Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
Подождите немного. Документ загружается.
276 8 Legendre and Bessel Functions
self-adjoint. The final step is to define
v
m
1 x
2
m/2
u
m
u
m
1 x
2
m/2
v
m
(8.64)
and to show that v
m
satisfies a self-adjoint equation. Allowing to evaluate
the derivatives
ux_1 x
2
m/2
vx
u
!
x
mx1 x
2
1
m
2
vx1 x
2
m/2
v
#
x
u
!!
x/ / Simpli fy
1 x
2
2
m
2
m
1 1 mx
2
vx
1 x
2
2mxv
#
x
1 x
2
v
##
x
and substituting into the differential equation for u
m
A 1 x
2
u
!!
x2xm 1u
!
x 1mm 1ux/ / Simpli fy
1 x
2
1
m
2
m
2
1 x
2
1
vx
1 x
2
2x v
#
x
1 x
2
v
##
x
A1 x
2
m/2
/ / Simpli fy
m
2
1 x
2
1
vx
1 x
2
2x v
#
x
1 x
2
v
##
x
1 x
2
we conclude that y
m
does indeed satisfy the associated Legendre equation for 0 m if
y
satisfies the ordinary Legendre equation.
The associated Legendre polynomials can now be expressed in terms of Rodrigues’
formula as
P
x
1
2
!
x
x
2
1
P
,m
x
m
1 x
2
m/2
2
!
x
m
x
2
1
(8.65)
or
P
,m
x
m
1 x
2
m/2
x
m
P
x (8.66)
where the Condon–Shortley phase
m
proves convenient for quantum mechanics. Note
that P
,0
xP
x.Here is denoted the degree and m the order of P
,m
. Using Rodrigues’
formula we immediately find
P
,m
x
m
P
,m
x (8.67)
and
m 0 P
,m
10 (8.68)
Although our derivation assumed that m is nonnegative, one can show that the generalized
Rodrigues’ formula offers a valid solution any integer in the range m . However,
the solutions for negative m are related to those for positive m according to
P
,m
x
m
m!
m!
P
,m
x (8.69)
and thus are not independent and we do not obtain a normal completeness relation. Never-
theless, the solutions for negative m will be useful for spherical harmonics in which the Φ
dependence distinguishes the sign of m.
8.2 Legendre Functions 277
Some attention must be given to the phase convention for associated Legendre poly-
nomials. Many authors of textbooks on mathematical physics, such as Arfken or Butkov,
omit the factor of
m
in the relationship between P
and P
,m
on the grounds that it is an
unnecessary complication at this stage. However, this phase is convenient for applications
of spherical harmonics in quantum mechanics, which are based upon associated Legendre
functions. Therefore, other authors of recent textbooks, like Lea, do include this factor
in P
,m
. Hence, it is incumbent upon the reader to be cognizant of the conventions chosen
by the author. (Caveat emptor!) We have chosen to include the
m
phase for three practi-
cal reasons: (1) it is used by the most useful compilation of mathematical formulas, that by
Abramowitz and Stegun; (2) my favorite mathematical software, ,alsouses
this phase in its definition of associated Legendre functions; and (3) it provides a more
natural extension to complex arguments.
The orthogonality properties for associated Legendre functions can also be developed
by using Rodrigues’ formula to write
1
1
P
,m
xP
'
,m
xx
'
2
'
!
'
!
1
1
1 x
2
m
x
m
1 x
2
x
'
m
1 x
2
'
x (8.70)
where it is important to note that m is the same for both Legendre functions. We assume,
without loss of generality, that
'
and integrate by parts
'
m times to obtain
1
1
P
,m
xP
'
,m
xx
'
'
m
2
'
!
'
!
1
1
1 x
2
'
x
'
m
1 x
2
m
x
m
1 x
2
x (8.71)
where the integrated terms vanish at both limits. The integrand can be simplified using
Leibnitz’s expansion of multiple derivatives of a product
x
n
f xgx
n
m0
n
m
f
nm
xg
m
x (8.72)
in terms of binomial coefficients and products of derivatives. Thus, we obtain
x
'
m
1 x
2
m
x
m
1 x
2
'
m
k0
'
m
k
x
'
mk
1 x
2
m
x
mk
1 x
2
(8.73)
Notice that because 1 x
2
Μ
has only 2Μ nonvanishing derivatives, nonvanishing terms
require
'
m k 2m
m k 2
^
M
M
_
M
M
`
'
,
'
m k m (8.74)
278 8 Legendre and Bessel Functions
Yet we assumed
'
. Therefore, this integral vanishes for
'
, demonstrating the
orthogonality of associated Legendre functions with the same m but different values.
Furthermore, the normalization for
'
can be obtained from the single term with k
m, such that
1
1
P
,m
x
2
x
m
2
!
2
m
m
1
1
1 x
2
x
2m
1 x
2
m
x
2
1 x
2
x
1
2
!
2
m
m
2m!2!
1
1
1 x
2
x
(8.75)
Although the remaining integral can be evaluated most simply as a finite sum by expanding
the integrand, more clever techniques are needed to obtain the simple closed-form result
1
1
1 x
2
x
2
21
!
2
2 1!
(8.76)
and are relegated to the problems at the end of the chapter. Collecting these results, we
finally obtained the orthogonality integral
1
1
P
,m
xP
'
,m
xx
2
2 1
m!
m!
∆
,
'
(8.77)
Having two indices, the associated Legendre functions exhibit a wider variety of recur-
sion relations than the ordinary Legendre functions. Some recursion relations vary the
degree, others the order, and many others vary both. Among these are
P
,m2
xm 1
2x
1 x
2
P
,m1
x
1mm 1
P
,m
x0 (8.78)
xP
,m
m 11 x
2
1/ 2
P
,m1
P
1,m
0 (8.79)
P
1,m1
2 11 x
2
1/ 2
P
,m
P
1,m1
0 (8.80)
m 1P
1,m
2 1xP
,m
mP
1,m
0 (8.81)
but we leave the derivations as exercises. Note that the choice of phase affects some of the
signs in these recursion relations. A simple method for deriving recursion relations for m 2
0 is to begin either with the differential equation or with one of the recursion relations for
Legendre polynomials, differentiate m times, and multiply by a power of 1 x
2
1/ 2
to
produce an analogous relation for associated Legendre functions. With further analysis
one can show that the same relations apply under more general conditions, complex x ! z
and nonintegral or even complex ,m!Ν, Μ.
Rather than develop further properties of associated Legendre functions, I prefer to
proceed directly to the spherical harmonics in which the Φ dependence distinguishes the
sign of m.
8.2 Legendre Functions 279
8.2.6 Spherical Harmonics
For problems in three dimensions it is convenient to define angular basis functions with a
simple orthonormality with respect to solid angle, such that
Y
,m
ˆrY
'
,m
'
ˆrE
2Π
0
1
1
Y
,m
Θ, ΦY
'
,m
'
Θ, Φcos ΘΦ∆
,
'
∆
m,m
'
(8.82)
The dependence of the spherical harmonics Y
,m
Θ, Φ upon the polar angle is obviously
related to the associated Legendre functions while the dependence upon the azimuthal
angle is proportional to Fourier functions
mΦ
. Thus, we define
Y
,m
Θ, Φ
2 1
4Π
m!
m!
1/ 2
P
,m
cos Θ
mΦ
(8.83)
where the ugly factor provides the desired normalization. Using the standard definition
for P
,m
with m<0, we obtain the conjugation property
Y
,m
Θ, Φ
m
Y
,m
Θ, Φ (8.84)
Similarly, under inversion of the coordinate system
ˆr !ˆr Θ!ΠΘ, Φ!ΠΦ (8.85)
and using
P
,m
x
m
P
,m
x , ExpmΦ Π
m
ExpmΦ (8.86)
to write
Y
,m
Π Θ, ΠΦ
Y
,m
Θ, Φ (8.87)
we identify the parity
Y
,m
ˆr
Y
,m
ˆr
(8.88)
as
. It is also useful to know that
Y
,m
Θ, 0
2 1
4Π
1/ 2
P
cos Θ ∆
m,0
(8.89)
Y
,m
ˆz
Y
,m
0, 0
2 1
4Π
1/ 2
∆
m,0
(8.90)
The degree is usually related to orbital angular momentum while the order m is the
projection of the total angular momentum on the polar axis.
280 8 Legendre and Bessel Functions
8.2.7 Multipole Expansion
Recognizing that the Legendre functions are complete for the polar angle while the trigono-
metric functions are complete for the azimuthal angle, we expect the spherical harmonics
to provide a complete orthonormal basis for functions defined on the unit sphere, such that
f ˆr
0
m
f
,m
Y
,m
ˆr (8.91)
f
,m
f ˆrY
,m
ˆr
E
2Π
0
1
1
f Θ, ΦY
,m
Θ, Φ
cos ΘΦ
2Π
0
Π
0
f Θ, ΦY
,m
Θ, Φ
SinΘ Θ Φ (8.92)
where the coefficients f
,m
are constants. Suppose that the angular delta function
∆ˆr ˆr
'
∆cos Θcos Θ
'
∆Φ Φ
'
0
m
A
,m
ˆr
'
Y
,m
ˆr (8.93)
is expanded with respect to unprimed variables using expansion coefficients which are
functions of the primed variables, here treated as parameters. Using the orthogonality prop-
erties, the expansion coefficients become
A
,m
ˆr
'
∆ˆr ˆr
'
Y
,m
ˆr
E Y
,m
ˆr
'
(8.94)
Therefore, the completeness relation for spherical harmonics takes the form
∆ˆr ˆr
'
0
m
Y
,m
ˆrY
,m
ˆr
'
,m
Y
,m
ˆrY
,m
ˆr
'
(8.95)
where the final sum uses a short-hand notation that indicates summation with respect to
both variables over their natural ranges, namely 0 < , m .
Notice that it does not matter which of the spherical harmonics in the completeness
relation is conjugated because this combination is real. The summation over m resembles
the summation over components used to evaluate the scalar product of two vectors. In fact,
Y
,m
represents a component of a spherical tensor of rank and this summation represents
a scalar product between two spherical tensors of the same rank. Thus, it is convenient to
define a scalar product of the form
Y
ˆr,Y
ˆr
'
[
m
Y
,m
ˆrY
,m
ˆr
'
(8.96)
and to express the completeness relation in the more compact form
∆ˆr ˆr
'
Y
ˆr,Y
ˆr
'
(8.97)
8.2 Legendre Functions 281
An interesting current application of multipole expansions is to angular variations of
the cosmic microwave background. The relic microwave background is approximately
uniform, with an average temperature of 2.72 K, but small fluctuations carry information
about the expansion of the early universe and can be used to test theories of inflation. Thus,
one writes
TΘ, Φ
T
,m
Y
Θ, Φ ,A
m
T
,m
2
(8.98)
where 4ΠA
0
represents the average temperature, A
1
is related to the motion of the sun
relative to the preferred reference frame, and higher multipole strengths carry the desired
cosmological information.
Next consider a scalar function of
r expanded in the form
f
r
0
m
f
,m
rY
,m
ˆr (8.99)
f
,m
r
f
rY
,m
ˆr
E (8.100)
where the coefficients are radial functions. For example, if f
r represents a density,
the expansion coefficients would be described as multipole densities.If f is real, such
that
f
f f
,m
m
f
,m
(8.101)
we find that the multipole densities share the conjugation properties of spherical harmon-
ics.
8.2.8 Addition Theorem
Suppose that two vectors,
r and
r
'
, are described by polar and azimuthal angles (Θ, Φ) and
(Θ
'
, Φ
'
) and that Γ represents the angle between them, such that cos Γˆr , ˆr
'
.Usingthe
completeness of spherical harmonics, we can expand P
cos Γ in terms Y
,m
Θ, Φ according
to
P
cos Γ
'
0
'
m
'
A
'
,m
Θ
'
, Φ
'
Y
'
,m
Θ, Φ (8.102)
where the expansion coefficients
A
'
m
Θ
'
, Φ
'
Y
'
,m
Θ, Φ
P
cos Γ E (8.103)
282 8 Legendre and Bessel Functions
depend upon the other pair of angles. Recognizing that P
cos Γ is real and using the
conjugation property of spherical harmonics
P
cos Γ P
cos Γ
'
0
'
m
'
A
'
,m
Θ
'
, Φ
'
Y
'
,m
Θ, Φ
'
0
'
m
'
m
A
'
,m
Θ
'
, Φ
'
Y
'
,m
Θ, Φ
'
0
'
m
'
m
A
'
,m
Θ
'
, Φ
'
Y
'
,m
Θ, Φ
(8.104)
we observe that the coefficients
A
'
,m
Θ
'
, Φ
'
m
A
'
,m
Θ
'
, Φ
'
(8.105)
share the same conjugation property. (Orthogonality of spherical harmonics ensures that
this relation applies term by term.) Furthermore, ˆr , ˆr
'
is symmetric with respect to the
exchange of primed and unprimed coordinates. Therefore, the expansion should take the
form
P
ˆr , ˆr
'
'
0
'
m
'
B
'
,m
Y
'
,m
Θ, ΦY
'
,m
Θ
'
, Φ
'
(8.106)
where the coefficients are independent of angles and this combination of spherical har-
monics is real. Finally, because P
ˆr , ˆr
'
is a scalar function its value must be independent
of the orientation of the coordinate system. Suppose that we choose a coordinate system
in which ˆr
'
ˆz such that
P
ˆr , ˆr
'
P
cos Θ
'
0
'
m
'
B
'
,m
Y
'
,m
Θ, Φ
2
'
1
4Π
1/ 2
∆
m,0
'
0
B
'
,m
2
'
1
4Π
P
'
cos Θ
(8.107)
The orthogonality of Legendre functions limits the rhs to the single term with
'
.
Therefore, we obtain the addition theorem
P
ˆr , ˆr
'
4Π
2 1
m
Y
,m
ˆrY
,m
ˆr
'
(8.108)
or, using the scalar product,
P
ˆr , ˆr
'
4Π
2 1
Y
ˆr,Y
ˆr
'
(8.109)
8.2 Legendre Functions 283
The addition theorem proves invaluable in a wide variety of problems. Some of the
most important examples are based upon the multipole expansion of
1
r
r
'
0
r
<
r
1
>
P
ˆr , ˆr
'
(8.110)
which we can now express in the form
1
r
r
'
0
4Π
2 1
r
<
r
1
>
Y
ˆr,Y
ˆr
'
(8.111)
8.2.8.1 Example: Far-Field Solution to Poisson’s Equation
The electrostatic potential Ψ
r around a charge distribution Ρ
r satisfies Poisson’s equa-
tion
+
2
Ψ
r4ΠΡ
rΨ
r
G
r,
r
'
Ρ
r
'
3
r
'
(8.112)
where the Green function for a point charge is simply
+
2
G
r,
r
'
4Π∆
r
r
'
G
r,
r
'
1
r
r
'
0
4Π
2 1
r
<
r
1
>
Y
ˆr,Y
ˆr
'
(8.113)
such that
Ψ
r
Ρ
r
'
r
r
'
3
r
'
(8.114)
The multipole expansion of the charge density takes the form
Ρ
r
,m
Ρ
,m
rY
,m
ˆr (8.115)
Ρ
,m
r
Ρ
rY
,m
ˆrE (8.116)
Suppose that r is much larger than any r
'
with appreciable charge density. A far-field solu-
tion for the electrostatic potential of a localized charge distribution can then be expanded
as
r D r
'
Ψ
r
,m
4Π
2 1
q
,m
r
1
Y
,m
ˆr (8.117)
where the spherical multipole moments q
,m
are given by
q
,m
r
Y
,m
ˆrΡ
r
3
r
0
Ρ
,m
rr
2
r (8.118)
Recognizing that the contributions of higher multipole moments are suppressed by increas-
ing powers of r, one expects that it is sufficient for very large r to employ only the lowest
few multipoles. Therefore, the multipole expansion provides a very efficient solution for
asymptotically large distances.
284 8 Legendre and Bessel Functions
8.2.9 Legendre Functions of the Second Kind
Generally a second-order differential equation must possess two linearly independent solu-
tions, but our anticipated application to the polar angle dependence of physics problems
with x CosΘ in the range 1 x 1 selects P
n
x as the most relevant of the two solu-
tions. The (sometimes) implicit requirement of analyticity at x 1 is actually a boundary
condition that forces the eigenvalue in the differential equation
1 x
2
y
''
x2xy
'
xΛyx0 (8.119)
to assume the values Λnn1 where n is an integer and selects the corresponding eigen-
functions P
n
x that are regular at the singular points of the differential equation (where the
coefficient of y
''
vanishes). However, if the polar angle is limited to a more restricted
range xa<1, the second solution is required. Linear homogeneous boundary condi-
tions at the smaller angular limits still yield an eigenvalue problem, but the eigenvalues are
more general and the eigenfunctions are linear combinations of the two solutions.
We apply the Frobenius method to construct solutions from power series about the
origin. Let
yxx
r
n0
a
n
x
n
(8.120)
where the appropriate value of r has yet to be determined. Direct substitution gives
n0
a
n
n rn r 11 x
2
x
nr2
2n rx
nr
Λx
nr
0 (8.121)
or
n2
a
n2
n r 2n r 1x
nr
n0
a
n
n rn r 1Λ
x
nr
(8.122)
The left-hand side has two powers that are absent from the right, namely n 2, 1,but
the coefficients of both vanish if we choose r 0. In order that this equation be satisfied
for all x within the radius of convergence, we equate the coefficients for like powers of x
to obtain a recursion relation
a
n2
a
n
nn 1Λ
n 2n 1
(8.123)
that determines two solutions, one for even and another for odd n. Therefore, the general
solution takes the form
yxa
0
1 Λ
x
2
2!
6 ΛΛ
x
4
4!
6 Λ20 ΛΛ
x
6
6!
,,,
a
1
x 2 Λ
x
3
3!
2 Λ12 Λ
x
5
5!
2 Λ12 Λ30 Λ
x
7
7!
,,,
(8.124)
where a
0
and a
1
are arbitrary.
8.2 Legendre Functions 285
Convergence is a key issue. The ratio test
lim
n!
a
n2
a
n
R
2
< 1 R<1 (8.125)
informs us that solutions for arbitrary Λ diverge on the unit circle. (Although the ratio test
is technically inconclusive at x1, the integral test is conclusively divergent.) However,
divergence is avoided when Λ1 for nonnegative integer causes the power series to
terminate in a polynomial of order that is identified as the familiar Legendre polynomi-
als P
x.If is even, the function associated with a
0
is proportional to P
while the function
associated with a
1
diverges at x 1, whereas if is odd a
1
produces P
while a
0
produces
a divergent function. The Legendre functions of the second kind, Q
x, are identified with
the divergent series except for a conventional normalization yet to be specified.
The lowest few Q
are illustrated in Fig. 8.2. Obviously, these functions become more
oscillatory as increases and have parity Q
x
1
Q
x opposite that of the corre-
sponding P
. For integer order one can factor the logarithmic divergence at the end points
from polynomial portions to obtain
Q
z
1
2
P
z Log
1 z
1 z
"/ 2#
m0
2 4m 1
2m 1!! m
P
12m
z (8.126)
Closer analysis shows that P
Ν
11 for all Ν,butP
Ν
x diverges logarithmically at
x !1forΝ. However, Q
Ν
x exhibits the opposite behavior: it is divergent at both
singular points of the differential equation, x 1, for Ν but is finite at x 1
for Ν2 1/ 2.
8.2.10 Relationship to Hypergeometric Functions
With suitable variable transformations, the hypergeometric equation
t1 tu
''
t
ΓΑΒ1t
u
'
tΑΒut0
ytAFΑ, Β, Γ,tBt
1Γ
F1 ΓΑ, 1 ΓΒ, 2 Γ,t (8.127)
contains as special cases many of the second-order differential equations of interest to
mathematical physics. Using standard methods, one can show that the power series
FΑ, Β, Γ,t
n0
Α
n
Β
n
Γ
n
t
n
n!
(8.128)
where
Α
0
1 (8.129)
Α
n
ΑΑ1Α n 1 (8.130)
converges for t < 1. Furthermore, when Α or Β is a nonpositive integer, the series termi-
nates and the hypergeometric function F reduces to a polynomial. Although neither time