MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps

For complex vector spaces properties 1-7 still hold; how- ever, the inner product is now defined as We now define a vector space L2, whose elements are complex valued functions of a rea

Adding to the above properties a scalar or an inner product enriches the vector space concept significantly and makes physical applications easier In Cartesian coordinates the inner product, also called the dot product, is defined

as

( 1 1.365)

Generalization to arbitrary dimensions is obvious The inner product makes

it possible to define the norm or magnitude of a vector as

274 CONTlNUOUS GROUPS AND REPRESENTATlONS

11.14.3 Four-Vector Space

In Section 10.10 we have extended the vector concept to Minkowski spacetime

as four-vectors, where the elements of the Lorentz group act on four-vectors and transform them into other four-vectors For four-vector spaces properties (1)-(7) still hold; however, the inner product of two four-vectors A" and B"

is now defined as

where goo is the Minkowski metric

11.14.4 Complex Vector Space

Allowing complex numbers, we can also define complex vector spaces in the complex plane For complex vector spaces properties (1)-(7) still hold; how- ever, the inner product is now defined as

We now define a vector space L2, whose elements are complex valued functions

of a real variable IL', which are square integrable in the interval [a, b] L 2 is also called the Hilbert space By square integrable it is meant that the integral

Function Space and Hilbert Space

00

f(z) = c cm.llm(z) (1 1.378)

m=O

Orthogonality of {u,(z)} is expressed as

(urntun) = uL(z)un(z)dz = &n, , (11.379) where we have taken ~ ( z ) = 1 for simplicity Using the orthogonality relation

we can free the expansion coefficients under the summation sign in Equation (11.378) to express them as

Lb

(1 1.380)

In physical applications {urn(.)} is usually taken as the eigenfunction set of

a Hermitian operator Substituting Equation (11.380) back into Equation (11.378) a formal expression for the completeness of the set {um (z)} is ob-

tained as

00

c u; (z’) u, (z) = qz - z’) ( 1 1.381) m=O

11.14.6

Proof of the completeness of the eigenfunction set is rather technical for our

purposes and can be found in Courant and Hilbert (p 427, vol 1) What is important for us is that any sufficiently well-behaved and at least piecewise

Completeness of the Set of Eigenfunctions { U r n (s))

276 CONTINUOUS GROUPS AND REPRESENTATIONS

continuous function, F (x) , can be expressed as a n infinite series in terms of

the set {urn (z)} as

00

(11.382)

m=O Convergence of this series to F (z) could be approached via the variation technique, and it could be shown that for a Sturm-Liouville system the limit (Mathews and Walker, p 338)

N

(11.385)

m=O However, for most practical situations convergence in the mean will accom- pany point-to-point convergence and will be sufficient We conclude this sec- tion by quoting a theorem from Courant and Hilbert (p 427)

Expansion Theorem: Any piecewise continuous function defined in the

fundamental domain [a, b] with a square integrable first derivative could

be expanded in an eigenfunction series F (z) = amum (z), which converges absolutely and uniformly in all subdomains free of points of

discontinuity At the points of discontinuity it represents the arithmetic mean of the right- and the left-hand limits

00

m=O

In this theorem the function does not have to satisfy the boundary con- This theorem also implies convergence in the mean; however, the ditions

converse is not true

11.15 HILBERT SPACE AND QUANTUM MECHANICS

In quantum mechanics a physical system is completely described by giving its

CONTINUOUS GROUPS AND SYMMETRIES 277

there corresponds a Hermitian differential operator acting on the functions in Hilbert space Because of their Hermitian nature these operators have real eigenvalues, which are the allowed physical values of the corresponding observ- able These operators are usually obtained from their classical definitions by replacing position, momentum, and energy with their operator counterparts

In position space the replacements

F -f 7,

y + -ativ, ( 1 1.386)

a

E -$ iti- at have been rather successful Using these, the angular momentum operator is obtained as

(1 1.387) -+

L = ? x Y

= -ah? x a

f

In Cartesian coordinates components of L are given as

where Li satisfies the commutation relation

[Li, Lj] = i h E i j k L k

(1 1.388) (1 1.389) (11.390)

(11.391)

11.16 CONTINUOUS GROUPS AND SYMMETRIES

In everyday language the word symmetry is usually associated with familiar operations like rotations and reflections In scientific applications we have a

broader definition in terms of general operations performed in the parame- ter space of a given system Now, symmetry mezlns that a given system is

invariant under a certain operation A system could be represented by a La- grangian, a state function, or a differential equation In our previous sections

we have discussed examples of continuous groups and their generators The theory of continuous groups was invented by Lie when he was studying sym- metries of differential equations He also introduced a method for integrating differential equations once the symmetries are known In what follows we dis- cuss extension (prolongation) of generators of continuous groups so that they could be applied to differential equations

278 CONTINUOUS GROUPS AND REPRESENTATIONS

11.16.1

In two dimensions general point transformations can be defined as

One-Parameter Point Groups and Their Generators

- z = 2(z, y)

where x and y are two variables that are not necessarily the Cartesian coordi- nates All we require is that this transformation form a continuous group so that finite transformations can be generated continuously from the identity

element We assume that these transformations depend on a t least on one

parameter, E ; hence we write

which corresponds to counterclockwise rotations about the z-axis by the amount

E If we expand Equation (11.394) about E = 0 we get

If we define the operator

we can write Equation (11.395) as

- z(z, y; &) = 2 + EXZ +

(11.396)

(1 1.397)

(11.398)

CONTINUOUS GROUPS AND SYMMETRIES 279

Operator X is called the generator of the infinitesimal point transformation For infinitesimal rotations about the z-axis this agrees with our previous result [Eq (11.34)] as

We have given the generators in terms of the (x,y) variables [Eq (11.398)]

However, we would also like to know how they look in another set of variables, Say

Transformation of Generators and Normal Forms

When substituted in Equation (11.404) this gives the generator in terms of

the new variables as

x = [ az- - L3 (1 1.407)

(11.408)

280 CONTINUOUS GROUPS AND REPRESENTATIONS

In other words, the coefficients in the definition of the generator can be found

by simply operating on the coordinates with the generator; hence we can write

x = (Xxi), d = (X?) d

We now consider the generator for rotations about the z-axis [Eq (11.400)]

in plane polar coordinates:

( 11.4 16)

We will not go into the proof, but the answer to this question is yes, where

CONTINUOUS GROUPS AND SYMMETRIES 281

11.16.3

Transformations can also depend on multiple parameters

transformations with m parameters we write

The Case of Multiple Parameters

We have already seen that the action of the generators of the rotation group

R(3) on a function f(r) are given as

Action of Generators on Functions

where the generators are given as

x2 = - ( 23- 8x1 -XI-) 3x3

(11.420) (1 1.421)

(1 1.422)

The minus sign in Equation (11.421) means that the physical system is rotated

clockwise by 0 about an axis pointing in the fi direction Now the change in

f(r) is given as

sf(r) = - (X .;i) f(r)se (11.423)

282 CONTINUOUS GROUPS AND REPRESENTATIONS

If a system represented by f ( r ) is symmetric under the rotation generated by

(x 6) , that is, it does not change, then we have

For rotations about the z-axis, in spherical polar coordinates this means

(11.425)

that is, f(r) does not depend on 4 explicitly

For a general transformation we can define two vectors

CONTINUOUS GROUPS AND SYMMETRIES 283

284 CONTINUOUS GROUPS AND REPRESENTATIONS

To find the coefficients /3rnl we can use Equation (11.433) in F4uation (11.431)

and then Equation (11.432) to obtain

This can also be written as

which for the first two terms gives us

= - + y ' ( - & - z ) - y ap ap aa I2 - aa

and

(1 1.439) (1 1.440)

(11.441) (11.442)

(11.443)

CONTINUOUS GROUPS AND SYMMETRIES 285

For the infinitesimal rotations about the z-axis the extended generator can now be written as

- (3y”Z + 4y’y”’)- ay,” +

For the extension of the generator for translations along the z-axis we obtain

(11.446) 11.16.6 Symmetries of Differential Equations

We are now ready t o discuss symmetry of differential equations under point transformations, which depend on a t least one parameter To avoid some singular cases (Stephani) we confine our discussion to differential equations,

D(z,y’, y”, , 9‘”)) = 0, (11.447)

which can be solved for the highest derivative as

-

D = y(n) - D(x, y’, y”, , y(np ’)) = 0 (1 1.448)

For example, the differential equation

286 CONTINUOUS GROUPS AND REPRESEN JATIONS

For infinitesimal transformations we keep only the linear terms in E :

In the presence of symmetry Equation (11.452) must be true for all E ; thus the left-hand side of Equation (11.454) is zero, and we obtain a formal expression for symmetry as

XD = 0 (11.455) Note that the symmetry of a differential equation is independent of the choice

of variables used Using an arbitrary point transformation only changes the form of the generator We now summarize these results in terms of a theorem (for special cases and alternate definitions of symmetry we refer the reader to

Stephani)

Theorem: An ordinary differential equation, which could be written as

D = g(") - G(z,$,$', .,g("-')) 1 0, (1 1.456) admits a group of symmetries with the generator X if and only if

CONTINUOUS GROUPS AND SYMMETRIES 287

which gives

(1 1.461) (1 1.462)

Hence if X generates a symmetry of a given differential equation, which can

be solved for its highest derivative as

then we can write

D = (y” - y’ + x)’ = 0,

all the first-order derivatives are zero for D = 0

d D ,,,,

288 CONTINUOUS GROUPS AND REPRESENTATlONS

Thus XD = 0 holds for any linear operator, and in normal coordinates even

= 0, we can no longer say that D does not depend on 5 explicitly

Problems

11.1 Consider the linear group in two dimensions

x' = ax + by

y' = cx +dy

Show that the four infinitesimal generators are given as

and find their commutators

11.2 Show that

det A = det eL = eTrL ,

where L is an n x n matrix Use the fact that the determinant and the trace of a matrix are invariant under similarity transformations Then make

a similarity transformation that puts L into diagonal form

11.3 Verify the transformation matrix

PROBLEMS 289

make a multipole expansion of the potential and evaluate all the nonvanishing

multipole moments What is the potential for large distances?

11.6 Show that d i , m ( P ) satisfies the differential equation

m2 + mI2 - 2mm‘cosp I

11.7 Using the substitution

in Problem 11.6 show that the second canonical form of the differential equa- tion for d&,m(,B) (Chapter 9) is given as

a2Y(A, m’, m, P) +

w2

11.8 Using the result of Problem 11.7, solve the differential equation for

d i m , (P) by the factorization method

a) Considering m as a parameter, find the normalized s t e p u p and s t e p down operators O+ (m + 1) and 0- (m), which change the index m while keeping the index m’ fixed

b)Considering m‘ as a parameter, find the normalized s t e p u p and s t e p down operators Oi(m’ + 1) and OL(m’), which change the index m‘ while keeping the index m fixed Show that Irnl 5 1 and lm’l 5 1

c) Find the normalized functions with m = m’ = 2

d) For 1 = 2, construct the full matrix &mtm(,B)

e) By transforming the differential equation for dk,, (p) into an appropriate

form, find the step-up and stepdown operators that shift the index 1 for fixed

m and m’, giving the normalized functions d i m , (p)

That is, express this as a combination of dkm,(p) with 1’ = 1 f 1,

discussed in Chapter 9.)

11.9 Show that

f)Using the result of Problem 11.8.5, derivea recursion relation for (cosp) dA,,(P)

(Note This is a difficult problem and requires knowledge of the material

a)

and

290 CONTINUOUS GROUPS AND REPRESEN TATIONS

Hint Use the invariant

11.10 For I = 2 construct the matrices

for L = 0,1,2,3,4, and show that the matrices with L 2 5 can be expressed

as linear combinations of these Use this result to check the result of Problem 11.8.4

11.11 We have studied spherical harmonics x,(6,4), which are single- valued functions of (0, #) for 1 = 0,1,2, However, the factorization method

also gave us a second family of solutions corresponding to the eigenvalues

x = J ( J + 1) with

M = J, ( J - l), , 0, , -(J - I), -J, where J = 0,1/2,3/2,

For J = 1/2, find the 2 x 2 matrix of the y component of the angular momentum operator, that is, the generalization of our [LY],,# Show that the matrices for Li, L;, Li, are simply related to the 2 x 2 unit matrix and the matrix [LYIMM, Calculate the &function for J = 1/2:

dJ='/2 (P)

M M ' with M and M' taking values +l/2 or -1/2

11.12 Using the following definition of Hermitian operators:

J IIr;Lc92dx = J (LWI)*c92dx,

PROBLEMS 291

show that

11.13 Convince yourself that the relations

e -iOL,, = e-iaL,e-iBLueiaL, and

e-iTLz, = ,-iBLu, e-i7Lz,eiPL,, 9

used in the derivation of the rotation matrix in terms of the original set of

axes are true

11.14 Show that the Di,,,(R) matrices satisfy the relation

c [DAt,,,(R)] [DA,,,(K 1 )] = fimJm

292 CONTINUOUS GROUPS AND REPRESEN TATlONS

11.18 Using induction, show that

of Laplace equation in two dimensions

2 The method of analytic continuation is very useful in finding solutions

of differential equations and evaluating some definite integrals

3 Infinite series, infinite products, asymptotic solutions, and stability cal-

culations are other areas, in which complex techniques are very useful

4 Even though complex techniques are very helpful in certain problems

of physics and engineering, which are essentially problems defined in the real domain, complex numbers in quantum mechanics appear as an essential part of the physical theory

294 COMPLEX VARIABLES AND FUNCT/ONS

x = TCOSO, y = r s i n 0 and

z = r ( c o s O + i s i n O ) or z = r e i * (12.3) The modulus of z is defined as