Mathematical Physics with Partial Differential Equations (Second Edition)

474 Pages·2018·3.488 MB·English

by  Kirkwood, James

Mathematical
Physics with Partial
Differential Equations
Second Edition
James Kirkwood
Professor of Mathematical Sciences,
Sweet Briar College, Sweet Briar, VA, USA
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Preface
Themajorpurposesofthisbookaretopresentpartialdifferentialequations(PDEs)
and vector analysis at an introductory level. As such, it could be considered a
beginning text in mathematical physics. It is also designed to provide a bridge
fromundergraduatemathematicstothefirstgraduatemathematicscourseinphysics,
appliedmathematics,orengineering.Inthesedisciplines,itisnotunusualforsuch
agraduatecoursetocovertopicsfromlinearalgebra,ordinarydifferentialequations
and partial differential equations (PDEs), advanced calculus, vector analysis,
complex analysisand probability,and statistics at ahighly accelerated pace.
Inthistextwestudyindetail,butatanintroductorylevel,areducedlistoftopics
important to the abovementioned disciplines. In PDEs, we consider Green’s func-
tions, the Fourier and Laplace transforms, and how these are used to solve PDEs.
We also study using separation of variables to solve PDEs in great detail. Our
approachistoexaminethethreeprototypicalsecond-orderPDEsdLaplace’sequa-
tion,theheatequation,andthewaveequationdandsolveeachequationwitheach
method. The premise is that in doing so, the reader will become adept at each
method and comfortablewith each equation.
Theotherprominentareaofthetextisvectoranalysis.Whiletheusualtopicsare
discussed, an emphasis is placed on viewing concepts rather than formulas. For
example, we view the curl and gradient as properties of a vector field rather than
assimplyequations.Asignificantportionofthisareadealswithcurvilinearcoordi-
nates toreinforcethe idea of conversionof coordinate systems.
Reasonable prerequisites for the course are a course in multivariable calculus,
familiarity with ordinary differential equations to the point of being able to solve
a second-order boundary problem with constant coefficients, and some experience
with linear algebra.
Indealingwithordinarydifferentialequations,weemphasizethelinearoperator
approach.Thatis,weconsidertheproblemasbeinganeigenvalue/eigenvectorprob-
lemforaself-adjointoperator.Inadditiontoeliminatingsometediouscomputations
regarding orthogonality, this serves as a unifying theme and a more mature
viewpoint.
Thelevelofthetextgenerallyliesbetweenthatoftheclassicencyclopedictexts
ofBoasandKreyszigandthenewertextbyMcQuarrie,andthePDEbooksofWein-
berg and Pinsky. Topics such as Fourier series are developed in a mathematically
rigorous manner. The section on completeness of eigenfunctions of a Sturme
Liouville problem is considerably more advanced than the rest of the text and can
be omitted ifonewishes tomerelyacceptthe result.
Thetextiswrittenatalevelwhereitcanbeusedasaself-containedreference
as well as an introductory text. There was a concerted effort to avoid situations
where filling in details of an argument would be a challenge. One thought in
writing the text was that it would serve as a source for students in subsequent
courses that felt “I know I’m supposed to know how to derive this, but I don’t.”
xiii
xiv Preface
AcoupleofsuchexamplesarethefundamentalsolutionofLaplace’sequationand
the spectrum of the Laplacian.
The major changes from the first edition are first that a chapter on generating
functions has been added. This gives a nice way of considering solutions to
LaGuerre equations, Hermite equations, Legendre’s equations, and Bessel equa-
tions. Second, in depth analyses for the rigid rotor, one-dimensional quantum me-
chanical oscillator, and the hydrogen atom are presented. Also, more esoteric
coordinate systemshave been movedtoan appendix.
CHAPTER
1
Preliminaries
1.1 SELF-ADJOINT OPERATORS
Thepurposeofthistextistostudysomeoftheimportantequationsandtechniques
ofmathematicalphysics.Itisafortuitousfactthatmanyofthemostimportantsuch
equationsarelinear,andwecanapplythewell-developedtheoryoflinearoperators.
Weassumeknowledgeofbasiclinearalgebrabutreviewsomedefinitions,theorems,
and examplesthat will beimportant tous.
Definition:
Alinearoperator(orlinearfunction)fromavectorspaceVtoavectorspaceW,
isa functionL:V/W for which
Lða vb þa vb Þ¼a Lðvb Þþa Lðvb Þ
1 1 2 2 1 1 2 2
for all vb ;vb ˛V and scalars a anda .
1 2 1 2
One of the most important linear operators for uswill be
L½y(cid:2)¼a ðxÞyðxÞþa ðxÞy0ðxÞþa ðxÞy00ðxÞ
0 1 2
wherea (x), a (x) anda (x)are continuousfunctions.
0 1 2
Definition:
IfL:V/V isalinearoperator,thenanonzerovector vbisaneigenvectorofL
with eigenvaluelif LðvbÞ¼lvb.
b
Note that 0 cannot bean eigenvector,but0 can be aneigenvalue.
Example:
d
ForL¼ ,we have
dx
d
LðeaxÞ¼ ðeaxÞ¼aeax
dx
so eaxisan eigenvectorofL with eigenvalue a.
d2
Anextremely important exampleisL¼ .Among its properties are
dx2
d2
LðsinnxÞ¼ ðsinnxÞ¼(cid:3)n2 sinnx and
dx2
d2
LðcosnxÞ¼ ðcosnxÞ¼(cid:3)n2cosnx.
dx2
1
MathematicalPhysicswithPartialDifferentialEquations.https://doi.org/10.1016/B978-0-12-814759-7.00001-6
Copyright©2018ElsevierInc.Allrightsreserved.
2 CHAPTER 1 Preliminaries
WeleaveitasExercise1toshowthatifvbisaneigenvectorofLwitheigenvalue
l, then avbisalsoan eigenvectorofL with eigenvalue l.
Definition:
Aninnerproduct(alsocalledadotproduct)onavectorspaceVwithscalarfield
F (which is the real number or the complex number ) is a function
h ; i:V(cid:4)V/F such that for all f, g, heVand aeF
haf;gi¼ahf;gi;
hf;agi¼ahf;gi; where xþiy¼x(cid:3)iy;
hf þg;hi¼hf;hiþhg;hi;
hf;gi¼hg;fi;
hf;fi(cid:5) 0 with equality if and only iff¼0.
Avectorspacewith an innerproductiscalled aninnerproductspace.
IfV ¼ℝn, the usual inner product for ab¼ða ;.;a Þ; bb¼ðb ;.;b Þ is
1 n 1 n
D E
ab;bb ¼a b þ/þa b .
1 1 n n
If the vector space is ℂn, then we must modify the definition, because, for
example,underthisdefinition, if ab¼ði;iÞ,then
hab;abi¼i2þi2 ¼(cid:3)2:
Thus, onℂn,for ab¼ða ;.;a Þ; bb¼ðb ;.;b Þ, we define
1 n 1 n
D E
ab;bb ¼a b þ/þa b .
1 1 n n
We use the notation hf;fi¼kfk2, which is interpreted as the square of the
length off,and kf (cid:3)gk isthe distance from ftog.
We shall be working primarily with vector spaces consisting of functions that
satisfysomepropertysuchascontinuityordifferentiability.Inthissetting,oneusu-
allydefinesthe innerproductusing an integral.Acommon inner product is
Z
b
hf;gi¼ fðxÞgðxÞdx;
a
where a or bmay be finite orinfinite.There might be aproblem with somevector
spaces in that hf;fi¼0 with fs0. This problem can be overcome by a minor
modificationofthevectorspaceorbyrestrictingthefunctionstobeingcontinuous
and will not affect our work. We leave it as Exercise 4 to show that the function
defined aboveisan innerproduct.
On some occasions it will be advantageous to modify the inner product above
with aweight function w(x). Ifw(x)(cid:5)0 on [a,b],then
Z
b
hf;gi ¼ fðxÞwðxÞgðxÞdx
w
a
isalso aninnerproductas we showin Exercise5.
1.1 Self-Adjoint Operators 3
Definition:
A linear operator A on the inner product space V is self-adjoint if
hAf;gi¼hf;Agi forall f;gεV.
Self-adjoint operators are prominent in mathematical physics. One example is
the Hamiltonian operator. It is a fact (Stone’s theorem) that energy is conserved if
and only if the Hamiltonian is self-adjoint. Another example is shown below. Part
of the significance of thisexampleisdue toNewton’s law F¼ma.
Example:
d2
The operator isself-adjoint on the inner product space
dx2
V ¼ffjf has a continuous second derivative and is periodic on ½a;b(cid:2)g;
with innerproduct
Z
b
hf;gi¼ fðxÞgðxÞdx.
a
We mustshow
hf00;gi¼hf;g00i;
that is,
Z Z
b b
f00ðxÞgðxÞdx¼ fðxÞg00ðxÞdx.
a a
Todo this, we integratebypartstwice.Let
u¼gðxÞ du¼g0ðxÞ
dv¼f00ðxÞ v¼f0ðxÞ
so
Z (cid:2) Z
(cid:2)b b
uv(cid:3) vdu¼gðxÞf0ðxÞ(cid:2) (cid:3) f0ðxÞg0ðxÞdx.
a a
(cid:2)
(cid:2)b
The periodicityoffand g forces gðxÞf0ðxÞ(cid:2) ¼0. Thus,
Z Z a
b b
f00ðxÞgðxÞdx¼(cid:3) f0ðxÞg0ðxÞdx:
a a
Integrating the integralon the rightby parts with
u¼g0ðxÞ du¼g00ðxÞ
dv¼f0ðxÞ v¼fðxÞ
4 CHAPTER 1 Preliminaries
we have
2 3
Z (cid:2) Z Z
b (cid:2)b b b
(cid:3) f0ðxÞg0ðxÞdx¼(cid:3)4fðxÞg0ðxÞ(cid:2)(cid:2) (cid:3) fðxÞg00ðxÞdx5¼ fðxÞg00ðxÞdx.
a a a a
Noticethatif[a,b]isoflength2p,thenfsinðnxÞcosðnxÞjn˛ZgisasubsetofV.
Wenextprovetwoimportant facts about self-adjoint operators.
Theorem:
IfL:V/V isaself-adjoint operator,then
1. The eigenvalues ofL are real;
2. Eigenvectorsof L with different eigenvalues are orthogonal; that is, their inner
productis0.
Proof:
1. Suppose thatfisaneigenvectorof Lwith eigenvalue l. Then
hLf;fi¼hlf;fi¼lhf;fi
and
hf;Lfi¼hf;lfi¼lhf;fi.
Since L isself-adjoint,
hLf;fi¼hf;Lfi so lhf;fi¼lhf;fi
andsincehf;fis0, we have l¼l, solis real.
2. Suppose
Lf ¼l f and Lg¼l g with l sl .
1 2 1 2
Then
hLf;gi¼hl f;gi¼l hf;gi
1 1
and
hLf;gi¼hf;Lgi¼hf;l gi¼l hf;gi.
2 2
So
l hf;gi¼l hf;gi
1 2
andthus hf;gi¼0 because l sl .
1 2
Example:
Wehave
Z
2p
sinðnxÞcosðmxÞdx¼0;
0