Laguerre functions

SPECIAL FUNCTIONS
Show that the th spherical Bessel function is given by
1 d
f (x) = (−1) x
f0 (x),
x dx
(18.106)
where f (x) denotes either j (x) or n (x).
The recurrence relation (18.93) for Bessel functions of the first kind reads
d −ν
Jν+1 (x) = −xν
x Jν (x) .
dx
Thus, on setting ν = + 12 and rearranging, we find
d x−1/2 J+1/2
,
x−1/2 J+3/2 (x) = −x
dx
x
which on using (18.102) yields the recurrence relation
d
j+1 (x) = −x [x− j (x)].
dx
We now change + 1 → and iterate this result:
d
j (x) = −x−1 [ x−+1 j−1 (x) ]
dx d
d −+2
x−+1 (−1)x−2
j−2 (x)
= −x−1
x
dx
dx
d
1
d
x
= (−1)2
x−+2 j−2 (x)
x dx x dx
= ···
1 d
= (−1) x
j0 (x).
x dx
This is the expression for j (x) as given in (18.106). One may prove the result (18.106) for
n (x) in an analogous manner by setting ν = − 12 in the recurrence relation (18.92) for
Bessel functions of the first kind and using the relationship Y+1/2 (x) = (−1)+1 J−−1/2 (x). Using result (18.106) and the expressions (18.104) and (18.105), one quickly
finds, for example,
3
sin x cos x
1
3 cos x
,
j2 (x) =
−
,
sin x −
j1 (x) = 2 −
x
x
x3
x
x2
3
cos x sin x
1
3 sin x
,
n2 (x) = −
n1 (x) = − 2 −
−
.
cos x −
x
x
x3
x
x2
Finally, we note that the orthogonality properties of the spherical Bessel functions
follow directly from the orthogonality condition (18.88) for Bessel functions of
the first kind.
18.7 Laguerre functions
Laguerre’s equation has the form
xy + (1 − x)y + νy = 0;
616
(18.107)
18.7 LAGUERRE FUNCTIONS
it has a regular singularity at x = 0 and an essential singularity at x = ∞. The
parameter ν is a given real number, although it nearly always takes an integer
value in physical applications. The Laguerre equation appears in the description
of the wavefunction of the hydrogen atom. Any solution of (18.107) is called a
Laguerre function.
Since the point x = 0 is a regular singularity, we may find at least one solution
in the form of a Frobenius series (see section 16.3):
y(x) =
∞
am xm+σ .
(18.108)
m=0
Substituting this series into (18.107) and dividing through by xσ−1 , we obtain
∞
[(m + σ)(m + σ − 1) + (1 − x)(m + σ) + νx] am xm = 0.
(18.109)
m=0
Setting x = 0, so that only the m = 0 term remains, we obtain the indicial
equation σ 2 = 0, which trivially has σ = 0 as its repeated root. Thus, Laguerre’s
equation has only one solution of the form (18.108), and it, in fact, reduces to
a simple power series. Substituting σ = 0 into (18.109) and demanding that the
coefficient of xm+1 vanishes, we obtain the recurrence relation
am+1 =
m−ν
am .
(m + 1)2
As mentioned above, in nearly all physical applications, the parameter ν takes
integer values. Therefore, if ν = n, where n is a non-negative integer, we see that
an+1 = an+2 = · · · = 0, and so our solution to Laguerre’s equation is a polynomial
of order n. It is conventional to choose a0 = 1, so that the solution is given by
(−1)n n n2 n−1 n2 (n − 1)2 n−2
x
Ln (x) =
+
− · · · + (−1)n n! (18.110)
x − x
n!
1!
2!
n
n!
xm ,
=
(−1)m
(18.111)
(m!)2 (n − m)!
m=0
where Ln (x) is called the nth Laguerre polynomial. We note in particular that
Ln (0) = 1. The first few Laguerre polynomials are given by
L0 (x) = 1,
3!L3 (x) = −x3 + 9x2 − 18x + 6,
L1 (x) = −x + 1,
4!L4 (x) = x4 − 16x3 + 72x2 − 96x + 24,
2!L2 (x) = x2 − 4x + 2,
5!L5 (x) = −x5 + 25x4 − 200x3 + 600x2 − 600x + 120.
The functions L0 (x), L1 (x), L2 (x) and L3 (x) are plotted in figure 18.7.
617
SPECIAL FUNCTIONS
10
L2
5
L0
1
2
4
3
5
7
6
x
L3
−5
L1
−10
Figure 18.7 The first four Laguerre polynomials.
18.7.1 Properties of Laguerre polynomials
The Laguerre polynomials and functions derived from them are important in
the analysis of the quantum mechanical behaviour of some physical systems. We
therefore briefly outline their useful properties in this section.
Rodrigues’ formula
The Laguerre polynomials can be expressed in terms of a Rodrigues’ formula
given by
Ln (x) =
ex dn n −x xe
,
n! dxn
(18.112)
which may be proved straightforwardly by calculating the nth derivative explicitly
using Leibnitz’ theorem and comparing the result with (18.111). This is illustrated
in the following example.
618
18.7 LAGUERRE FUNCTIONS
Prove that the expression (18.112) yields the nth Laguerre polynomial.
Evaluating the nth derivative in (18.112) using Leibnitz’ theorem, we find
ex n dr xn dn−r e−x
Cr
n! r=0
dxr dxn−r
n
Ln (x) =
ex n!
n!
xn−r (−1)n−r e−x
n! r=0 r!(n − r)! (n − r)!
n
=
=
n
(−1)n−r
r=0
n!
xn−r .
r!(n − r)!(n − r)!
Relabelling the summation using the index m = n − r, we obtain
Ln (x) =
n
m=0
(−1)m
n!
xm ,
(m!)2 (n − m)!
which is precisely the expression (18.111) for the nth Laguerre polynomial. Mutual orthogonality
In section 17.4, we noted that Laguerre’s equation could be put into Sturm–
Liouville form with p = xe−x , q = 0, λ = ν and ρ = e−x , and its natural interval
is thus [0, ∞]. Since the Laguerre polynomials Ln (x) are solutions of the equation
and are regular at the end-points, they must be mutually orthogonal over this
interval with respect to the weight function ρ = e−x , i.e.
∞
Ln (x)Lk (x)e−x dx = 0
if n = k.
0
This result may also be proved directly using the Rodrigues’ formula (18.112).
Indeed, the normalisation, when k = n, is most easily found using this method.
Show that
∞
I≡
Ln (x)Ln (x)e−x dx = 1.
(18.113)
0
Using the Rodrigues’ formula (18.112), we may write
∞
dn
(−1)n ∞ dn Ln n −x
1
Ln (x) n (xn e−x ) dx =
x e dx,
I=
n! 0
dx
n!
dxn
0
where, in the second equality, we have integrated by parts n times and used the fact that the
boundary terms all vanish. When dn Ln /dxn is evaluated using (18.111), only the derivative
of the m = n term survives and that has the value [ (−1)n n! n! ]/[(n!)2 0!] = (−1)n . Thus
we have
∞
1
xn e−x dx = 1,
I=
n! 0
where, in the second equality, we use the expression (18.153) defining the gamma function
(see section 18.12). 619
SPECIAL FUNCTIONS
The above orthogonality and normalisation conditions allow us to expand any
(reasonable) function in the interval 0 ≤ x < ∞ in a series of the form
f(x) =
∞
an Ln (x),
n=0
in which the coefficients an are given by
∞
f(x)Ln (x)e−x dx.
an =
0
We note that it is sometimes convenient to define the orthonormal Laguerre functions φn (x) = e−x/2 Ln (x), which may also be used to produce a series expansion
of a function in the interval 0 ≤ x < ∞.
Generating function
The generating function for the Laguerre polynomials is given by
∞
G(x, h) =
e−xh/(1−h)
=
Ln (x)hn .
1−h
(18.114)
n=0
We may prove this result by differentiating the generating function with respect to
x and h, respectively, to obtain recurrence relations for the Laguerre polynomials,
which may then be combined to show that the functions Ln (x) in (18.114) do
indeed satisfy Laguerre’s equation (as discussed in the next subsection).
Recurrence relations
The Laguerre polynomials obey a number of useful recurrence relations. The
three most important relations are as follows:
(n + 1)Ln+1 (x) = (2n + 1 − x)Ln (x) − nLn−1 (x),
Ln−1 (x) =
Ln−1 (x)
−
Ln (x),
(18.115)
(18.116)
xLn (x) = nLn (x) − nLn−1 (x).
(18.117)
The first two relations are easily derived from the generating function (18.114),
and may be combined straightforwardly to yield the third result.
Derive the recurrence relations (18.115) and (18.116).
Differentiating the generating function (18.114) with respect to h, we find
(1 − x − h)e−xh/(1−h)
∂G
=
nLn hn−1 .
=
∂h
(1 − h)3
Thus, we may write
(1 − x − h)
Ln hn = (1 − h)2
nLn hn−1 ,
and, on equating coefficients of hn on each side, we obtain
(1 − x)Ln − Ln−1 = (n + 1)Ln+1 − 2nLn + (n − 1)Ln−1 ,
620