Orthogonal Polynomials 1 Introduction
Orthogonal Polynomials
Intro duction
Mathematically orthogonal means p erp endicular that is at right angles For example the set of vectors
fx y z g in three dimensional sppace are orthogonal The concept of orthogonality can b e formalised with
resp ect to vectors as
x y x z y z
that is the dot pro duct of two orthogonal vectors is zero Indeed the set of vectors fx y z g is an orthogonal
set If fx y z g are all unit vectors wesaythat fx y z g is an orthonormal set of vectors Orthogonal vectors
are linearly indep endent and fx y z g therefore spans three dimensional space Thus any p oint in space
may b e written in terms of its comp onents in the x y and z directions or indeed in terms of comp onents
in the directions of any three orthogonal vectors In this w ay fx y z g forms a basis for three dimensional
space
Two functions f x and f x are orthogonal on an interval a b if
i j
Z
b
f x f x dx
i j
a
Mathematically wesay that the inner product of the functions f x andf x is zero The functions are
i j
orthonormal if
Z
b
f x f x dx
i j ij
a
where
i j
ij
i j
is the Kronecker delta function
A set of orthogonal p olynomials is an in nite sequence of p olynomials p x p x p x where p x
n
has degree n and anytwo p olynomials in the set are orthogonal to each other that is
Z
b
f x f x dx for i j
i j
a
The set of p olynomials is orthonormal if
Z
b
f x f x dx
i j ij
a
The interval a b is called the interval of orthogonalityandmay b e in nite at one or b oth ends
Any in nite sequence of p olynomials fp g with p having degree n forms a basis for the in nite dimensional
n n
vector space of all p olynomials Such a sequence can b e turned into an orthogonal basis using the Gram
Schmidt orthogonalisation pro cess by pro jecting out the comp onents of each p olynomial that are orthog
onal to the p olynomials already chosen
Hilb ert Space
Hilb ert David Hilb ert space generalises the idea of Euclidean space that is three dimensional
vector space etc to in nite dimensional spaces Mathematically a Hilb ert space is an inner product space
that is complete
Hilb ert spaces typically arise as in nite dimensional function spaces An element of a Hilb ert space can
be expressed byits co ordinates with resp ect to an orthonormal basis of the space directly analogous to
fx y z g co ordinates in three dimensional vector space
Note
Complete means that as a sequence of functions approaches some limit then the limit is also part of the
Hilb ert space For example if a molecular wave function is expanded in terms of orthogonal basis functions
of a Hilb ert space then the molecular wavefunction is also an element of the Hilb ert space
The concept of Hilb ert space o ers one of the best mathematical form ulations of quantum mechanics
Quantum mechanical states wavefunctions are describ ed byvectors in a Hilb ert space quantum mechan
ical observables can b e expressed by linear op erators and quantum measurement is related to orthogonal
pro jection
Prop erties of Orthogonal Polynomials
All sets of orthogonal p olynomials havea numb er of fascinating prop erties
Any p olynomial f x of degree n can be expanded in terms of p p p that is there exist
n
P
n
co e cients a such that f x a p x
i i i
i
g each p olynomial p x is orthogonal to Given an orthogonal set of p olynomials fp x p x
k
any p olynomial of degree k
Any orthogonal set of p olynomials fp x p x g has a recurrence formula that relates any three
consecutive p olynomials in the sequence that is the relation p a x b p c p exits
n n n n n n
where the co e cients a b and c dep end on n Such a recurrence formula is often used to generate
higher order memb ers in the set
Each p olynomial in fp x p x g has all n of its ro ots real distinct and strictly within the
interval of orthogonality ie not on its ends This is an extremely unusual prop erty It is particularly
imp ortant when considering the classes of p olynomials that arise as quantum mechanical solutions to
agiven Hamiltonian or other Hermitian op erator see b elow
th th
Furthermore the ro ots of the n degree p olynomial p lie strictly inside the ro ots of the n
n
degree p olynomial p
n
Examples of Orthogonal Polynomials
The eld of orthogonal p olynomials was develop ed in the late th century and many of the sets of orthog
onal p olynomials describ ed arose from descriptions of sp eci c physical problems
Legendre Polynomials
These p olynomials are orthogonal on the interval They arise as solutions to Legendre s di erential
equation Adrien Marie Legendre
P x P x
x P x with n n x
x x
This equation is frequently encountered in Physics particularly when spherical p olar co ordinates are used
and where the problem has cylindrical symmetry Legendre p olynomials form part of the quantum me
chanical solution for rotational motion
P x
P x x
P x x
P x x x
P x x x
P x x x x
P x x x x
Laguerre Polynomials
The Laguerre p olynomials L are solutions to Laguerre s di erential equation Edmond Laguerre
n
L L
n
n
x x L with n
n
L x
n
This equation arises in the radial part of the Hamiltonian for one electron atoms
L x
L x x
x x L x
L x x x x
L x x x x x
L x x x x x x x
L x x x x x x x
x
The Laguerre p olynomials are orthogonal over the interval with resp ect to the weight function e
Indeed the radial comp onent of the wavefunction of the one electron atom is obtained bymultiplying the
x
Laguerre p olynomials bye
Hermite Polynomials
Hermite p olynomials H x arise as part of the solution to the quantum harmonic oscillator Hamiltonian
n
They are solutions to Hermite s equation Charles Hermite
H
n
xH H with n
n n
H
n
The Hermite p olynomials are orthogonal over the interval with resp ect to the weight function
x
e Indeed the quantum harmonic oscillator wavefunctions are obtained by multiplying the Hermite
x
p olynomials bye
H x
H x x
H x x x
H x x x
x x x H
H x x x x
H x x x x
Op erators
Every observable physical quantity can b e characterised byanop erator
An op erator is a mathematical device that converts one function into another It is one step up from a
function which is a device that converts one numb er or collection of numbers into another
In general a caret or hat is used to denote an op erator
The simplest form of op erator arises for physical observables that are just functions of p osition co ordinates
These just multiply the functions on which they op erate
Example The classical p otential energy of a harmonic oscillator is kx The asso ciated quantum
kx thatis mechanical op erator V just multiplies any function by
V x kx x
Amoreinteresting typ e of op erator is a di erential op erator
d
For instance the op erator changes the function sin x to cos x
dx
d
sin x cos x
dx
If a particle s p osition is describ ed by co ordinates x y z then its momentum in the x direction is given
by the op erator
p i h
x
x
Where partial derivatives have b een used b ecause there are often several co ordinates to consider The form
of the quantum op erator for momentum p is unusual it do es not app ear to haveany corresp ondance with
x
classical momentum mv This form represents another of the basic axioms on whichquantum mechanics
has b een develop ed
If wecan write some physical quantityin terms of p osition and momentum variables then the rules for
constructing op erators are
All p osition variables remain unchanged
A momentum in the direction of a co ordinate q is replaced by the op erator i h
q
Notes
We almost always work with momentum in quantum mechanics rather than velo city
This second rule applies to angular co ordinates as well as Cartesian ones For example rotation
ab out the z axis is often describ ed using a p olar co ordinate Angular momentum ab out the z axis
is then describ ed by the op erator i h
Exp ectation Values
In quantum mechanics every physical observable Q can be characterised by an op erator Q and the
exp ectation value of Q is calculated as
R
Q d
R
hQi
d
R
Here d is a conventional notation meaning that weintegrate over all the variables and over all space
The order of the factors in the numerator is imp ortant op erators op erate to the right only so in this
case Q op erates only on not on This do es not matter for p osition op erators which op erate by
multiplication but is crucial for di erential op erators
Dirac Notation
The notation j i is often used for the wavefunction Often we abbreviate writing j i as jni j i or
n n
jni is called a ket vector
h j is called a bra vector The bra notation implies the complex conjugate
a complete bra ket expression like h j i or h jQj i implies integration over all space
Thus
Z
h j i d
Z
j j d
Z
h jQj i Q d
and
h jQj i
hQi
h j i
Eigenfunctions and Eigenvalues
Supp ose that wehave an op erator Q andawavefunction that satisfy the equation
q
Q q
q q
where q is just a numb er with the appropriate dimensions That is if dep ends on some set of variables
q
x x then q is indep endent of all these variables
Equation is an eigenvalue equation The wavefunction is an eigenfunction and q is the corre
q
sp onding eigenvalue
Much of the e ort in practical quantum chemistry involves nding eigenfunctions and eigenvalues of inter
esting op erators
The op erator for energy is called the Hamiltonian and its symbol is H Its eigenvalue equation is
Schr odinger s time indep endent equation
H E
In general this equation has many solutions
H E n
n n n
and wehave a set of eigenfunctions for Hamiltonian In such cases the symbol do es not carry any
n
information and the lab el n is imp ortant so it is common to use the Dirac notation jni which gives the
lab el more prominence
A general prop ertyof the set of eigenfunctions of an op erator like the Hamiltonian is that they are or
thogonal
Z
a
hmjni dx if m n
n
m
Another imp ortant prop erty is that the eigenfunctions form a complete set pro vided that E as
n
n That is any function of the same variables with the same b oundary conditions can b e expressed
as a linear combination of the
n
X
c
n n
n
Hermiticity
This is an imp ortant technicality Hermiticity is a generalisation of complex conjugation under complex
y
conjugation an op erator Q b ecomes its Hermitian conjugate Q
Consider the complex conjugation
Z
Q d h jQj i
Z
Q d
hQ j i
y
However we can consider the bra vector hQ j in terms of a new op erator Q Where we de ne
y
fQj ig jQ i hQ j h jQ
Therefore we can write
y
h jQj i hQ j i h jQ j i
y
Thus every op erator Q has a Hermitian conjugate Q which can be de ned using eq or its
integral equivalent
Z Z
y
Q d Q d
An op erator that is equal to its Hermitian conjugate is said to b e Hermitian
d d
Example Find the Hermitian conjugates of the op erators and i
dx dx
Z Z
d d
dx dx
dx dx
Z
d
dx
dx
d d
y
therefore
dx dx
Z Z
d d
i i dx dx i
dx dx
Z
d
dx i
dx
d d
y
therefore i i
dx dx
d d
Thus i is Hermitian cf the momentum op erator but is not
dx dx
Prop erties of Hermitian op erators
Their eigenvalues are always real
Eigenfunctions corresp onding to di erent eigenvalues are orthogonal
Pro of Show the eigenfunctions of a Hermitian op erator are real and that eigenfunctions with di erent
eigenvalues are orthogonal
Supp ose and are eigenfunctions of an Hermitian op erator B with eigenvalues b and b Then using
Dirac notation
B j i b j i a
B j i b j i b
Premultiply eq a by and integrate over all space this is a standard trick and is used in most pro ofs
hanics in quantum mec
h jB j i b h j i
Nowtake the complex conjugate
h j i h jB j i b
Now use Hermiticity
h jB j i b h j i
Premultiply eq b by and integrate over all space
h jB j i b h j i
Combining the last two results
b b h j i
Now if we had started with and b b then wewould have
b b h j i
But the integral of j j over all space is non zero the basic p ostulate of quantum mechanics that j j is
prop ortional to probability so for the equation to b e true wemust have b b that is the eigenvalues
must b e real
Alternately ifwe had started with b b then wemust conclude that the integral h j i Hence the
eigenfunctions of a Hermitian op erator with di erenteigenvalues are orthogonal This is one side of an if
and only if pro of it can also b e shown that if an op erator has all its eigenvalues real and its eigenfunctions
orthogonal it must b e Hermitian
Other useful prop erties of Hermitian op erators are if A and B are Hermitian
y y y
If A is invertible so is A and A A
y y y
A B A B
y y
A where is the complex conjugate of the scalar A
y y y
AB B A
All observable measurable prop erties of a system must b e real and hence the op erators asso ciated with
observable prop erties likethe Hamiltonian momentum dip ole momentetc are Hermitian Thus when
wedealwithsuchoperators we can make use of all the useful prop erties of Hermitian op erators
Another prop erty of op erators is that if two observables are to havesimultaneously precisely de ned values
their corresp onding op erators must commute If you would like more details on this please let me know if
nothing else it will tell me if anyone has actually read this far
This non examinable description of some of the mathematics b ehind quantum mechanics touches on the
th century advances in mathematical theory Most of the maths we use is th century or earlier
The most elegant derivation of quantum mechanics to me anyway comes from the recognition that the
quantum mechanical commutator identi ed as the Lie bracket generates a Lie Algebra The eigenvalues
of the op erators are the roots of the Lie Algebra As such the recurrence relations between the various
eigenfunctions and step up and step down op erators arise as the op erators that step from one ro ot of the
Lie algebra to the next