(Pure and applied physics 5) EUGENE p WIGNER (eds ) group theory and its application to the quantum mechanics of atomic spectra academic press (1959)

This is Volume 5 in
P U R E A N D APPLIED PHYSICS
A Series of Monographs and Textbooks
Consulting Editors: H . S. W . M A S S E Y AND K E I T H A . B R U E C K N E R
A complete list of titles in this series appears at the end of this volume.


GROUP THEORY
AND ITS APPLICATION TO THE
QUANTUM MECHANICS OF ATOMIC SPECTRA

EUGENE

P. W I G N E R

Palmer Physical Laboratory, Princeton University
Princeton, New Jersey

TRANSLATED

FROM

THE GERMAN BY

J. J. GRIFFIN
University of California,

Los Alamos Scientific Laboratory

Los Alamos, N e w Mexico

EXPANDED

AND IMPROVED

EDITION

1959

A C A D E M I C
A Subsidiary

New

York

of Harcourt

London

Brace

Toronto

P R E S S
Jovanovich,

Sydney

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Publishers

Sen

Frencieco


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10003

United Kingdom Edition published by
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PRINTED IN THE UNITED STATES OF AMERICA
82

9

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Author's Preface
The purpose o f this b o o k is to describe the application o f group theoretical
methods t o problems o f quantum mechanics with specific reference t o atomic
spectra. The actual solution o f quantum mechanical equations is, in general,
so difficult that one obtains b y direct calculations only crude approximations
to the real solutions. It is gratifying, therefore, that a large part o f the
relevant results can be deduced b y considering the fundamental symmetry
operations.
When the original German version was first published, in 1931, there was
a great reluctance among physicists toward accepting group theoretical
arguments and the group theoretical point o f view. It pleases the author
that this reluctance has virtually vanished in the meantime and that, in fact,
the younger generation does not understand the causes and the basis for
this reluctance. Of the older generation it was probably M. v o n Laue w h o
first recognized the significance o f group theory as the natural tool with
which t o obtain a first orientation in problems o f quantum mechanics. V o n
Laue's encouragement of both publisher and author contributed significantly
t o bringing this book into existence. I like t o recall his question as t o which
results derived in the present volume I considered most important. M y

answer was that the explanation o f Laporte's rule (the concept o f parity) and
the quantum theory of the vector addition model appeared t o me most
significant. Since that time, I have come t o agree with his answer that the
recognition that almost all rules of spectroscopy follow from the symmetry
of the problem is the most remarkable result.
Three new chapters have been added in translation. The second half o f
Chapter 24 reports on the work o f Racah and of his followers. Chapter 24
of the German edition now appears as Chapter 25. Chapter 26 deals with
time inversion, a symmetry operation which had not yet been recognized
at the time the German edition was written. The contents o f the last part
of this chapter, as well as that o f Chapter 27, have not appeared before in
print. While Chapter 27 appears at the end o f the book for editorial reasons,
the reader may be well advised t o glance at it when studying, in Chapters
17 and 24, the relevant concepts. The other chapters represent the translation
of Dr. J. J. Griffin, t o w h o m the author is greatly indebted for his ready
acceptance o f several suggestions and his generally cooperative attitude. H e
also converted the left-handed coordinate system originally used t o a righthanded system and added an Appendix on notations.
ν

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VI

AUTHOR'S

PREFACE

The character o f the book—its explicitness and its restriction to one
subject only, viz. the quantum mechanics of atomic spectra—has not been

changed. Its principal results were contained in articles first published in the
Zeitschrift für Physik in 1926 and early 1927. The initial stimulus for these
articles was given b y the investigations o f Heisenberg and Dirac on the
quantum theory of assemblies of identical particles. W e y l delivered lectures
in Zürich on related subjects during the academic year 1927-1928. These
were later expanded into his well-known book.
W h e n it became known that the German edition was being translated,
many additions were suggested. It is regrettable that most o f these could not
be followed without substantially changing the outlook and also the size of
the volume. Author and translator nevertheless are grateful for these
suggestions which were very encouraging. The author also wishes to thank
his colleagues for many stimulating discussions on the role o f group theory in
quantum mechanics as well as on more specific subjects. H e wishes to record
his deep indebtedness t o Drs. Bargmann, Michel, Wightman, and, last but
not least, J. v o n Neumann.
E . P . WLGNER

Princeton, New Jersey
February, 1959

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Translator's Preface
This translation was initiated while the translator was a graduate student
at Princeton University. It was motivated b y the lack of a good English
work on the subject of group theory from the physicist's point o f view. Since
that time, several books have been published in English which deal with
group theory in quantum mechanics. Still, it is perhaps a reasonable hope
that this translation will facilitate the introduction o f English-speaking

physicists to the use o f group theory in modern physics.
The book is an interlacing o f physics and mathematics. The first three
chapters discuss the elements of linear vector theory. The second three deal
more specifically with the rudiments o f quantum mechanics itself. Chapters
7 through 16 are again mathematical, although much o f the material covered
should be familiar from an elementary course in quantum theory. Chapters
17 through 23 are specifically concerned with atomic spectra, as is Chapter 25.
The remaining chapters are additions t o the German text; they discuss
topics which have been developed since the original publication of this b o o k :
the recoupling (Racah) coefficients, the time inversion operation, and the
classical interpretations of the coefficients.
Various readers may wish t o utilize the book differently. Those who are
interested specifically in the mathematics o f group theory might skim over
the chapters dealing with quantum physics. Others might choose t o deemphasize the mathematics, touching Chapters 7, 9, 10, 13, and 14 lightly for
background and devoting more attention t o the subsequent chapters.
Students o f quantum mechanics and physicists who prefer familiar material
interwoven with the less familiar will probably apply a more even distribution
of emphasis.
The translator would like t o express his gratitude t o Professor E . P. Wigner
for encouraging and guiding the task, to Drs. Robert Johnston and John
McHale who suggested various improvements in the text, and t o Mrs.
Marjorie Dresback whose secretarial assistance was most valuable.
J.

Los Alamos, New Mexico
February, 1959

vii

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J.

GRIFFIN


I.

VECTORS AND MATRICES

LINEAR TRANSFORMATIONS

An aggregate of η numbers (o t) , X> , · * * , V ) is called an ^-dimensional
vector, or a vector in ^-dimensional space; the numbers themselves are the
components o f this vector. The coordinates of a point in η-dimensional space
can also be interpreted as a vector which connects the origin of the co­
ordinate system with the point considered. Vectors will be denoted b y bold
face German letters; their components will carry a roman index which
specifies the coordinate axis. Thus v is a vector component (a number), and
V is a vector, a set o f η numbers.
Two vectors are said to be equal if their corresponding components are
equal. Thus
v

3

2

n

k

v = m

(L.I)

is equivalent to the η equations
Ü! =

mi,

Υ =

ΠΓ,

2

· · · ;

O

N

vo .

=

n

A vector is a null vector if all its components vanish. The product ct) of a

number c with a vector TL is a vector whose components are c times the
components o f T>, or (cv) — CO . Addition of vectors is defined b y the rule
that the components of the sum are equal to the sums o f the corresponding
components. Formally
k

k

(T> + tt>) =
fc

R> + N V

(1.2)

FC

In mathemtaical problems it is often advantageous to introduce new
variables in place of the original ones. I n the simplest case the new variables
linear functions of the old ones, x x , · · · , x . That is
v

x[ =

χ

αι1

τ

Η

2

n

ax

+

ln

n

(1.3)
x

n

=

nl l

a

X

+ *** +

nn n

a

X

or
η
Xi = 2,*üFkk= l
The introduction of new variables in this way is called linear
I

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(

L 3 a

)

transformation.


GROUP T H E O R Y A N D ATOMIC SPECTRA

2

The transformation is completely determined b y the coefficients α , · · · , a ,
and the aggregate of these n numbers arranged in a square array is called
of the linear transformation (1.3):
the matrix

η

w n

2

a

*12

l n \

*2W

*22

(1.4)

W e shall write such a matrix more concisely as ( a ) or simply a.
For E q . (1.3) actually to represent an introduction of new variables, it is
necessary not only that the x' be expressible in terms of the x , but also that
the χ can be expressed in terms of the χ . That is, if we view the x as un­
knowns in E q . (1.3), a unique solution t o these equations must exist giving
the χ in terms o f the x ' . The necessary and sufficient condition for this is
that the determinant formed from the coefficients a be nonzero:
i k

i

ik

Φ 0 .

(1.4a)

Transformations whose matrices have nonvanishing determinants are referred
t o as proper
transformations, but an array of coefficients like (1.4) is always
called a matrix, whether or not it induces a proper transformation. Bold­
face letters are used to represent matrices; matrix coefficients are indicated
b y affixing indices specifying the corresponding axes. Thus α is a matrix,
an array of n numbers; a is a matrix element (a number).
2

jk

T w o matrices are equal if all their corresponding coefficients are equal.
Thus
α = β
(1.5)
is equivalent t o the n

2

equations

«i* = ß *

(j,fc=l,2,---,w).

Another interpretation can be placed on the equation

x

i

—

2
k = l

a

i k

x

k

(1.3a)

b y considering the Xp not as components o f the original vector in a new
coordinate system, but as the components o f a new vector
in
the
original

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3

VECTORS AND MATRICES

coordinate system. W e then say that the matrix α transforms the vector χ
into the vector

or that α applied t o χ gives x'
x' = ax.

(1.3b)

This equation is completely equivalent to (1.3a).
A n w-dimensional matrix is a linear operator on w-dimensional vectors. It
is an operator because it transforms one vector into another vector; it is
linear since for arbitrary numbers a and b, and arbitrary vectors t and t),
the relation
a(ax + bv) = aar + bat)
is true.

(1.6)

T o prove (1.6) one need only write out the left and right sides

explicitly. The kth component o f at + bt) is ax + bv , so that the *th com­
k

k

ponent o f the vector on the left is :

η
Σ

*ik( k

bv ).

+

ax

k

k = l

But this is identical with the iih component o f the vector on the right side
of (1.6)
η

η

Σ *<λ + & Σ

a

k = l

α

Λ ·

k = l

This establishes the linearity o f matrix operators.
An n-dimensional matrix is the most general linear operator in η-dimensional vector
space. That is, every linear operator in this space is equivalent to a matrix. To prove
this, consider the arbitrary linear operator Ο which transforms the vector ei = (1, 0, 0,
• · · , 0) into the vector x , the vector e = (0, 1, 0, · · · , 0) into the vector ϊ, , and finally,
the vector e = (0, 0, 0, · · · , 1) into t. , where the components of the vector t, are
iifcj rfc, * * * , Xnk- Now the matrix (t ) transforms each of the vectors t fc > ' ' ' >*n
the same vectors,
t. , · · · , t. as does the operator O. Moreover, any w-dimensional
vector d is a linear combination of the vectors t\ £ , * * * , fc . Thus, both Ο and
(T ) (since they are linear) transform any arbitrary vector α into the same vector
«it.! + · · · + α Χ.η' The matrix (r ) is therefore equivalent to the operator 0.
ml

2

n

2

k

n

m

ik

2

2

lt

2

n

f

2

n

ik

η

ifc

The most important property o f linear transformations is that t w o o f them,
applied successively, can be combined into a single linear

transformation.

Suppose, for example, we introduce the variables x' in place o f the original
χ via the linear transformation (1.3), and subsequently introduce variables

x" via a second linear

transformation,
x\ = β χ[
η

.

+ β

1 2

4 +

h òi

ô

"n = ònl^l + ò ằ * 4 + * · * +

x

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A

·

(1-7)
ßnΑ·

t

o


GROUP T H E O R Y A N D ATOMIC SPECTRA

4

Both processes can be combined
into
a single
one, so that the x" are introduced
directly in place o f the χ b y one linear transformation. Substituting (1.3)
into (1.7), one finds
l

X

X

2

=

ß l l (

=

$n(
a

i A

x

+

* * ' +

« Ι A )+

+

* · · +

ΛχηΧη)

V Pln("nl l

+

X

+

* * * +

β

2

η

(

α

« Α

* ' * +

+

<*nn n)
x

Η

*nn n)
x

(1.8)

X

n

=

ß n l ( < * l A Η

+

<*1 A )

+

* * * +

Thus, the χ" are linear functions o f the x .

β ^ η (

α

η Α

+

\~

*ηη η)'
Χ

W e can write (1.8) more concisely

b y condensing (1.3) and (1.7)

η
χ

ι = Σ

α

ι Α

(j =

1,2, · · · , w )

(1.3c)

(i = l , 2 , - . - , n ) .

(1.7a)

k=r

x

i=lPa 'j
x

3

Then (1.8) becomes
η

η

< =Σ Σβ«
3= 1

α

ί A

(1.8a)

·

k= l

Furthermore, b y defining γ through
η

one obtains simply
η

< = ΣΥΑ·

(l-8b)

This demonstrates that the combination o f two linear transformations (1.7)
and (1.3), with matrices ( ß ) and ( a ) is a single linear transformation which
has the matrix
(y ).

The matrix (y ),
defined in terms of the matrices ( a ) and ( ß . ) according
to E q . (1.9), is called the product
of the matrices ( ß ) and (0L ).
Since ( a )
transforms the vector r into χ' = a t , and ( ß ) transforms the vector χ' into
χ" = ß t ' , the product matrix ( y ) b y its definition, transforms χ directly
into x" = y x . This method of combining transformations is called "matrix
multiplication," and exhibits a number o f simple properties, which we n o w
enumerate as theorems.
i f c

i

k

ik

ik

i

i f c

k

i Ä

ik

i

k

i k

i k

First o f all we observe that the formal rule for matrix multiplication is
the same as the rule for the multiplication o f determinants.
1.

The

determinants

determinant
of the two

of a product

of two

matrices

i s equal

factors.

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to the product

of

the


5

VECTORS A N D MATRICES

In the multiplication o f matrices, it is not necessarily true that
Α Β = ΒΑ.
For

(l.E.l)

example, consider the t w o matrices

Then

(l 0 G Ỵ K ί)

and

Ỵ î)(o ι) (ι
=

2)·

This establishes a second property o f matrix multiplication.

2. The product of two matrices depends in general upon the order of the factors.
I n the v e r y special situation when E q . ( l . E . l ) is true, the matrices Α and
Β are said to

commute.

I n contrast t o the commutative law,

3. The associative law of multiplication is valid in matrix multiplication.
That is,
Γ(ΒΑ)

=

(ΓΒ)Α.

(1.10)

Thus, it makes no difference whether one multiplies Γ with the product o f Β
and A , or the product o f Γ and Β with A . T o prove this, denote the
element o f the matrix on the left side o f (1.10) b y e .
i k

η
*ik =

η η

Σ Υ«(Ρ«)Λ =

3=1
The

i-kth element

= é,
ik

ΣΓ«Β*Ι »·
Α

(

L

1

0

A

)

(

L

1

0

B

)

j = l1 = 1

η

η η

Σ (YP)
Σ

1=1
ik

Σ

on the right side o f (1.10) is

<* =

Then e

i-kth.

Then

ΣΥ<ΑΑ*·

1=1 3=1

and (1.10) is established.

One can therefore write simply

Γ Β Α for both sides of (1.10).
The validity o f the associative law is immediately obvious if the matrices
are considered as linear operators.

Let Α transform the vector χ into χ' =

AT, Β the vector t' into R" = ßT', and Γ the vector χ" into χ'" = γχ". Then
the combination o f t w o matrices into a single one b y matrix multiplication
signifies simply the combination of t w o operations. The product Β Α trans­
forms X directly into R", and Γ Β transforms χ' directly into χ'". Thus both
( Γ Β ) Α and Γ ( Β Α ) transform R into χ'", and the t w o operations are equivalent.

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6

GROUP T H E O R Y A N D ATOMIC SPECTRA

4. The unit matrix
0
1
0
1

0
0
1

· •
· •
· •

0
0
0
(1.11)

=

V)

0

0

···

1'

plays a special role in matrix multiplication, just as the number 1 does in
ordinary multiplication. For every matrix A ,
A · 1 =

1 · A.

That is, 1 commutes with all matrices, and its product with any matrix is
just that matrix again. The elements of the unit matrix are denoted b y the
symbol ö , so that
ik

d

{ιφ k)

= 0

ik

d =l

(< = * ) .

ik

(1.12)

The ô

defined in this way is called the Kronecker delta-symbol. The matrix
1 induces the identity transformation, which leaves the variables
ik
unchanged.
I f for a given matrix A, there exists a matrix Β such that
ik

(ô ) =

β α = 1,

(1.13)

then Β is called the inverse, or reciprocal, of the matrix A. Equation (1.13)
states that a transformation via the matrix Β exists which combines with Α
to give the identity transformation. I f the determinant o f Α is not equal to
zero (|A | φ 0), then an inverse transformation always exists (as has been
mentioned on page 2 ) . T o prove this we write out the n equations (1.13)
more explicitly
IFE

2

lßifl*
3=1

=a

(<,*=1,2,.··,η).

ik

(1.14)

Consider now the η equations in which i has one value, say I. These are η
linear equations for η unknowns Β , Β , · · · , ß . They have, therefore, one
and only one solution, provided the determinant \oijk\ does not vanish. The
same holds for the other η — 1 systems o f equations. This establishes the
fifth property we wish to mention.
Α

Ί 2

ln

5. / / the determinant \aL \ Φ 0, there exists one and only one matrix Β such
that ΒΑ = 1.
jk

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VECTORS A N D MATRICES

Moreover, the determinant

|ß^.| is the

7

reciprocal o f

\oi \,
jk

since, according

to Theorem 1,

· |««| = \K\ = ιFrom this it follows that α has no inverse if
of a, must also have an inverse.
W e n o w show that if (1.13) is true, then

\a \ =
ik

(Lis)
0, and that

β, the

αβ = 1

inverse

(1.16)

is true as well. That is, if β is the inverse of a, then a is also the inverse o f
β. This can be seen most simply b y multiplying (1.13) from the right with β,
βαβ = β,

(1.17)

and this from the left with the inverse of β, which we call γ.

Then

γβαβ = γβ
and since, b y hypothesis γβ = 1, this is identical with (1.16). Conversely,
(1.13) follows easily from (1.16). This proves Theorem 6 (the inverse of α
is denoted b y a ).
-1

6. / / a is the inverse of a, then a is also the inverse of a .
It is clear that inverse matrices commute with one another.
Rule: The inverse o f a product αβγδ is obtained b y multiplying
-1

-1

inverses o f the individual factors in reverse order

(8~ ~ ~ ).
1

1

_1

1

the
That is

(S-y^ò- ô- ) Ã () = 1.
1

1

Another important matrix is

7. The null matrix, every element of which is zero.

0 =

(1.18)

Obviously one has
for any matrix a.
The null matrix plays an important role in another combination process
for matrices, namely, addition. The sum γ of t w o matrices α and β is the
matrix whose elements are

Υ« = α* + β«·
The n equations (1.19) are equivalent to the equation
2

γ= α+β

or

γ — α — β = 0.

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( )
L19


8

GROUP T H E O R Y A N D ATOMIC SPECTRA

Addition of matrices is clearly commutative.
Α + Β = Β + Α.

(1.20)

Moreover, multiplication b y sums is distributive.
Γ(Α +
(Α +

Β) =

ΓΑ +

ΓΒ

Β)Γ =

ΑΓ +

ΒΓ.

Furthermore, the product o f a matrix Α and a number α is defined t o be that
matrix Γ each element of which is α times the corresponding elements o f A.
aa .

(1.21)

ik

The formulas
(ab)a = a(ba);

ΑΑΒ =

ΑΑΒ;

a(A +

Β) =

AA + A ß

then follow directly.
Since integral powers of a matrix Α can easily be defined b y successive
multiplication
A

2

=

A · A;

A

3

=

A · A · A; . . .

(1.22)
polynomials with positive and negative integral exponents can also be defined
α_„Α~

1

Α~

+ α 1 + α Α +

1

0

(1.23)

Χ

The coefficients α in the above expression are not matrices, but numbers. A
junction of Α like (1.23) commutes with any other function of Α (and, in par­
ticular, with Α itself).
Still another important type of matrix which appears frequently is the
diagonal matrix.
8. A diagonal matrix is a matrix the elements of which are all zero except for
those on the main diagonal.
(D
0
···
0 \
0
D
0
1

9

(1.24)
0

D

0

n!

The general element of this diagonal matrix can be written

D* =

J>Af

(1.25)

All diagonal matrices commute, and the product of two diagonal matrices is again
diagonal. This can be seen directly from the definition o f the product.
(DD')

TT

= I

ViP*

= 1 DAP?*

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=

D D/O .
(

ik

(1.26)

VECTORS A N D

MATRICES

9

Conversely, if a matrix Α commutes with a diagonal matrix D , the diagonal
elements o f which are all different, then Α must itself be a diagonal matrix.
Writing out the product
AD =

DA

(*D) = a D = (Da) = D a .
ik

ik

k

ik

{

(1.27)

ik

That is

Φι — Dk) ik — 0
a

(1.27a)

and for a nondiagonal element (i φ k), Ό φ
D requires that a be zero.
Then Α is diagonal.
The sum o f the diagonal elements of a matrix is called the spur or trace
of the matrix.
ί

k

ik

Tr Α = Σ « „ = « , ! + « « + · · · + « « „ ·

(1-28)

3

The trace of a product Α Β is therefore
Tr Α Β =

Σ (αβ)„ = Σ
i
jk

=

T r

Ρ«·

(

L 2

9)

This establishes another property of matrices.

9. The trace of a product of two matrices does not depend on the order of the
two factors.
This rule finds its most important application in connection with similarity
transformations o f matrices. A similarity transformation is one in which the
transformed matrix Α is multiplied b y the transforming matrix Β from the
right and b y its reciprocal from the left. The matrix Α is thus transformed
into Β Α Β . A similarity transformation leaves the trace of the matrix unchanged,
since the rule above states that Β Α Β has the same trace as Α Β Β = Α .
_ 1

_ 1

- 1

The importance o f similarity transformations arises from the fact that

10. A matrix equation remains true if every matrix in it is subjected to the

same similarity transformation.
For example, transformation o f a product of matrices Α Β = Γ yields
Σ

- 1

ΑΣΣ

_ 1

ΒΣ =

Σ

αβ =

1,

ΒΣ =

Σ

-

ΓΣ

1

and if

Σ

_ 1

ΑΣΣ

_ 1

-

1

1 · Σ =

1.

W e can also see that relationships among sums of matrices and products o f

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GROUP T H E O R Y A N D ATOMIC

10

SPECTRA

matrices and numbers are also preserved under similarity transformation.
Thus, from
Γ =

Α +

Β

+

Β) =

σ*

it follows that
Σ

-ΐΓ =

Σ

-ΐ(

Α

1

+ Σ-!Β

and
Σ

ΓΣ =

_ 1

Σ

_ 1

ΑΣ+

Σ

_ 1

ΒΣ.

Likewise
Β =

α· Α

ΒΣ =

ΑΣ

implies
Σ

_ 1

_ 1

ΑΣ.

Theorem 10 therefore applies t o every matrix equation involving products
of matrices and numbers or other matrices, integral (positive or negative)
powers o f matrices, and sums o f matrices.
These ten theorems for matrix manipulation were presented in the very
first papers on quantum mechanics b y Born and Jordan, and are undoubtedly
already familiar t o many readers. They are reiterated here since a firm
command o f these basic rules is indispensable for what follows and for
practically every quantum mechanical calculation. Besides, they must very
often be used implicitly, or else even the simplest proofs become excessively
tedious.
1

2

LINEAR INDEPENDENCE OF VECTORS

The vectors T) T> , · · · , T) are said to be linearly independent if no relation­
ship o f the form
1?

2

FC

+ a T> + · · · + a *k =
2

(

0

n

2

L 3 0

)

exists except that in which every a α , · · · , a is zero. Thus n o vector in a
v

k

2

linearly independent set can be expressed as a linear combination o f the
other vectors in the set. In the case where one o f the vectors, say t) is a null
v

vector, the set can n o longer be linearly independent, since the relationship
1 · T>! + 0 · T> + · · · + 0 · v = 0
k

2

is surely satisfied, in violation o f the condition for linear independence.

AS AN EXAMPLE OF LINEAR DEPENDENCE, CONSIDER THE FOUR-DIMENSIONAL VECTORS: Q =
1

(1, 2, — 1 , 3),T> = (0, — 2 , 1, —1), ANDT> = (2, 2, — 1 , 5). THESE ARE LINEARLY DEPENDENT
2

3

SINCE
2T>! + T> - T> = 0.
2

ON THE OTHER HAND, T) AND T)
X

2

A

R

E

3

LINEARLY INDEPENDENT.

M . BORN AND P. JORDAN, Z. Physik 3 4 , 858 (1925).
FOR EXAMPLE, THE ASSOCIATIVE LAW OF MULTIPLICATION (THEOREM 3) IS USED IMPLICITLY
THREE TIMES IN THE DEDUCTION OF THE COMMUTABILITY OF INVERSES (THEOREM 6). (TRY

WRITING OUT ALL THE PARENTHESES!)
1

2

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11

VECTORS A N D MATRICES

If k vectors v t) , · * · , t ) are linearly dependent, then there can be found
among them k' vectors (k < k) which are linearly independent. Moreover,
all k vectors can be expressed as linear combinations o f these k' vectors.
In seeking k' vectors which are linearly independent we omit all null
vectors, since, as we have already seen, a null vector can never be a member
of a linearly independent set. W e then go through the remaining vectors one
after another, rejecting any one which can be expressed as a linear combina­
tion of those already retained. The k' vectors retained in this w a y are linearly
independent, since if none of them can be expressed as a linear combination
of the others, no relationship o f the type (1.30) can exist among them.
Moreover, each o f the rejected vectors (and thus all o f the k original vectors)
can be expressed in terms o f them, since this was the criterion for rejection.
The linear dependence or independence o f k vectors v t>, · · · , v is also
a property o f the vectors a i ) · · · , a t ) which result from them b y a proper
transformation a. That is,
v

2

k

f

v

v

+

2

k

k

αυ
2

2

+

h av
k

=

k

0

(1.31)

implies

+ 2 *2 + * · +
a

a

β

a

k

w

k

—0

(1.31a)

as can be seen b y applying α t o both sides o f (1.31) and using the linearity
property to obtain (1.31a). Conversely, (1.31a) implies (1.31). It also follows
that any specific linear relationship which exists among the

exists among
the at>t, and conversely.
N o more than η ^-dimensional vectors can be linearly independent. T o
prove this, note that the relation implying linear dependence
«!»!+···+«„+!»„+! = 0

(1.32)

is equivalent t o η linear homogeneous equations for the components o f the
vectors.

+ ûWl(*f*l)l =

«i(»i)i + · · · +

0

+ * ' ' + «»(*»)n + «η+ΐ(»η+ΐ)η = °'

(1.32a)

I f the coefficients a α , · · · , a , a
in these equations are viewed as un­
knowns, the fact that η linear homogeneous equations in η + 1 unknowns
always have nontrivial solutions implies at once that the relationship (1.32)
always exists. Thus, η + 1 ^-dimensional vectors are always linearly
dependent.
A n immediate corollary to the above theorem is the statement that any
η linearly independent η-dimensional vectors form a complete vector system; that
v

2

n

w + 1

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12

GROUP T H E O R Y AND ATOMIC SPECTRA

is, an arbitrary ^-dimensional vector vo can be expressed as a linear com­
bination of them. Indeed, the theorem states that some relationship
α

Λ + ' ' ' + n*n + OVO = 0
a

must exist among the η vectors and the arbitrary vector. Moreover, if
V V , * ' · , V are linearly independent, the coefficient b cannot be zero. Thus
any vector to can be written as a linear combination o f the v so that these
form a complete vector system.
V

2

n

i9

A row or a column o f an ^-dimensional matrix can be looked upon as a
vector. For example, the components o f the vector a, which forms the k i h
column are a , a , · · · , ci , and those of the vector a . which forms the ith
row are α , · · · , a .
A nontrivial linear relationship among the column
vectors α , · · · , α.
fc

l

k

2 k

α

nk

{

in

β1

η

ΐ ·1 + * * ' + η*·η =

α α

α

0

is simply a nonzero solution of the set o f linear homogeneous equations for
the a a , · · · , a .
v

2

n

«Al +

^ η«1η =

1«η1 +

\~ η*ηη = °·

α

α

0

α

The vanishing of the determinant |a | is the necessary and sufficient condition
that such a solution exist. Therefore, if this determinant does not vanish
(|affc| Φ 0), then the vectors a.1? · · · , a.TO are linearly independent and form a
complete vector system. Conversely, if the vectors V · · · , V are linearly
independent, the matrix which is formed b y taking them as its columns
must have a nonzero determinant. Of course, this whole discussion applies
equally well to the row-vectors o f a matrix.
fÄ

v

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n


2. Generalizations
1, W e n o w generalize the results o f the previous chapter. The first
generalization is entirely formal, the second one is o f a more essential nature.
T o denote the components o f vectors and the elements o f matrices, we have
affixed the appropriate coordinate axes as indices. So far, the coordinate
axes have been denoted b y 1, 2, 3, · · · , n . From n o w on we will name the
coordinate axes after the elements o f an arbitrary set. I f G is a set of objects
g, h , i , · · · , then the vector ο in the space o f the set G is the set o f numbers
V , T> , O , · · · . Of course only vectors which are defined in the same space
can be equated (or added, etc.) since only then do the components correspond
t o the same set.
g

h

t

A similar system will be used for matrices. Thus for a matrix α t o be
applied to a vector ν with components v , O , Ό , · · · , the columns o f α must
be labeled b y the elements of the same set G as that specifying the components
of v . In the simplest case the rows are also named after the elements
g, h , i , · · · o f this set, and α transforms a vector t) in the space o f G into a
vector at) in the same space. That is
g

h

{

leG

where j is an element o f the set G , and I runs over all the elements o f this set.

For example, the coordinate axes can be labeled by three letters x, y, z. Then v, with
components y) = 1, t) = 0, t) = —2, is a vector, and
x

y

2

y

2

ζ

3\

χ

5

-1 )

y

2

4k]

ζ

is a matrix. (The symbols for the rows and columns are indicated.) In this example
= 1, a
= 2, a
= 3.
Eq. (2.1) states that the ^-component of t>' = at> is given by

ct

xx

x y

x z

= 1 · 1 + 2 · 0 + 3(-2) -

-5.

The simple generalization above is purely formal; it involves merely
another system o f labeling the coordinate axes and the components of vectors
and matrices. T w o matrices which operate on vectors in the same space can
13

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GROUP THEORY AND ATOMIC SPECTRA

14

be multiplied with one another, just like the matrices in the previous chapter.
The expression
γ = βα
(2.2)
is equivalent to

Yjk = Σ ßjl lk >
l€G
a

where j and h are two elements of the set G, and I runs over all the elements
of this set.
2. A further generalization is that in which the rows and columns o f
matrices are labeled b y elements o f different sets, F and G. Then from (2.1),
m

i = Σ Ji°i >
l€G
a

(- )
2

la

where j is an element o f the set F, and I runs over all the elements o f the set G.
Such a matrix, whose rows and columns are labeled b y different sets is
called a rectangular matrix, in contrast with the square matrices o f the
previous chapter; it transforms a vector Q in the space o f G into a vector
tt> in the space o f F. In general the set F need not contain the same number
of elements as the set G. I f it does contain the same number o f elements, then
the matrix has an equal number o f rows and columns and is said t o be
''square in the broader sense."
Let the set G contain the symbols *, Δ> • > and the set F the numbers 1 and 2. Then
*



ô = (
\0

-1

5

7

ã
V
-2/ 2

is a rectangular matrix. (The labels of the rows and columns are again indicated.) It
= —2 into the vector
transforms a vector t)* = 1, D/\ = 0,

vo =

an.

The components χθχ and m are then
2

tX>i

= <*i* * + ι Δ Δ +

tö =
2

D

α

+ α

ϋ

2 Δ

ϋ

Δ

*ιΠ°Π

+ α

2 Π

υ

π

= 5 · 1 -f 7 · 0 -f 3( —2) = - 1

= 0 · 1 + (-1)(0) + ( - 2 ) ( - 2 ) - 4.

Two rectangular matrices β and α can be multiplied only if the columns of
the first factor and the rows of the second factor are labeled b y the same set
F; i.e., only if the rows o f the second factor and the columns o f the first

"match." On the other hand, the rows o f the first and the columns o f the
second factor can correspond to elements o f completely different sets, Ε and
G. Then
(2.2a)
γ = βα
is equivalent to

Yi* = Σ Pu*™
leF

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15

GENERALIZATIONS

where j is an element o f E, k an element o f G, and I runs over all the elements
of F. The rectangular matrix Α transforms a vector in the space o f G into
one in the space o f F; the matrix Β then transforms this vector into one in
the space of E. The matrix Γ therefore transforms a vector in the space o f G
into one in the space o f E.
LET G BE THE SET * , Δ > •

AGAIN, LET F CONTAIN THE LETTERS χ AND y, AND Ε THE NUMBERS

1 AND 2. THEN IF
x

/7

β = \

\9

y
8\ 1
I
3/ 2

AND

=

*



/2
\
\5

3
6

ã
4

\x
ijy

'

ONE HAS, FOR INSTANCE,
YI.

= ò I A .



= 2** + ò

2

+ òIVô .
V

A

A

A

= 7 Ã 2 + 8 · 5 = 54
= 9-3 +

3 - 6 - 4 5

AND
*

/54
Υ

~~ \ 3 3

Δ
69

•
84\ 1

45

57/ 2

3. W e n o w investigate h o w the ten theorems o f matrix calculus deduced
in Chapter 1 must be modified for rectangular matrices. W e see immediately
that they remain true for the generalized square matrix discussed at the
beginning o f this chapter, since the specific numerical nature o f the indices
has not been used anywhere in the first chapter.
Addition o f two rectangular matrices—just as that o f two vectors—pre­
supposes that they are defined in the same coordinate system, that is, that
the rows match the rows and the columns match the columns. In the equation
«

+

Β =

Γ

the labeling o f the rows o f the three matrices Α, Β , Γ must be the same, as
well as the labeling o f their columns. On the other hand, for multiplication
the columns o f the first factor and the rows o f the second factor must match;
only then (and always then) can the product be constructed. The resulting
product has the row labeling o f the first, and the column labeling o f the
second factor.
THEOREM 1. W e can speak o f the determinant o f a rectangular matrix if it
has the same number o f rows and columns, although these may be labeled
differently. For matrices "square in the broader sense" the rule that the
determinant o f the product equals the product o f the determinants is still
valid.
THEOREMS 2 and 3. The associative law also holds for the multiplication o f
rectangular matrices
(ΟΒ)Γ

=

Α(ΒΓ).

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(2.3)


16

GROUP T H E O R Y A N D ATOMIC SPECTRA

Clearly all multiplication on the right side can actually be carried out

provided it can be done on the left side, and conversely.
T H E O R E M S 4, 5, and 6. The matrix 1 will always be understood to be a
square matrix with rows and columns labeled b y the same set. Multiplication
by it can always be omitted.
Matrices which are square in the broader sense have a reciprocal only if
their determinant is nonvanishing. For rectangular matrices with a different
number of rows and columns, the inverse is not defined at all. I f Α is a matrix
which is square only in the broader sense, the equation
ΒΑ =

1

implies that the columns of Β match the rows o f A . Furthermore, the rows
of 1 must match the rows of Β , and its columns must match the columns o f A .
Since 1 is square in the restricted sense, the columns o f Α must also match the
rows of Β .
The
the

rows

of the matrix

elements

Β inverse

of the columns

of a,

to the matrix

its columns

Α are

by the same

labeled
elements

by the same
as the rows

set as
of A .

There exists for any matrix Α which is square in the broader sense and has a
nonvanishing determinant, an inverse Β such that
ΒΑ

=

(2.4)

1.

Moreover,
ΑΒ = 1.

(2.4a)

However, it should be noted that the rows and columns o f 1 in (2.4) are
labeled differently from those of 1 in (2.4a).
T H E O R E M 7. With respect to addition and the null matrix, the same rules
hold for rectangular matrices as for square matrices. However, the powers
of
rectangular
matrices
cannot
be constructed
since the multiplication o f Α w ith
Α presupposes that the columns of Α and the rows of Α match, i.e., that Α
is square.
T H E O R E M S 8 , 9, and 10. For rectangular matrices the concepts of diagonal
matrix and trace are meaningless; also, the similarity transformation is
undefined. Consider the equation
r

ΣΑΣ

-

—

1

Β.

This implies that the labeling of the rows of Β and Σ are the same. But this
is the same as the labeling of the columns of Σ , and thus o f the columns o f
Β . It follows that the matrix Β is square in the restricted sense; likewise, A ,
whose rows must match the columns of Σ and whose columns must match the
rows o f Σ , must be square in the restricted sense.
_ 1

- 1

On the other hand,
rows

of Α are

then

Σ itself

different

can
from

be square
those

of Β .

in

the broad

sense:

the columns

and

Similarity transformations which

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17

GENERALIZATIONS

change the labeling o f rows and columns are especially important. The
so-called transformation theory o f quantum mechanics is an example o f such
transformations.
The introduction o f rectangular matrices is very advantageous in spite of
the apparent complication which is involved, since substantial simplifications
can be achieved with them. The outline above is designed not as a rigid
scheme but rather to accustom the reader to thinking in terms o f these
entities. The use o f such more complicated matrices will always be explained
specifically unless the enumeration o f rows and columns is so v e r y clear b y
the form and definition o f the elements that further explanation is scarcely
desirable.
4. Quite frequently it occurs that the rows are named not with just one
number but with t w o or more numbers, for example

fa^b c d
abcd
abcd
\a b c d
1

1

abcd
abcd
abcd
a b c^d

1

1

2

1

1

2

1

1

1

2

2

1

1

1

1

1

2

1

2

1

2

2

1

1

2

2

2

abcd
abcd
abcd
abcd

2

1

1

2

1

1

2

2

1

2

1

2

1

2

2

2

1

abcd\
a^b c d
abcd
abcdl
1

1

2

2

2

2

2

2

1

2

2

2

2

2

2

(2.E.1)

The first column is called the "1,1 c o l u m n ; " the second, the "1,2 column;"
the third, the "2,1 c o l u m n ; " the fourth, the "2,2 column;" the rows are
designated in the same way. The elements o f (2.E.1) are

Yiy,kl = fifk i ·
a

d

For clarity, a semicolon is used to separate the labels o f the rows from those
of columns.
Among such matrices, the direct product Γ o f two matrices (a )
is especially important

and ( ß )

ik

Γ =

Equation (2.5) is equivalent t o



j 7

(2.5)

.

1

Yij;kl =

ôi*òil Ã

( Ã )

2

6

If the number o f rows in Α is n and the number o f columns, n , and the
corresponding numbers for Β are n[ and n2, then Γ has exactly nxnx rows and
n2n2 columns. In particular, if Α and Β are both square matrices then Α Χ Β
is also square.
1

1

2

THE FACTORS Α AND Α OF THE ORDINARY MATRIX PRODUCT ARE MERELY WRITTEN NEXT TO ONE

ANOTHER, ΑΑ. THE MATRIX (2.E.1) IS THE DIRECT PRODUCT OF THE TWO MATRICES

(

ac
1

1

a c
2

«IC \

/^I^I

acJ

\6 ^I

2

1

2

2

2

&Α\

^2^2/

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GROUP T H E O R Y A N D ATOMIC SPECTRA

18

1. If_aa = a and ß ß = β, and if α χ β = γ and α Χ β =

THEOREM

γ

then γ γ = α" Χ β.
(α Χ β)(α" Χ β) = α α Χ β β

(2.7)

That is, the matrix product of two direct products is the direct product of the
two matrix products. T o show this, consider

and
(α Χ ß)(ä Χ ß);* rfc" = Σ
( · )
2

;

8

But
(aäV

=

2

; (ßß)fcF' =

Σ

ß**'fW

and
(αα

= Σ α„< 5,, <. Σ ß * * ' ( W »
i'
k'

Χ

( · )
2

9

Therefore, from (2.8) and (2.9), one obtains Theorem 1, namely
(α Χ β) (a" X ß) = a ä Χ β β .

(2.7)

T H E O R E M 2. The direct product of two diagonal matrices is again a diagonal
matrix-, the direct product of two unit matrices is a unit matrix. This is easily
seen directly from the definition o f direct products.

In formal calculations with matrices it must be verified that the multi­
plication indicated is actually possible. In the first chapter where we had
square matrices with η rows and columns throughout, this was, o f course,

always the case. In general, however, it must be established that the rows o f
the first factor in matrix multiplication match the columns o f the second
factor, i.e., that they both have the same names or labels. The direct
product o f two matrices can always be constructed b y (2.6).
A generalized type o f matrix with several indices is referred to b y M. Born
and P. Jordan as a "super-matrix."
They interpret the matrix
as
a matrix (A ) whose elements A are themselves matrices. A is that matrix
in which the number a^. occurs in the jih row and the Zth column.
ik

ik

ik

fcợ

= ô = (A ),

THEOREM

where

(A^)^ = a

i j ; k l

.

(2.10)

3. If a = (A >) and β = ( B r ) , then α β = γ = ( C ) where
u

iV

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ir


19

GENERALIZATIONS

The

right-hand side o f (2.11) consists of a sum o f products o f

multiplications.

matrix

W e have

On the other hand
ίik\i"k'

and

2 (A 'B r)

= (c„.
ii")kk"

fi

k'

iV

Ä

k'

Therefore,

(ß)iÄ;;t*Ä — Yik;i"k">
ir

a

which proves Theorem 3. Of course, on the right side of (2.11) care must be
taken with regard to the order o f the factors, whereas in the corresponding
equation for the multiplication o f simple matrices this was not necessary.
With this single restriction, super-matrices can be multiplied according to the
rules which hold for simple matrices.
In the simplest case we might have two square matrices

«11 l2
21 22
a

"011 012 013 014 0is"
p
022 023 024 025

«15
l3 a
23 24 25

a

a

a

a

1 4

a

a

2 1

and

31 32 33 34 35
a
42 43 44 45
_ 51 52 53 54 55_
a

a

a

a

a

a

a

a

a

a

a

a

a

031 032 033 034 035
041 042 043 044 045
_051 052 053 054 055

4 1

a

(2.12)

W e can divide these into submatrices along the dotted lines, taking care that
the division o f columns in the first (2:3) coincides with the division of rows
in the second. W e then write for the t w o matrices (2.12) the abbreviated
forms

\A A j
'21 22/
al

\B
B /
\**21 22'
2 1

Λ

2 2

x,

The product of the two matrices (2.12) can be written

A12B21 AnB + A B \ _ /C
A B A B + A B /

/ 11 ιΐ
\ 21 n
Α

Β

A

B

22

21

21

12

12

22

12

22

22

12
c

n

u

22

On the other hand, the expression

/B
\B

n
21

B \/A A \ ^ / B A + B A
B / \A A / VB^Ai! + B A
'21 22'
12

22

n

21 Λ

12

n

22

n

12

21

22

21

A22 ί
B A + B12
B A
22^22/
n

12

21

12

19

Λ

is meaningless, since the number o f columns o f B , for example, differs
n

from the number o f rows of A .
n

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