MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

10.1 CARTESIAN COORDINATES In threedimensional Euclidean space a Cartesian coordinate system can be constructed by choosing three mutually orthogonal straight lines... 10.3 Motion in Ca

GEGENBAUER FUNCTIONS (TYPE A) 153

9.8 GEGENBAUER FUNCTIONS (TYPE A)

The Gegenbauer equation in general is given as

d2C;' (x) dC2' (x)

d x 2 - (2X' + 1)x- d x + n(n + 2X')C,"'(x) = 0 (9.250) (1 - x2)

For X = 1/2 this equation reduces to the Legendre equation

values of n its solutions reduce to the Gegenbauer or Legendre polynomials:

x = -cos6

U T ( X ) = Z?(Q)(sinQ)-rn-l

we can put Equation (9.252) into the second canonical form as

(9.255) (9.256)

(9.257)

m(m + 1) dQ2

On the introduction of

A" = x + (m + I ) ~ , (9.258) and comparing with, Equation (9.90), this is of type A with c = p = d = 0,

a = 1, and z = 8, and its factorization is given by

k(6, m) = m cot B

p(m) = m2

(9.259) (9.260)

154 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD

The solutions are found by using

and the formulas

(9.261)

(9.262)

Note that Z[n is the eigenfunction corresponding to the eigenvalue

A " = (2+112, 1 - m = 0 , 1 , 2 , , (9.263) that is, to

x = (1 + 1)2 - (m + 1)2

= ( 2 - m ) ( l + m + 2) (9.264)

9.9 SYMMETRIC TOP (TYPE A)

The wave equation for a symmetric top is encountered in the study of simple molecules If we separate the wave function as

U = @ ( U ) exp(iK4) exp(im+), (9.265) where 8,4, and + are the Euler angles and K and m are integers, O(8) satisfies the second-order ordinary differential equation

Y + (a + K~ + 1/4)Y = 0

(9.269) ( m - 1/2)(m+ 1/2)+rc2-2rn~cosU

sin2 u

BESSEL FUNCTIONS (TYPE C) 155

This equation is of type A, and we identify the parameters in Equation (9.90)

(9.272)

(9.273)

(9.274) (9.275)

(9.276)

9.10 BESSEL FUNCTIONS (TYPE C)

Bessel’s equation is given as

Multiplying this equation by l/z, we obtain the first canonical form as

(9.278) where

p(z) = z, and ~ ( z ) = 2 (9.279)

156 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD

A second transformation,

(9.280) (9.281) gives us the second canonical form

Because p ( m ) is neither a decreasing nor an increasing function of m, we have

no limit (upper or lower) to the ladder We have only the recursion relations

and

where

9, = X W m ( A 1 / 2 X )

9.11 HARMONIC OSCILLATOR (TYPE D)

The Schrodinger equation for the harmonic oscillator is given as

0+0_9x = (A - l)Ikx, (9.290)

PROBLEMS 157

where

(9.291)

Operating on Equation (9.289) with Of and on Equation (9.290) with 0-

we obtain the analog of Theorem I as

*A,, 0: o+*x (9.292) and

Moreover, corresponding to Theorem IV, we find that we can not lower the eigenvalue X indefinitely Thus we have a bottom of the ladder

A = 2 n f 1, n = 0,1,2 ,"' (9.294) Thus the ground state must satisfy

o-qo = 0, (9.295)

(9.296) Now the other eigenfunctions can be obtained from

158 STURM-LIOUVILLE SYSTEMS AND THE FACTORlZATlON METHOD

by using the transformations

Y(Z) = w [ W ( ~ ) P ( ~ ) I ~ ~ ~ and

9.3 Derive the normalization constant in

9.4 Derive Equation (9.195), which is given as

9.5 The general solution of the differential equation

is given as the linear combination

y(x) = C, sin f i x + C, cos A x

Show that factorization of this equation leads to the trivial result with

k ( x , m ) = 0, p ( m ) = 0, and the corresponding ladder operators just produce other linear combinations

of sin A x and cos dz

9.6 Show that taking

k ( z , m) = h ( z ) + kl(z)m + k2(z)m2

Show that as long as we admit a finite number of negative powers of m

is a periodic function of m with the period one

Use this result to verify

9.9 Derive the stepdown operator in

tions of the first kind:

Use the factorization method to show that the spherical Hankel func-

160 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD

9.12

normalized eigenfunctions y(n, E , T ) of the differential equation

Using the factorization method, find a recursion relation relating the

to the eigenfunctions with 1 f 1

Hint: First show that

1 = n - 1,n - 2, , 1 = integer

and the normalization is

9.13 The harmonic oscillator equation

PROBLEMS 161

and show that the E < 0 eigenvalues are not allowed

t o €,in and then use it t o express all the remaining eigenfunctions

iv) Using the factorization technique, find the eigenfunction corresponding Hint: Use the identity

9.14 Show that the standard method for the harmonic oscillator problem

leads t o a single ladder with each function on the ladder corresponding t o a

different eigenvalue A This follows from the fact that ~ ( z , m) is independent of

m The factorization we have introduced in Section 9.11 is simpler, and in fact the method of factorization originated from this treatment of the problem

9.15 The spherical Bessel functions jl(2) are related to the solutions of

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In this chapter we start with the Cartesian coordinates, their transforma- tions, and Cartesian tensors We then generalize our discussion to generalized coordinates and general tensors The next stop in our discussion is the coor- dinate systems in Minkowski spacetime and their transformation properties

We also introduce four-tensors in spacetime and discuss covariance of laws

of nature We finally discuss Maxwell’s equations and their transformation properties

10.1 CARTESIAN COORDINATES

In threedimensional Euclidean space a Cartesian coordinate system can be constructed by choosing three mutually orthogonal straight lines A point is defined by giving its coordinates, ( q , z 2 , q ) , or by using the position vector

163

164 COORDINATES AND TENSORS

fig 10.1 Cartesian coordinate system

+ r as

-f- r - IL& + z 2 G 2 + ~ 3 G 3 (10.1)

= (XI 7 2 2 , 2 3 ) , (10.2) where G; are unit basis vectors along the coordinate axis (Fig 10.1) Similarly,

an arbitrary vector in Euclidean space can be defined as

+ u = U , Z I + a2Z2 + a323, (10.3) where the magnitude is given as

CARTESIAN COORDINATES 165

t i?=izx*

Fig 10.2 Scalar and vector products

ii) Addition or subtraction is done by adding or subtracting the correspond- ing components of two vectors:

i?*T= (a1 k b l , a z f b 2 , a s f b 3 )

iii) Multiplication of vectors

There are two types of vector multiplication:

a) Dot or scalar product is defined as

(10.6)

( 10.7)

(a, b ) = i? f = abcos@ab

= aibi + a 2 b 2 + asbs,

where gab is the angle between the two vectors

b) Vector product is defined as

(10.8) -+

where the permutation symbol takes the values

+1 for cyclic permutations

t i j k = 0 when any two indices are equal

- 1 for anticyclic permutations

{

The vector product of two vectors is again a vector with the magnitude

where the direction is conveniently found by the right-hand rule (Fig 10.2)

166 COORDINATES AND TENSORS

Fig 10.3 Motion in Cartesian coordinates

10.1.2 Differentiation of Vectors

In a Cartesian coordinate system motion of a particle can be described by

giving its position in terms of a parameter, which is usually taken as the time (Fig 10.3), that is,

dt2 '

(10.12) (10.13)

(10.14)

(10.15) The derivative of a general vector is defined similarly Generalization of these equations to n dimensions is obvious

10.2 ORTHOGONAL TRANSFORMATIONS

There are many ways to chose the orientation of the Cartesian axes Symme- tries of the physical system often make certain orientations more advantageous than others In general, we need a dictionary to translate the coordinates

ORTHOGONAL TRANSFORMATIONS 167

Fig 10.4 Direction cosines

assigned in one Cartesian system to another A connection between the coor- dinates of the position vector assigned by the two sets of Cartesian axes with

a common origin can be obtained as (Fig 10.4)

(10.16) (10.17) where

This can also be written as

where cos 6ij are called the direction cosines defined as

168 COORDINATES AND TENSORS

h

cos eij = isi sj (10.20)

Note that the first unit basis vector is always taken as the barred system, that

is,

h

The transformation equations obtained for the position vector are also true for an arbitrary vector 3 as

- al = (cosell) al + (cose12)a2 + (cose13)a3

a2 = (cosQ21) a1 + (cosQ22)az + (cosQ23)a3

a3 = (cose3,) al + (cos~32)a2 + (COS~33)a3

(10.22)

-

-

The transformation equations given in Equation (10.22) are the special case

of general linear transformation, which can be written as

-

21 = allxl f a1222 + a13x3

2 2 = a2121 + a 2 2 2 2 + a2323 x3 = a31x1 + a3222 + (333x3

Unless otherwise stated, we use the Einstein summation convention Magni- tude of 7 in this notation is shown as

Using matrices, transformation Equations (10.23) can also be written as

- where r and f are represented by the column matrices

r = [ ii] a n d F = [ ii], (10.29)

From Equation (10.34) it is seen that linear transformations that preserve the

length of a vector must satisfy the condition

170 COORDINATES AND TENSORS

Taking the determinant of the orthogonality relation, we see that the deter- minant of transformations that preserve the length of a vector satisfies

lations Transformations with DetA = -1 are called improper transfor- mations, and they involve reflections

10.2.1 Rotations About Cartesian Axes

For rotations about the xa-axis the rotation matrix takes the form

10.3 FORMAL PROPERTIES OF THE ROTATION MATRIX

i) Two sequentially performed rotations, say A and B, is equivalent to

another rotation C as

ii) Because matrix multiplications do not commute, the order of rotations

is important, that is, in general

FORMAL PROPER TIES OF THE 170 JA JlON MATRIX 171

-

3

N X 1

fig 10.5 Direction cosines

However, the associative law,

A(BC) = (AB)C, (10.46)

holds between any three rotations A, B, and C

nality relation [Eq (lo.%)], it is equal to the transpose of A, that is, iii) The inverse transformation matrix A-' exists, and from the orthogo-

A - l = ; i (10.47)

Thus for orthogonal transformations we can write

172 COORD/NATES AND TENSORS

10.4 EULER ANGLES AND ARBITRARY ROTATIONS

The most general rotation matrix has nine components (10.30) However, the

orthogonality relation AA - = I, written explicitly as

(10.49)

gives six relations among these components Hence, only three of them can be

independent In the study of rotating systems to describe the orientation of a

system it is important to define a set of three independent parameters There

are a number of choices The most common and useful are the three Euler

angles They correspond to three successive rotations about the Cartesian axes so that the final orientation of the system is obtained The convention we

follow is the most widely used one in applied mechanics, in celestial mechanics, and frequently, in molecular and solid-state physics For different conventions,

we refer the reader to Goldstein et al

The sequence starts with a counterclockwise rotation by q5 about the x3-axis

of the initial state of the system as

This is followed by a counterclockwise rotation by 8 about the xi of the

intermediate axis as

Finally, the desired orientation is achieved by a counterclockwise rotation about the x!-axis by $I as

EULER ANGLES AND ARBITRARY ROTATIONS 173

A(+), B(6), and C(+) are the rotation matrices for the corresponding trans-

formations, which are given as

cos+cos + - cos8 sin +sin +

- sin + cos + - cos 6sin +cos + - cos+ sin + sin sin + + + + cos8cos+sin cos 6cos +cos + + cos +sin 8 sin+sin6

I

all(t) al2(t) al3(t) a3l(t) a32(t) a33(t) a21(t) a22(t) a23(t) (10.59) Using trigonometric identities it can be shown that

A(t2 + t i ) = A(t2)A(ti) (10.60) Differentiating with respect t o t2 and putting t2 = 0 and tl = t, we obtain a

result that will be useful to us shortly as

174 COORDINATES AND TENSORS

2

+ r

f

Fig 10.6 Passive and active views of the rotation matrix

10.5 ACTIVE A N D PASSIVE INTERPRETATIONS OF ROTATIONS

It is possible t o view the rotation matrix A in

as an operator acting on r and rotating it in the opposite direction, while keeping the coordinate axes fixed (Fig This is called the active view The case where the coordinate axes are rotated is called the passive view In principle both the active and passive views lead to the same result

However, as in quantum mechanics, sometimes the active view may offer some advantages in studying the symmetries of a physical system

In the case of the active view, we also need to know how an operator A

transforms under coordinate transformations Considering a transformation represented by the matrix B, we multiply both sides of Equation (10.62) by

(10.67)

-

r' = BF

mation, we then write

176 COORDINATES AND TENSORS

r ( t ) N (I + Xt)r(O), (10.81)

(10.82)

Sr ~ l i Xtr(O), (10.83) where X is called the generator of the infinitesimal transformation

Using the definition of X in Equation (10.76) and the rotation matrices [Eqs (10.42) and (10.43)] we obtain the generators of infinitesimal rotations about the z1- ,z2- , and z~-axes, respectively as

x = XIS, + X2S2 + X3S3

= ( X l , x2, X3) (10.86) and the unit vector

h 1

n = dwi [ i,] (10.87)

we can write Equation (10.85) as

r ( t ) = (I + X.iit)r(O), (10.88) where

This is an infinitesimal rotation about an axis in the direction i? by the amount

t For finite rotations we write

INFINITESIMAL TRANSFORMATIONS 177

10.6.1 Infinitesimal Transformations Commute

Two successive infinitesimal transformations by the amounts tiand t2 can be written as

178 COORDINATES AND TENSORS

which again proves that infinitesimal rotations commute

10.7 CARTESIAN TENSORS

Certain physical properties like temperature and mass can be described com- pletely by giving a single number They are called scalars Under orthogonal transformations scalars preserve their value Distance, speed, and charge are other examples of scalars On the other hand, vectors in three dimensions require three numbers for a complete description, that is, their ~ 1 ~ x 2 , and z 3

components Under orthogonal transformations we have seen that vectors transform as

a! 2 = A e3 .a 3' (10.99) There are also physical properties that in three dimensions require nine components for a complete description For example, stresses in a solid have nine components that can be conveniently represented as a 3 x 3 matrix:

t 23 - [ ::: ::: 2 ] (10 100)

t 3 1 t 3 2 t 3 3

Components t i j correspond to the forces acting on a unit area element, that

is, t i j is the ith component of the force acting on a unit area element when its normal is pointing along the j t h axis Under orthogonal transformations stresses transform as

Stresses, vectors, and scalars are special cases of a more general type of objects called tensors

Cartesian tensors in general are defined in terms of their transformation properties under orthogonal transformations as

All indices take the values 1,2,3, , n, where n is the dimension of space An important property of tensors is their rank, which is equal to the number

of free indices In this regard, scalars are tensors of zeroth rank, vectors are tensors of first rank, and stress tensor is a second-rank tensor

10.7.1 Operations with Cartesian Tensors

i) Multiplication with a constant is accomplished by multiplying each com- ponent of the tensor with that constant