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

If it does contain the same number of elements, then the matrix has an equal number of rows and columns and is said to be ''square in the broader sense." Let the set G contain the symbo

PURE 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 BRUECKNER

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

NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,

BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS

LIBRARY OF CONGRESS CATALOG CARD N U M B E R : 5 9 - 1 0 7 4 1

PRINTED IN THE UNITED STATES OF AMERICA

82 9

The purpose of this book is to describe the application of group theoretical methods to 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 of 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 von Laue who 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

to bringing this book into existence I like to recall his question as to which results derived in the present volume I considered most important My answer was that the explanation of Laporte's rule (the concept of parity) and the quantum theory of the vector addition model appeared to me most significant Since that time, I have come to 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 of 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 of the last part

of this chapter, as well as that of 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 to glance at it when studying, in Chapters

17 and 24, the relevant concepts The other chapters represent the translation

of Dr J J Griffin, to whom the author is greatly indebted for his ready acceptance of several suggestions and his generally cooperative attitude H e also converted the left-handed coordinate system originally used to a right-handed system and added an Appendix on notations

ν

The character of 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 of 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

When it became known that the German edition was being translated, many additions were suggested It is regrettable that most of 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 of group theory in quantum mechanics as well as on more specific subjects H e wishes to record his deep indebtedness to Drs Bargmann, Michel, Wightman, and, last but not least, J von Neumann

E P WLGNER

Princeton, New Jersey

February, 1959

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 of 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 of English-speaking physicists to the use of group theory in modern physics

The book is an interlacing of 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 of 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 to the German text; they discuss topics which have been developed since the original publication of this book: the recoupling (Racah) coefficients, the time inversion operation, and the classical interpretations of the coefficients

Various readers may wish to utilize the book differently Those who are interested specifically in the mathematics of group theory might skim over the chapters dealing with quantum physics Others might choose to de-emphasize the mathematics, touching Chapters 7, 9, 10, 13, and 14 lightly for background and devoting more attention to the subsequent chapters Students of 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 to express his gratitude to 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 to Mrs Marjorie Dresback whose secretarial assistance was most valuable

J J G R I F F I N

Los Alamos, New Mexico

February, 1959

vii

LINEAR TRANSFORMATIONS

An aggregate of η numbers (o v t)2, X> 3 , · * * , V n ) is called an ^-dimensional

vector, or a vector in ^-dimensional space; the numbers themselves are the

components of 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 k 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

is equivalent to the η equations

Ü! = mi, Υ2 = ΠΓ, · · · ; 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 of T>, or (cv) k — CO k Addition of vectors is defined b y the rule

that the components of the sum are equal to the sums of the corresponding

components Formally

In mathemtaical problems it is often advantageous to introduce new

variables in place of the original ones In the simplest case the new variables

linear functions of the old ones, x v x 2 , · · · , x n That is x[ = αι1 χ τ Η + a ln x n

I

The transformation is completely determined by the coefficients αη, · · · , aw n,

and the aggregate of these n 2 numbers arranged in a square array is called

W e shall write such a matrix more concisely as ( a i k ) 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 i as un­

knowns in E q (1.3), a unique solution to these equations must exist giving

the χ in terms of the x ' The necessary and sufficient condition for this is

that the determinant formed from the coefficients a ik be nonzero:

Transformations whose matrices have non vanishing determinants are referred

to 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

by affixing indices specifying the corresponding axes Thus α is a matrix,

an array of n 2 numbers; a jk is a matrix element (a number)

Two matrices are equal if all their corresponding coefficients are equal

by considering the Xp not as components of the original vector in a new

coordinate system, but as the components of a new vector in the original

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

into the vector or that α applied to χ gives x'

x' = ax (1.3b)

This equation is completely equivalent to (1.3a)

An 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) (1.6)

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

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

ponent of the vector on the left is :

η

k = l But this is identical with the iih component of the vector on the right side

of (1.6)

η η

a Σ * < λ + & Σ α Λ ·

k = l k = l

This establishes the linearity of 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 ml , the vector e2 = (0, 1, 0, · · · , 0) into the vector ϊ,2, and finally,

the vector en = (0, 0, 0, · · · , 1) into t.n, where the components of the vector t,k are

iifcj r 2fc, * * * , Xnk- Now the matrix (t ik ) transforms each of the vectors t lt fc2> ' ' ' >*nm t o

the same vectors, t.2, · · · , t.n as does the operator O Moreover, any w-dimensional

vector d is a linear combination of the vectors t\ f £2, * * * , fcn Thus, both Ο and

(T ik ) (since they are linear) transform any arbitrary vector α into the same vector

«it.! + · · · + α η Χ.η' The matrix (rifc) is therefore equivalent to the operator 0

The most important property of linear transformations is that two o f them,

applied successively, can be combined into a single linear transformation

Suppose, for example, we introduce the variables x' in place of the original

χ via the linear transformation (1.3), and subsequently introduce variables

x" via a second linear transformation,

x\ = β η χ[ + β1 2 4 + h ßi A

« · (1-7)

x "n = ßnl^l + ß » * 4 + * · * + ß n Α ·

Both processes can be combined into a single one, so that the x" are introduced

directly in place of the χ b y one linear transformation Substituting (1.3)

into (1.7), one finds

This demonstrates that the combination of two linear transformations (1.7)

and (1.3), with matrices ( ß i f c ) and ( a i k ) is a single linear transformation which

has the matrix (y ik )

The matrix (y ik ), defined in terms of the matrices ( a i k ) and ( ß i Ä ) according

to E q (1.9), is called the product of the matrices ( ß i f c ) and (0L ik ). Since ( a i k )

transforms the vector r into χ' = a t , and ( ß i k ) transforms the vector χ' into

χ" = ß t ' , the product matrix ( y i k ) 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

First of all we observe that the formal rule for matrix multiplication is

the same as the rule for the multiplication o f determinants

1 The determinant of a product of two matrices i s equal to the product of the

determinants of the two factors

In the multiplication o f matrices, it is not necessarily true that

This establishes a second property of matrix multiplication

2 The product of two matrices depends in general upon the order of the factors

In the very special situation when E q ( l E l ) is true, the matrices Α and

Β are said to commute

I n contrast to the commutative law,

3 The associative law of multiplication is valid in matrix multiplication

That is,

Γ ( Β Α ) = ( Γ Β ) Α (1.10)

Thus, it makes no difference whether one multiplies Γ with the product of Β

and A, or the product of Γ and Β with A. T o prove this, denote the i-kth

element o f the matrix on the left side of (1.10) b y e i k Then

Then e ik = é ik , and (1.10) is established One can therefore write simply

Γ Β Α for both sides of (1.10)

The validity of 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 two matrices into a single one b y matrix multiplication

signifies simply the combination of two 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 two operations are equivalent

4 The unit matrix

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 ik , so that

dik = 0 {ιφ k)

The ô ik defined in this way is called the Kronecker delta-symbol The matrix

(ôik) = 1 induces the identity transformation, which leaves the variables

unchanged

If for a given matrix A, there exists a matrix Β such that

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 If the determinant of Α is not equal to

zero (|A I F E| φ 0), then an inverse transformation always exists (as has been

mentioned on page 2 ) T o prove this we write out the n 2 equations (1.13)

more explicitly

lßifl* = aik ( < , * = 1 , 2 , · · , η ) (1.14)

3 = 1

Consider now the η equations in which i has one value, say I These are η

linear equations for η unknowns ΒΑ , Β Ί 2, · · · , ß ln They have, therefore, one

and only one solution, provided the determinant \oijk\ does not vanish The

same holds for the other η — 1 systems of equations This establishes the

fifth property we wish to mention

5 / / the determinant \aLjk\ Φ 0, there exists one and only one matrix Β such

that ΒΑ = 1

Moreover, the determinant |ß^.| is the reciprocal of \oijk\, since, according

to Theorem 1,

From this it follows that α has no inverse if \aik\ = 0, and that β, the inverse

of a, must also have an inverse

W e n o w show that if (1.13) is true, then

αβ = 1 (1.16)

is true as well That is, if β is the inverse of a, then a is also the inverse of

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

It is clear that inverse matrices commute with one another

Rule: The inverse o f a product αβγδ is obtained b y multiplying the

inverses of the individual factors in reverse order (8~1

Another important matrix is

7 The null matrix, every element of which is zero

0 =

Obviously one has

(1.18)

for any matrix a

The null matrix plays an important role in another combination process

for matrices, namely, addition The sum γ of two matrices α and β is the

matrix whose elements are

)

The n 2 equations (1.19) are equivalent to the equation

γ = α + β or γ — α — β = 0

ΑΑΒ = Α Α Β ; a(A + Β) = AA + A ß

The formulas

(ab)a = a(ba);

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 ~ 1 + α0 1 + αΧ Α + (1.23)

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

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

( D D ' ) T T = I ViP* = 1 DAP?* = D ( D/O ik (1.26)

Addition of matrices is clearly commutative

Α + Β = Β + Α (1.20)

Moreover, multiplication by sums is distributive

Γ ( Α + Β ) = Γ Α + Γ Β (Α + Β ) Γ = Α Γ + Β Γ Furthermore, the product of a matrix Α and a number α is defined to be that

matrix Γ each element of which is α times the corresponding elements of A

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

A D = D A

(*D)ik = aikDk = (Da)ik = D{aik. (1.27) That is

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 _ 1 Α Β A similarity transformation leaves the trace of the matrix unchanged,

since the rule above states that Β _ 1 Α Β has the same trace as Α Β Β - 1 = Α

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 of a product of matrices Α Β = Γ yields

matrices and numbers are also preserved under similarity transformation

Theorem 10 therefore applies t o every matrix equation involving products

of matrices and numbers or other matrices, integral (positive or negative)

powers of 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,1 and are undoubtedly

already familiar t o many readers They are reiterated here since a firm

command of 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.2

LINEAR INDEPENDENCE OF VECTORS

The vectors T) 1? T> 2 , · · · , T) FC are said to be linearly independent if no relation­

ship o f the form

+ a2 T> 2 + · · · + a n *k = 0

) exists except that in which every a v α2, · · · , a k is zero Thus no vector in a

linearly independent set can be expressed as a linear combination o f the

other vectors in the set In the case where one of the vectors, say t) v is a null

vector, the set can no longer be linearly independent, since the relationship

1 · T>! + 0 · T>2 + · · · + 0 · v k = 0

is surely satisfied, in violation of the condition for linear independence

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

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

SINCE

2T>! + T>2 - T>3 = 0

ON THE OTHER HAND, T)X AND T)2 A E LINEARLY INDEPENDENT

1 M BORN AND P JORDAN, Z Physik 3 4 , 858 (1925)

2 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!)

If k vectors v v t) 2, · * · , t ) k are linearly dependent, then there can be found

among them k' vectors (k f < k) which are linearly independent Moreover,

all k vectors can be expressed as linear combinations of 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 way are linearly

independent, since if none of them can be expressed as a linear combination

of the others, no relationship of 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 of them, since this was the criterion for rejection

The linear dependence or independence o f k vectors v v t> 2, · · · , v k is also

a property of the vectors a i ) v · · · , a t ) k which result from them b y a proper

transformation a That is,

as can be seen b y applying α to 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

No more than η ^-dimensional vectors can be linearly independent T o

prove this, note that the relation implying linear dependence

If the coefficients a v α2, · · · , a n , aw + 1 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

An immediate corollary to the above theorem is the statement that any

η linearly independent η-dimensional vectors form a complete vector system; that

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 + ' ' ' + a

n*n + OVO = 0

must exist among the η vectors and the arbitrary vector Moreover, if

V V V 2 , * ' · , V n are linearly independent, the coefficient b cannot be zero Thus

any vector to can be written as a linear combination o f the v i9 so that these form a complete vector system

A row or a column of an ^-dimensional matrix can be looked upon as a

vector For example, the components of the vector a, fc which forms the k i h

column are a l k , a 2 k , · · · , ci nk , and those of the vector a { which forms the ith

row are α α, · · · , a in A nontrivial linear relationship among the column

The vanishing of the determinant |a fÄ | 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 · · · , V n 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 of a matrix

1, W e now 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 now on we will name the

coordinate axes after the elements of an arbitrary set I f G is a set of objects

g, h , i , · · · , then the vector ο in the space of the set G is the set of numbers

V g , T> h , O t , · · · Of course only vectors which are defined in the same space

can be equated (or added, etc.) since only then do the components correspond

to the same set

A similar system will be used for matrices Thus for a matrix α to be

applied to a vector ν with components v g , O h , Ό { , · · · , the columns of α must

be labeled by 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 , · · · of this set, and α transforms a vector t) in the space of G into a

vector at) in the same space That is

leG

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

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

y ζ

5 - 1 ) y

is a matrix (The symbols for the rows and columns are indicated.) In this example

ct xx = 1, a x y = 2, a x z = 3 Eq (2.1) states that the ^-component of t>' = at> is given by

= 1 · 1 + 2 · 0 + 3 ( - 2 ) - - 5

The simple generalization above is purely formal; it involves merely

another system of 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

be multiplied with one another, just like the matrices in the previous chapter

2 A further generalization is that in which the rows and columns o f

matrices are labeled b y elements of different sets, F and G. Then from (2.1),

where j is an element of the set F, and I runs over all the elements of 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 of G into a vector

tt> in the space of F In general the set F need not contain the same number

of elements as the set G If it does contain the same number of elements, then

the matrix has an equal number of rows and columns and is said to be

''square in the broader sense."

Let the set G contain the symbols *, Δ> • > and the set F the numbers 1 and 2 Then

* Δ •

\0 - 1 - 2 / 2

is a rectangular matrix (The labels of the rows and columns are again indicated.) It

transforms a vector t)* = 1, D/\ = 0, = —2 into the vector

vo = an

The components χθχ and m2 are then

tX>i = <*i*D* + αι ΔϋΔ + *ιΠ°Π = 5 · 1 -f 7 · 0 -f 3( —2) = - 1

tö2 = + α2 ΔϋΔ + α2 Πυπ = 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 of the second factor and the columns of the first

"match." On the other hand, the rows o f the first and the columns of the

second factor can correspond to elements of completely different sets, Ε and

where j is an element of E, k an element of 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 of 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 of G

into one in the space of E

LET G BE THE SET * , Δ > • AGAIN, LET F CONTAIN THE LETTERS χ AND y, AND Ε THE NUMBERS

* Δ • / 5 4 69 8 4 \ 1

Υ ~~ \ 3 3 4 5 5 7 / 2

3 W e now investigate h o w the ten theorems of 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 of this chapter, since the specific numerical nature of the indices

has not been used anywhere in the first chapter

Addition of 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 of the rows of the three matrices Α, Β, Γ must be the same, as

well as the labeling of their columns On the other hand, for multiplication

the columns of the first factor and the rows of the second factor must match;

only then (and always then) can the product be constructed The resulting

product has the row labeling of the first, and the column labeling of the

second factor

THEOREM 1 W e can speak of the determinant of a rectangular matrix if it

has the same number of rows and columns, although these may be labeled

differently For matrices "square in the broader sense" the rule that the

determinant of the product equals the product of the determinants is still

valid

THEOREMS 2 and 3 The associative law also holds for the multiplication o f

rectangular matrices

( Ο Β ) Γ = Α ( Β Γ ) (2.3)

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 If Α is a matrix

which is square only in the broader sense, the equation

Β Α = 1

implies that the columns of Β match the rows of A Furthermore, the rows

of 1 must match the rows of Β , and its columns must match the columns of A

Since 1 is square in the restricted sense, the columns o f Α must also match the

rows of Β

The rows of the matrix Β inverse to the matrix Α are labeled by the same set as

the elements of the columns of a, its columns by the same elements as the rows 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

Moreover,

Α Β = 1 (2.4a) However, it should be noted that the rows and columns of 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

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

Σ Α Σ - 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 _ 1 , and thus of 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 of Σ - 1 , must be square in the restricted sense

On the other hand, Σ itself can be square in the broad sense: the columns and

change the labeling of rows and columns are especially important The

so-called transformation theory of quantum mechanics is an example of such

transformations

The introduction of 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 of these

entities The use of such more complicated matrices will always be explained

specifically unless the enumeration of rows and columns is so very clear b y

the form and definition of 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 two or more numbers, for example

The first column is called the "1,1 column;" the second, the "1,2 column;"

the third, the "2,1 column;" the fourth, the "2,2 column;" the rows are

designated in the same way The elements of (2.E.1) are

If the number of rows in Α is n 1 and the number of columns, n 2 , 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 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

(a 1 c 1 « I C 2 \ /^I^I & Α \

a 2 c 1 a 2 c 2 J \ 62 ^ I ^2^2/

T H E O R E M 1 If_aa = a and ß ß = β, and if α χ β = γ and α Χ β = 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

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 of direct products

In formal calculations with matrices it must be verified that the multi­

plication indicated is actually possiblẹ In the first chapter where we had

square matrices with η rows and columns throughout, this was, of course,

always the casẹ In general, however, it must be established that the rows of

the first factor in matrix multiplication match the columns of the second

factor, ịẹ, that they both have the same names or labels The direct

product of two matrices can always be constructed b y (2.6)

A generalized type of 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 ik ) whose elements A ik are themselves matrices A ik is that matrix

in which the number ậ f c î occurs in the jih row and the Zth column

= « = (A < J b), where (Â)^ = a i j ; k l (2.10)

T H E O R E M 3 If a = (A u >) and β = ( Bi Vr ) , then αβ = γ = ( Ci r) where

The right-hand side of (2.11) consists of a sum o f products of matrix multiplications W e have

On the other hand

In the simplest case we might have two square matrices

22'

is meaningless, since the number o f columns of B n , for example, differs from the number o f rows of A n

In the first chapter we established a very important property of similarity

transformations They leave the trace of a matrix unchanged;1 the matrix

α has the same trace as σ-1

ασ. Is the trace o f a matrix the only invariant under similarity transformation? Clearly not, since, for example, the deter­

minant |σ_1

ασ| is also equal to the determinant |a|. In order to obtain further

invariants, we consider the determinantal equation of the nth order for λ

Clearly the determinant |σ_1

(α — Α1)σ| is also equal to zero; this can be written

Ισ-1

Equation (3.4) shows that the η roots of the secular equation |β — λΐ\ — 0

are identical2 with the η roots of the secular equation |α — λΐ\ = 0 The

roots of the secular equation, the so-called eigenvalues of the matrix, are invariant

under similarity transformations W e shall see later that in general a matrix

has no other invariants Also, the trace is the sum, and the determinant is

the product of the eigenvalues, so that their invariance is included in the

theorem stated above

1

The matrix which undergoes a similarity transformation must always be a square

matrix For this reason we again denote the rows and columns with the numbers,

W e now consider one eigenvalue, λ ν The determinant of the matrix

(a — Ajl) is zero, so that the linear homogeneous equations

have a solution A linear homogeneous system o f equations like (3.5) can

be written for each of the η eigenvalues X k W e denote the solutions o f

this system, which are determined only up to a common constant factor, b y

The set of η numbers rl f c, r2 f c, ' * * , V n k is called an eigenvector t mJc of the matrix

a; the eigenvector t, k belongs to the eigenvalue X k Equation (3.5a) can

then be written

The matrix transforms an eigenvector into a vector which differs from the

eigenvector only b y a constant factor; this factor is the eigenvalue itself

The eigenvectors t v r 2 , · · · , t n can be combined into a matrix ρ in

such a way that t k is the &th column of this matrix

Pik =

(**k)i — *ik *

Then the left side of (3.5a) consists of the (ik) element of αρ The right side

also can be interpreted as the (ik) element of a matrix, the matrix ρΛ,

where Λ is a diagonal matrix with diagonal elements λ ν λ 2 , ' ' ' , λ η

A similarity transformation by a matrix whose columns are the η eigen­

vectors transforms the original matrix into the diagonal form; the diagonal

elements are the eigenvalues of the matrix. T w o matrices which have the same

eigenvalues can always be transformed into one another since they can both

be transformed into the same matrix The eigenvalues are the only invariants

under a similarity transformation

This is true, of course, only if ρ has a reciprocal, that is, if the η vectors

t i t 2 · · ' , t n are linearly independent This is generally the case, and is always true if the eigenvalues are all different Nevertheless, there are exceptions, as is shown, for example, b y the matrices

These cannot be brought into diagonal form b y any kind of similarity transformation The theory of elementary divisors deals with such matrices; however, we need not go into this, since we shall have always to deal with matrices which can be brought into the diagonal form (3.6a) (e.g., with unitary and/or Hermitian matrices)

The conditions for the commutability of two matrices can be reviewed very well from the viewpoint developed above I f two matrices can be brought into diagonal form b y the same transformation, i.e., if they have the

commute after the similarity transformation; therefore they must also commute in their original form

In the first chapter we defined the rational function of a matrix

/ ( A ) = · · · a _ 3A ~ 3 + a_ 2ar 2 + a ^ A - 1 + a 0l + aXA + a 2 A 2 + a3 A 3 + · · ·

To b r i n g / ( A ) into diagonal form it is sufficient to transform Α to the diagonal

Σ - γ ( Α ) Σ = Σ - Χ ( · · · a _ 2 a r 2 + a ^ a r 1 + a 0 l + α Χ Α + α 2 Α 2

+ · · · ) Σ ,

and this is itself a diagonal matrix I f X k is the Arth diagonal element in

Λ — ( A i k ) — (ô ik X k ), then (X k ) p is the Arth diagonal element in ( A ) P and

* ' * 0 - 2 * Γ 2 + n - i K 1 + o + < h h + α 2 λ 1 + = f ( K )

is the Arth diagonal element i n / ( A )

A rational function / ( A ) of a matrix Α can be brought into diagonal form

by the same transformation which brings Α into diagonal form The diagonal elements, the eigenvalues of / ( A ) , are the corresponding functions

/ ( A ^ / ^ g ) , * * * , / ( λη) of the diagonal elements λ ν A2, * * · , λ η of Α W e assume

that this law holds not only for rational functions but also for arbitrary functions F (a) of Α and consider this as the definition of general matrix functions

3 N o t e t h a t t h e e i g e n v a l u e s c a n d i f f e r a r b i t r a r i l y

or

= · · · α_ 2 Λ~ 2 + α - Ι Λ - 1 + α 0 1 + + α 2 Α 2 + · · · = / ( Α )

Special Matrices

One can obtain from a square matrix α a new matrix α', in which the roles

of rows and columns are interchanged The matrix a' so formed is called

which verifies (3.7a)

The matrix which is formed b y replacing each of the n 2 elements with its

complex conjugate is denoted b y a*, the complex conjugate of a I f a = a*

all the elements are real

By interchanging the rows and columns and taking the complex conjugate

as well, one obtains from α the matrix a*' = a'* This matrix is called the

B y assuming various relationships between a matrix α and its adjoint,

transpose, and reciprocal, special kinds of matrices can be obtained Since

their names appear frequently in the literature we will mention them all;

in what follows, we shall use only unitary, Hermitian, and real orthogonal

matrices

If α = α* (i.e., a i k = a^*), the matrix is said to be real, and all n 2

elements a i k are real I f α = —α* (oL ik = — α$*), then the matrix is purely

imaginary

If S = S' {S ik = & ki ), the matrix is symmetric; if S = —S' (S ik = —S ki ) }

it is skew- or anti-symmetric

If H = H* (H iÄ = Η^·), the matrix is said to be Hermitian; if A = —A*,

skew- or anti-Hermitian

If α is real as well as symmetric, then α is Hermitian also, etc

If 0' = 0 _ 1, then 0 is complex orthogonal A matrix U, for which = U- 1 ,

is said to be a unitary matrix I f R 1 " = R - 1 , and R = R* (real), then

R' = R*' — Rt — R-i and R' = R - 1 ; R is said to be real orthogonal, or

simply orthogonal

Unitary Matrices and the Scalar Product

Before discussing unitary matrices, we must introduce one more new

concept In the very first chapter we defined the sum of two vectors and a

constant multiple of a vector Another important elementary concept is the

scalar product of two vectors The scalar product o f a vector ft with a vector

b is a number W e shall distinguish between the Hermitian scalar product

and the simple scalar product

αΛ + a2b2 + · · · + a n b n = ((ft, b)) (3.9a) Unless we specify otherwise, we always refer to the Hermitian scalar product

rather than the simple scalar product If the vector components a 1? a 2, * * · , a n

are real, both products are identical

If (a, b) = 0 = (b, ft), then a and b are said to be orthogonal to one another

If (a, a) = 1, it is said that α is a unit vector, or that it is normalized The

product (ft, ft) is always real and positive, and vanishes only when all the

components of α vanish This holds only for the Hermitian scalar product,

in contrast to the simple scalar product; for example, suppose α is the

two-dimensional vector (1, i) Then ((a, a)) = 0, but (a, a) = 2 In fact (a, a) = 0

implies that a = 0; but this does not follow from ((a, ft)) = 0

Simple Rules for Scalar Products:

1 Upon interchange of the vectors

On the other hand

(eft, b) = c*(ft, b) whereas ((eft, b)) = c((ft, b))

(3.10)

(3.10a)

(3.11)

3. The scalar product is linear in the second factor, since

(a, bb + cc) = b(a, b) + c(a, c) (3.12)

It is, however, "antilinear" in the first factor

(aa + bb, C) = α*(α, C) + b*(b, C) (3.12a)

4 Furthermore, the important rule

(α, ab) = (οΛι, b) or (ßa, b) = (a, ßf b) (3.13)

is valid for arbitrary vectors α and b, and every matrix a T o see this,

Instead of applying the matrix a to one factor of a scalar product, its adjoint

a* can be applied to the other factor

For the simple scalar product the same rule holds for the transposed

matrix; that is

((α, ab)) = ((a'a, b))

5 W e now write the condition = U - 1 for the unitarity of a matrix

somewhat more explicitly: U^U = 1 implies that

ί (Ut)„U,f c = I U|U,fc = ίΛ; (U.,, V.k) = aÄ (3.14)

// £Λβ η columns of a unitary matrix are looked upon as vectors, they comprise η

orthogonal unit vectors. Similarly, from UU^ = 1, it follows, that

j The η rows of a unitary matrix also form η unit vectors which are mutually

orthogonal

6 A unitary transformation leaves the Hermitian scalar product un­

changed; in other words, for arbitrary vectors α and b,

(Ua, Ub) = (a, Uf

Conversely, if (3.15) holds for a matrix U for every pair of arbitrary vectors

α and 6, then U is unitary, since then Eq (3.15) holds also for α = t i t and

b = t k (where (t k ) l = ô kl ) But in this special case (3.15) becomes

The same rule applies to complex orthogonal matrices, with respect to the

simple scalar product

7 The product UV of two unitary matrices U and V is unitary

The reciprocal U _ 1 of a unitary matrix is also unitary

(U-i)t = (Ut)t = U = (U-1

)-1

The Principal Axis Transformation for Unitary and Hermitian Matrices

Every unitary matrix V and every Hermitian matrix Η can be brought into

diagonal form by a similarity transformation with a unitary matrix U For

such matrices, the exceptional case mentioned on page 22 cannot occur

First of all, we point out that a unitary (or Hermitian) matrix remains

unitary (or Hermitian) after a unitary transformation Since it is a product

of three unitary matrices, U - 1

To bring V or Η to the diagonal form, we determine an eigenvalue of V

or H Let this be λ χ ', the corresponding eigenvector, U.x = (U n · · · U wl ) is

determined only up to a constant factor W e choose the constant factor

so that

(TJ.1; U.i) = 1

This is always possible since (U el , U el ) can never vanish W e now construct

a unitary matrix U of which the first column is Ue l.4

With this unitary matrix we now transform V or Η into U^VU or U_ 1

HU. For example, in

XJ-1

VU, we have for the first column

xr l = (U-!VU)rl = (Utvu)rl = 2KI ν,μνμι = Σ ΚΚ^Λ = δΛλν

ν μ ν

since Ue l is already an eigenvector of V. W e see that λ1 occurs in the first

row of the first column, and all the other elements of the first column are zero

Obviously, this holds true not only for U_1

VU, but also for U_ 1

HU. Since

4 See the lemma at the end of the proof

U _ 1 HXJ is Hermitian, the first row is also zero, except for the very first

element; thus U _ 1 H U has the form

But XJ _ 1 VU must have exactly the same form! Since X is a unitary matrix,

its first column X # 1 is a unit vector, and from this it follows that

| X u | 2 + | X 2 1 | 2 + · · · + | X M L | 2 = L^L 2 = Ι ( 3 E 2 ) The same argument applies to the first row, X 1 # of X The sum of the squares

is given b y

| X n | 2 + |Χχ 2 | 2 + · · · + |Xm| 2 = W 2 + | X 1 2 | 2 + | X 1 3 | 2 + · · · + |Xi„| 2 = I

which implies that X 1 2 , X 1 3 , · · · , X l w all vanish

Therefore, every unitary or Hermitian matrix can be transformed into

the form (3.E.1) b y a unitary matrix The matrix (3.E.1) is not yet a diagonal

matrix, as it cannot be, since we have used the existence of only one eigen­

value It is however more like a diagonal matrix than the original matrix V,

or H It is natural to write (3.E.1) as a super-matrix

\ 0 VJ \0 B.J ( 3 E 3 )

where the matrix V x or has only Η — 1 rows and columns W e can then

transform ( 3 E 3 ) b y another unitary matrix o f the form

A ( Η

LO I Y

where JJ 1 has only Η — 1 rows and columns

Under this process (3.E.1) assumes the form

The procedure above can be applied again and JJ 1 can be chosen so that

U L V I U I O R ΥΐΗΧΫΧ has the form

where V 2 or H 2 are only Η

has the form

Clearly, repetition of this procedure will bring V or H entirely into diagonal form, so that the theorem is proven

This theorem is not valid for symmetric or complex orthogonal matrices,

as the second example on page 22 shows (the second matrix is symmetric and complex orthogonal) However, it is valid for real symmetric or real orthogonal matrices, which are just special cases o f Hermitian or unitary matrices

Lemma I f (U#1 , U t ) = 1, then a unitary matrix can be constructed (in

many different ways), whose first column is Μ Λ — (un, U2i> · · * , Unl)

W e first construct in general a matrix the first column o f which is U #1 and which has a nonvanishing determinant Let the second column of this matrix

be t>.2 = (o 12 , t)2 2, · · · , v n2 ), the third t>.3 , etc

The vectors tt#1, t) 2 , υ 3 , · · · are then linearly independent since the deter­

minant does not vanish Since we also wish them to be orthogonal, we use the Schmidt procedure to "orthogonalize" them First substitute U.2 = «2iUe l + t> 2 for t> 2 ; this leaves the determinant unaltered Then set

(U v tt.2) = 0 = a 21 {U v U.i) + (U 1? t> 2) = a 21 + (u v t> 2 )

and determine a 21 from this Next write U 3 in place of t> 3 with tt.3 =

α3ΐ**.ι + α32**.2 + tt.3 a n

d determine a 31 and a 32 so that

Proceeding in this way, we finally write U n in place of t) w , with n w =

U w _ x + V n , and determine a nl , a n2 , a n3 , · · · ,

a n,n-V s o ^ a t

0 = « , ) = a 3l (U v «.!> + («.!, t> 3 )

0 = (U.2, II.3) = a32(M.2, U.2) + (U.2,W.3)

0 = (n v U.J = a nl (U v tt.a) + ( « ! , » „ ) ,

0 = (U.2> «.„) = an2(U.2,1l.2) + (U.2, V.n),

0 = (u n _ v «.„) = «„.„^(tt.^, u n ^) + (u.^, o.n)

In this way, with the help of the \N(N — 1) numbers A we succeed in sub­ stituting the vectors u for the vectors V The U are orthogonal and non- null b y virtue o f the linear independence o f the V Assume, for example, that U N = 0 This implies

and since the tt#1, TI 2 > " ' ' , tt.w are linear combinations o f the ttel, T> 2 > ' * * >

V n _ v one could write T) N in terms of these Η — I vectors, in contradiction

to their linear independence

Finally, we normalize the U 2 , U 3, · · * , U N , thereby constructing a unitary

matrix whose first column is U# 1

This "SCHMIDT ORTHOGONALIZATION PROCEDURE" shows how t o construct from

any set o f linearly independent vectors an orthogonal normalized set in

which the KTH unit vector is a linear combination o f just the first K o f the original vectors I f one starts with Η η-dimensional vectors which form a complete set of vectors, one obtains a complete ORTHOGONAL system

If a unitary matrix V or a Hermitian matrix Η is brought t o the diagonal

form this way, then the resulting matrix A V or A H is also unitary or Hermitian

It follows that

THE ABSOLUTE VALUE OF EACH EIGENVALUE OF A UNITARY MATRIX 5 IS 1; THE EIGENVALUES

OF A HERMITIAN MATRIX ARE REAL This follows directly from (3.19), which states that for the eigenvalues Λ Ν of the unitary matrix, Λ Ν Λ* = 1 ; for those o f a Hermitian matrix, X H = Λ* The eigenvectors o f V, and of H, as columns

of the unitary matrix U, can be assumed t o be orthogonal

Real Orthogonal and Symmetric Matrices

Finally, we investigate the implications of the requirement that V, or H,

be complex orthogonal (or symmetric), as well as unitary (or Hermitian) I n

this case, both V and Η are real

From U^VU = AV, we obtain the complex conjugate υ*^ν*υ* = (U*)îVU* = Λ*. Since the eigenvalues, as roots of the secular equation, are

independent of how the matrix is diagonalized (i.e., whether b y U or U*), the

diagonal form A V can also be written as Λ* Thus the numbers Λ Ν Λ 2 ,·'·,Λ Η

are the same as the numbers Λ*, Λ*, * · · , A* This implies that THE COMPLEX

EIGENVALUES o f a real orthogonal matrix V OCCUR IN CONJUGATE PAIRS Moreover

since VV — 1, they all have absolute value 1 ; the real eigenvalues are

therefore ^ 1 I n an odd-dimensional matrix at least one eigenvalue must

be real

1, or AH = A{ (3.19)

AS EQUATION (3.E.2) ALREADY SHOWS

If D is an eigenvector for the eigenvalue λ, then t)* is an eigenvector for

the complex conjugate value A* T o see this write Vt) — At); then V*t)* =

= Vt)* Moreover, if λ* is different from λ, then (t>*, t>) = 0 = ((t), t>));

the simple scalar product of an eigenvector with itself vanishes if the corre­sponding eigenvalue is not real (not ± 1 ) · Conversely, real eigenvectors (for which the simple scalar product does not vanish) correspond to the eigen­

values J^l Also, let t) be the eigenvector for λ ν let υ* be that for Af, and

3 that for A2 Then if λ χ φ λ 2 , it follows that

0 = (»*, J) = ((», }))·

The simple scalar product of two eigenvectors of a real orthogonal matrix i s always zero if the corresponding eigenvalues are not complex conjugates; when the eigenvalues are complex conjugates, the corresponding eigenvectors are them­ selves complex conjugates

W = 1; it follows that the determinant of V multiplied with that of V must give 1 The determinant of V, however, is equal to that of V, so that

both must be either + 1 or — 1

If Η is real, the Eq (3.5) is real, since the X h are real. The eigenvectors of

up to a constant factor, they can also be multiplied b y a complex factor.)

Thus, the unitary matrix U in U _ 1H U = A H can be assumed real

1 In the years before 1925 the development o f the then new "Quantum

Mechanics" was directed primarily toward the determination of the energy

of stationary states, i.e., toward the calculation of the energy levels The

older "Separation Theory" o f Epstein-Schwarzschild gave a prescription for

the determination o f the energy levels, or terms, only for systems whose

classical mechanical motions had the very special property of being periodic,

or at least quasi-periodic

An idea of W Heisenberg, which attempted a precise statement of the

Bohr correspondence principle, corrected this deficiency It was proposed

independently b y M Born and P Jordan, and b y P A M Dirac Its essence

is the requirement that only motions which later would be seen as quantum

mechanically allowed motions should occur in the calculation In carrying

through this idea these authors were led to introduce matrices with infinite

numbers of rows and columns as a formal representation of position and

momentum coordinates, and formal calculations with "q-numbers" obeying

the associative but not the commutative law

Thus, for example, the equation for the energy H of the linear oscillator1

1 Ο Κ Β

2m 2

is obtained b y formally substituting the matrices Ρ and Q for THE MOMENTUM

AND POSITION COORDINATES Ρ and Q in the HAMILTONIAN FORMULATION of the

classical expression for the energy It is required that Η be a diagonal

matrix The diagonal terms H T O N then give the possible energy values, the

stationary levels o f the system On the other hand, the absolute squares o f

elements Q NK of the matrix Q are proportional to the probability of a spon­

taneous transition from a state with energy H N N to one with energy VI KK They

| | J_|

give, therefore, the intensity of the line with frequency ω = — - k

All

Η

of this follows from the same considerations which suggest the introduction

of matrices for Ρ and Q

In order to specify the problem completely, one had still to introduce a

1 THE M IS THE MASS OF THE OSCILLATING PARTICLE, AND Κ THE FORCE CONSTANT; Q AND Ρ

ARE THE POSITION AND MOMENTUM COORDINATES

31

''commutation relation" between ρ and q This was assumed to be

p q - q p - τ ΐ (4.2)

where % is Planck's constant divided b y 2π

Calculations with these quantities, although often fairly tedious, led very

rapidly to beautiful and important results of a far-reaching nature Thus,

the "selection rules" for angular momentum and certain "sum rules" which

determine the relative intensity of the Zeeman components of a line could be

calculated in agreement with experiment, an achievement for which the

Separation Theory was inadequate

E Schrödinger, b y an approach which was independent of Heisenberg's

point of view, arrived at results which were mathematically equivalent to

those mentioned above His method bears deep resemblance to the ideas o f

L DeBroglie The discussion to follow is based on Schrödinger 's approach

Consider a many-dimensional space with as many coordinates as the system

considered has position coordinates Every arrangement of the positions o f

the particles of the system corresponds to a point in this multidimensional

"configuration space." This point will move in the course of time, tracing

out a curve b y which the motion o f the system can be completely described

classically There exists a fundamental correspondence between the classical

motion of this point, the system point in configuration space, and the motion

of a wave-packet, also considered in configuration space,2 if only we assume

that the index of refraction for these waves is [2m(E — V)]li2

/E Ε is the

total energy of the system; V, the potential energy as a function o f the

configuration

The correspondence consists in the fact that the smaller the ratio between

the wavelengths in the wave-packet and the radius of curvature of the path

in configuration space, the more accurately the wave-packet will follow that

path On the other hand if the wave-packet contains wavelengths as large

as the classical radius of curvature of the path in configuration space then

important differences between the two motions exist, due to interference

among the waves

Schrödinger assumes that the motion of the configuration point corre­

sponds to the motion of the waves, and not to the classically calculated

The development of the text follows Schrödinger's ideas more closely than is

customary at present (remark of translator)

ψ = ψΕ exp (^—i ^ tj, (4.4)

where ψ Ε is independent o f t H e thus obtains the eigenvalue equation

where ψ Ε is a function of the particle position coordinates x v x 2) · · · , x f

It is necessary to require that ψ Ε be square-integrable, i.e., the integral

OO 00

j " ' J | V E ( * „ % , · • ' * , ) | !

* » , · dx,

— 00 — 00

over all configuration space must be finite In particular, ψ must vanish at

infinity The values of Ε for which the determination o f such a function,

ψΕ, is possible are called the "eigenvalues" o f (4.5); they give the possible

energy values of the system. The corresponding square integrable solution o f

(4.5) is called the eigenfunction belonging t o the eigenvalue E

Equation (4.5) is also written in the form

where Η is a linear operator (The Hamiltonian, or energy operator)

\2m1 dx\ 2m2 dx\

+ V(xvx2, - · · ,xf). (4.5b)

The last term means multiplication b y V(x v x 2f - · -, x f )

It transforms one function o f x v x 2 , · · · , x f into another function The

function ψ o f (4.4) fulfills the relationship

dt

The total energy o f the system does not appear explicitly in (4.6), so that it

where the position coordinates o f the particles in

the system considered, m1, m 2 , · · · , m f , the corresponding masses, and

V(x v x 2 , * · · , x f ) is the potential energy in terms of the coordinates of the

individual particles 00 ^ y 00 g 5 J *^^ ·

The total energy o f the system appears explicitly in (4.3) On the other

hand the frequency, or the period o f the waves, is still unspecified

Schrödinger assumes that the frequency o f a wave which is associated with

the motion of a system with total energy Ε is given b y hco = Ε H e therefore

substitutes into (4.3)

applies generally t o all motions, independent o f the energy o f the system;

it is called the TIME-DEPENDENT SCHRÖDINGER EQUATION

The two Eqs (4.5) (or (4.5a), (4.5b)) and (4.6) are the basic equations o f quantum mechanics The latter specifies the change of a configuration wave

in the course o f time—to which, as we will see, a far-reaching physical reality is attributed; (4.5), (or (4.5a), (4.5b)) is the equation for the fre­

quency Ω = E/H, the energy Ε, and the periodic time-dependence o f the wave function Ψ Indeed, (4.5a) results from (4.6) and the assumption that

All the simple calculational rules o f Chapter 3 apply t o this scalar product

Thus, if A 1 and A 2 are numerical constants,

(Ψ, (H9I + a 29 2 ) = A I(<P> 9ι) + Α 2 (Φ, G 2 ),

and

(<P>9) = (9>

Ψ)*-(Φ, Φ) is real and positive and vanishes only IF Φ = 0 IF Ψ)*-(Φ, Φ) = 1, then

Ψ is said to be normalized I f the integral

oo

is finite, then Φ can always be normalized b y multiplication b y a constant

^1/c in the case above, since ^— ,— J = i j T w o FUNCTIONS are orthogonal

if their scalar product is zero

The scalar product given in the Eq (4.7) is constructed by considering the

functions Φ(χ 1 · · · X F ), G(X L ···#/) of X LT X 2 , · · · , X F as vectors, whose components are

labeled by / continuous indices The function vector Φ{χ χ · · · X F ) is defined in an /-fold

infinite-dimensional space Each system of values of X X · · · X F , i.e., each configuration,

corresponds to one dimension Then the scalar product of Φ and G, in vector language, is

{ψ* 9) = 2 ' ' ' χ Α*9( χ ι " ' x f )

for which the integral (4.7) was substituted