormal set of functions on (−1, 1). To prove (8.1), we use the recursion relation (5.8b), namely,

lP l (x) = xP ′ l (x) − P ′ l−1 (x).

Multiply (8.2) by P l (x) and integrate to get

(8.3) l [P

2 l (x)] dx =

xP l (x)P ′ l (x) dx −

P l (x)P ′ l−1 (x) dx.

The last integral is zero by Problem 7.4. To evaluate the middle integral in (8.3), we integrate by parts:

xP l (x)P

l (x) dx = [P l (x)]

[P l (x)] dx

[P l (x)] 2 dx

(see Problem 2.2). Then (8.3) gives

2 1 l 1 [P l (x)] dx = 1 − [P l (x)] 2 dx

580 Series Solutions of Differential Equations Chapter 12

which simplifies to (8.1). We can combine (7.1) and (8.1) to write

P l (x)P m (x) dx =

PROBLEMS, SECTION 8

Find the norm of each of the following functions on the given interval and state the normalized function.

1. cos nx on (0, π) 2. P 2 (x) on (−1, 1) 3. xe −x/2 on (0, ∞)

4. e −x 2 /2 on (−∞, ∞) 5. xe −x 2 /2 on (0, ∞) Hint: See Chapter 4, Section 12. 6. Give another proof of (8.1) as follows. Multiply (5.8e) by P l (x) and integrate from

−1 to 1. To evaluate the middle term, integrate by parts. Then use Problem 7.4. 7. Using (8.1), write the first four normalized Legendre polynomials and compare with

the answers we found by a different method in Chapter 3, Section 14, Example 6.