Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
Подождите немного. Документ загружается.
11.6
PROBLEMS
323
11.33. Find the asymptotic dependeoce
of
the modified Bessel function
of
the
second kind:
K
(a)
==!
[
e-(aI2)(z+1/z)~
IJ 2 Jc
zv+l'
where C starts at
00,
approaches the origin and circles it, and goes back to
00.
Thus the positive real axis is excluded from the domain of analyticity.
Additional Reading
I. Denoery, P.and Krzywicki, A.Mathematics for Physicists, Harper andRow,
1967.
2. Lang, S.
ComplexAnalysis, 2nd ed., Springer-Verlag, 1985. Contains a very
lucid discussion of analytic continuation.
I
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
Part IV _
Differential Equations
12 _
Separation
of
Variables in Spherical
Coordinates
The laws of physics are almost exclusively writteu in the form of differential
equations (DEs).
In
(point) particle mechanics there is only one independentvari-
able, leading to
ordinary
differential
equations
(ODEs).
In
otherareas of physics
in which extended objects such as fields are studied, variations with respect to po-
sition are also important. Partial derivatives with respect to coordinate variables
show up in the differential equations, which are therefore called
partial
differen-
tiai eqnations (PDEs). We list'the most common PDEs
of
mathematical physics
in the following.
12.1 PDEs of Mathematical Physics
In
electrostatics, where time-independent scalar fields such aspotentialsand vector
fields such as electrostatic fields are studied, the law is described by Poisson's
Poisson's
equation
equation,
V
2<1>(r)
=
-41fp(r).
(12.1)
Laplace's
equation
In
vacuum, where
p(r)
= 0, Equation (12.1) reduces to
Laplace's
equation,
(12.2)
Many electrostatic problems involve couductors held at constant potentials and
situatedinvacuum. In the space betweensuchconductingsurfaces, the electrostatic
potential obeys Equation
(12.2).
heat
equation
The most simplified version of the
heat
equation
is
(12.3)
328 12.
SEPARATION
OF
VARIABLES
IN
SPHERICAL
COOROINATES
where T is the temperature aud a is a constautcharacterizing the medium io which
heat is flowing.
One of the most frequently recurriog PDEs encountered in mathematical
wave
equation
physics is the wave eqnation,
(12.4)
This equation (or its simplification to lower dimensions) is applied to the vibration
of
striogs aud drums; the propagation of sound io gases, solids, aud liquids; the
propagation
of
disturbauces in plasmas; aud the propagation
of
electromagnetic
waves.
The Schriidinger
equation,
describiog nonrelativistic quautum phenomena,
Schrodinger
equation
is
where m is the mass of asubatomicparticle, Ii is Plauck's constaut (divided by 2".),
V is the potential energy
of
the particle, aud IW(r,
t)I
2
is the probability density
of
findiog the particle at r at time t.
A relativistic generalization
of
the Schrodinger equation for a free particle
of mass
,m is
the
Klein-Gordon
eqnation,
which, in terms of the natural units
(Ii = 1 = c), reduces to
Klein-Gordon
equation
~
aw
-
-V
2
w+
V(r)W
=
-Ui-,
2m at
2 2 a
2
q,
Vq,-mq,=-2'
at
(12.5)
(12.6)
time
is
separated
from
space
Equations (12.3-12.6) have partial derivatives with respect to time. As a first
step toward solviog these PDEs aud as au iotroduction to similar techniques used
io the solution of PDEs not iovolviog time,
1
let
us separate the time variable. We
will denote the functions io all four equations by the generic symbol W(r,
t).
The
basic idea is to separate the r aud t dependence ioto factors: W(r, t) es
R(r)T(t).
This factorization permits us to separate the two operations of space differentiation
aud time differentiation.
Let
L staud for all spatial derivative operators aud write
all the relevaut equations either as
LW
=
aw
jat
or as
L\j1
= a
2w
jat
2
•
With this
notation
andthe
above
separation,
we have
L(RT)
=
T(LR)
=
{RdTjdt,
Rd
2Tjdt
2
1See [Hass99]fora
thorough
discussionof
separation
in
Cartesian
and
cylindrical
coordinates.
Chapter
19of thisbookalso
contains
examples
of
solutions
tosome
second-order
linear
DEs
resulting
from
such
separation.
12.1
PDES
DF
MATHEMATICAL
PHYSICS
329
Dividing both sides by RT, we obtain
(12.7)
Now comes the crucial step in the process
of
separation
of
variables. The LHS
of Equation
(12.7) is a function
of
position alone, and the RHS is a function of
time alone. Since r and t are iudependent variables, the only way that (12.7) can
hold is for
both sides to be constant, say a:
1
-LR
= a
=}
LR =
«R
R
1
dT dT
--=a
=}
-=aT
T
dt
dt
We have reduced the original time-dependent POE to an ODE,
dT
=aT
dt
or
(12.8)
and a POE involving only the position variables, (L -
a)R
=
O.
The most general
formofL-a
arising from Equations (12.3-12.6) is
L-a
sa V
2
+
j(r).
Therefore,
Equations
(12.3-12.6) are equivalent to (12.8), and
V
2R
+
j(r)R
=
O.
(12.9)
ToincludePoisson'sequation,we replacethe zero on theRHS by
g(r)
==
-41!'p(r),
obtaining V
2R
+
j(r)R
=
g(r).
With the exception
of
Poisson's equation (an
inhomogeneous POE), in all theforegoing equations the term on the RHS is
zero.'
We will restrict ourselves to this so-called homogeneous case and rewrite (12.9)
as
V
2W(r)
+
j(r)W(r)
=
O.
(12.10)
Depending on the geometry
of
the problem, Equation (12.10) is further separated
into ODEs each involving a single coordinate of a suitable coordinate system. We
shall see examples of all major coordinate systems (Cartesian, cylindrical, and
210most cases, a is chosentobe real.In thecaseof theSchrodinger equation, it is moreconvenientto choosea to be
purely
imaginary
sothatthei inthe
definition
of Lcanbe
compensated.
Inall cases,theprecise
nature
ofa is
determined
by
boundary
conditions.
3Techniques
forsolving
inhomogeneous
PDEsarediscussed in
Chapters
21 and22.
330
12.
SEPARATION
OF
VARIABLES
IN
SPHERICAL
COORDINATES
spherical)
in
Chapter
19.
For
the
rest
of
this
chapter,
we
shall
concentrate
on
some
general
aspects
of
the
spherical
coordinates.
Jean
Le
Rond
d'A1embert (1717-1783) was the illegitimate son of a famous sa-
lon hostess
of
eighteenth-century Paris and a cavalry officer. Abandoned by his mother,
d'Alembert was raised by a foster family and later educated by the arrangement
of
his
father at a nearby church-sponsored school, in
which
he received instructionin the classics
and above-average instructionin mathematics. Afterstudying law and medicine, he
finally
choseto pursuea careerin mathematics.In the 17408he
joined
the ranks
of
the philosophes,
a growing group
of
deistic and materialistic thinkers and writers
who
actively questioned
the social and intellectual standards
of
the day. He traveled little (he leftFrance only once,
to visit the court
of
Frederick the Great), preferring instead the company
of
his friends in
the salons, among
whom
he was well known for his wit and laughter.
D'Alembert turned his mathematical and philosophical
talents to many
of
the outstanding scientific problems
of
the
day, with mixed success. Perhaps his most famous scien-
tific work, entitled 'Iraite de dynamique, shows his appre-
ciation that a revolution was taking place in the science of
mechanics-the
fonnaIization
of
the principlesstatedby New-
ton into a rigorous mathematical framework. The philoso-
phy to which d'Alembert subscribed, however, refused to ac-
knowledge the primacy
of
a concept as unclear and arbitrary
as "force,"introducinga certainawkwardnessto his treatment
and perhaps causing
him
to overlook the important principle
of conservation of
energy.
Later,d'Alembertproduceda treatiseon fluidmechanics (the
priority of which is still debated by historians), a paper dealing with vibrating sltings (iu
which the wave equation makes its first appearance in physics), and a skillful treatment
of
celestial mechanics. D' Alembert is also credited with
use
of
the first partial differential
equation as well as the first solution
to such an equation using
separation
of
variables.
(One should be careful interpreting ''first'': many of d'Alembert's predecessors and con-
temporaries gave similar. thoughless satisfactory. treatments
of
these milestones.) Perhaps
his
most
well-known contribution to mathematics (at least among students) is the ratio test
for the convergence
of
infinite series.
Much
of
the work for which d' Alembert is remembered occurred outside mathemat-
ical physics. He was chosen as the science editor
of
the Encyclopedie, and his lengthy
Discours Preliminaire in that volume is considered one
of
the defining documents
of
the
Enlightenment. Other works included writings on law, religion, and music.
Since d'Alembert's final years
were
not especially happy ones, perhaps this account
of
his life should eud with a glimpse at the bwnanity his philosopby often gave his work.
Like many
of
his contemporaries, he considered the problem
of
calculating the relative
risk
associated
with
the new practice
of
smallpox inoculation. which in rare cases caused
the disease it was designed to prevent. Although
not
very successful in the mathematical
sense.he was careful to pointout that the probability
of
accidentalinfection,howeverslight
or elegantly derived, would be small consolation
to a father whose child died from the
12.2
SEPARATION
OFTHEANGULAR PARTOFTHE
LAPLACIAN
331
inoculation.
It
is greatlyto his creditthat d'Alembertdidnot believesuchconsiderations
irrelevant to the problem.
12.2 Separationofthe Angular
Part
ofthe Laplacian
angular
momentum
operator
commutation
relations
between
components
01
angular
momentum
operator
With Cartesian and cylindrical variables, the boundary conditions are important
in determining the nature
of
the solutions
of
the
ODE
obtained from the POE.
In
almost all applications, however, the angular
part
of
the spherical variables can
be separated and studied very generally. This is because the angular part
of
the
Laplacian in the spherical coordinate system is closely related to the operation
of
rotation
and
the angular momentum, which are independent
of
any particular
situation.
The
separation
of
the angular
part
in spherical coordinates can be done in a
fashion exactly analogous to the separation
of
time by writing Ij1 as a product
of
three functions, each depending on ouly one
of
the variables. However, we
will follow an approach that is used in quantum mechanical treatments
of
angular
momentum. This approach, which is based on the operator algebra
of
Chapter 2
and is extremelypowerfuland elegant, gives solutionsfor the angular
part
in closed
form.
Define the vector operator
pas p=
-iV
so that its
jth
Cartesian component
is
Pj =
-i8/8xj,
for j =
1,2,3.
In quantum mechanics p(multiplied by
Ti)
is
the momentum operator.
It
is easy tu verify that"
[Xj,
Pk] =
i8jk
and
[Xj,
xk]
=
0=
[Pj'
Pk]·
We can also define the
angular
momentum
operator
as C= r x
p.
This
is expressed in components as
Li = (r X P)i = EijkXjPk for i =
1,2,3,
where
Einstein's summation convention (summing over repeated indices) is utilized.f
Using the commutation relations above, we obtain
[Lj, Lk] = iEjk/L/.
We will see shortly that Ccan be written solely in terms
of
the angles eand
tp, Moreover, there is one factor
of
pin the definition
of
C,
so
if
we square
C,
we
will get two factors
of
p,
and
a Laplacian may emerge in the expression for C.
L.
In this marmer, we may be able to write
'11
2
in terms
of
L
2
, which depends only on
4Theseoperatorsact on the space of functions possessingenough"nice"propertiesas to render
the
spacesuitable. The operator
xj
~ply
multiplies functions, while
p'
differentiates them.
It
IS assumed that the readerisfamiliar with vector algebrausing indices and such objects as
8ij
and
Eijk'
For
an introductory
treatment, sufficientfor
our
presentdiscussion, see [Hass 99]. A more advanced treatment
of
these objects (tensors) can be found
in
Part
VII
of
this book.
332 12.
SEPARATION
OF
VARIABLES
IN
SPHERICAL
COORDINATES
angles.
Let
us try this:
3
2
""
"
L =
L·
L =
.!...J
LiL; =
EijkX
jPkEimllXmPn
=
EijkEimn
X
jPkXmP
n
;=1
=
(8jm8kn
-
8jn8km)XjPkXmPn
=
XjPkXjPk
-
XjPkXkPj'
Weneedto write this expressionin such a way that factors with the same index are
next to each other, to give a dot product. We must also
try, when possible, to keep
the
pfactors to the right so that they can operateon functions without intervention
from the
x factors. We do this using the commutation relations between the
x's
and the p's:
L
2
=
Xj(XjPk
-
i8kj)Pk
-
(PkXj
+i
8kj)XkPj
=
XjXjPkPk
-
iXjPj
-
PkXkXjPj
-
iXjPj
=
XjXjPkPk
-
2ixjPj
-
(XkPk
-
i8kk)XjPj'
Recalling that
8kk
=
Ei~l
8kk
= 3 and
XjXj
=
E]~l
XjXj
= r·r= r
2
etc., we
can write
L
2
= r
2
p.p+
rr·
p-
(r.
p)(i'.
p),
which, if we make the substitution
p=
-iV,
yields
V
2
=
_r-
2
L
2
+
r-
2(r.
V)(r·
V)
+
r-
2
r · V.
Letting both sides act on the function "'(r,
e,
<p),
we get
2 I 2 1 1
V
'"
=
--L
'"
+
-(r·
V)(r·
V)'"
+
-r·
V"'.
r
2
r
2
r
2
(12.11)
(12.12)
Laplacian
separated
into
angular
and
radial
parts
But we note that
r·
V =
reT'
V =
ra/ar.
We thus get the final form of
\72",
in
spherical coordinates:
V2",
=
_~L2",
+
~~
(r
a"')
+
~
a",.
r
2
r
ar
ar
r
ar
It
is important to note that Equation (12.11) is a general relation that holds in
all.coordinate systems. Although all the manipulations leading to it were done in
Cartesian
coordinates,
sinceitis
written
in vector
notation,
there
is no
indication
in the final form that it was derived using specific coordinates.
Equation (12.12) is the spherical version of (12.11) and is the version we shall
use. We will first make the simplifying assumption that in Equation (12.10), the
master equation,
fer)
is a function
of
r only. Equation (12.10) then becomes
1 2 1 a (a"') I
a",
--L
'"
+
--
r-
+
--
+
f(r)'"
=
O.
r
2
r
ar ar
r
ar
Assuming, for the time being, that L
2
depends only on eand
<p,
and separating
'"
into a product of two functions,
"'(r,
e,
<p)
=
R(r)Y(e,
<p),
we can rewrite this