Group theory; the application to quantum mechanics

Every function of continuous variables and particularly the wave function in SchrOdinger mecbanics can be represented by vectors in a /UllCtion space in which the "umber of dimensions i

Ch 1 § 3J REDUcnON TO MAIN AXES 13

there are three rectangular axes for which one has

Di = ai E" (i == 1, 2, 3)

the Bl are the main dielectric constants of the medium Simllarly in the general case in which we are actually interested, we have to search for a unitary system of axes ei which brings the matrix A' (the transformed of ,4 ) into the

diagonal form (1.13) This means solving the following problem: find n

directions ei, e~, .• , e~ such that every vector x parallel to one of the

directions ,; will be transformed by A into a vector y 'which is parallel to %!

where (X is a constant

Writing the "components" of this vector equationt we obtain n linear homogeneous equations of the form

a'lxl + (an-«)x,+ , , a",x" := 0 (i == 1, 2, , n), (1.14a)

which are only compatible with each other if the undetermined constant Of

is a root of the secular equation 1

This equation has in general n roots.l ::: (Xl II «, (which may be distinct

or not) to which correspond the 11 directions of the axes given by' equation (1.141.) We know only the directions of the vectors :I but ftot their'magnitude becaUse (1.144) determines omy the ratios between the oomponents The roots' «, of (I IS) are the eigenvalues, propervalues, or characteristic con stallts of the matrix A They are real in the' Hermitian case; in the unitary

case 'their absolute value is 1 The er are the eiaenvector or principal

diree-tiollSa~ When (1.1S) represents a multiple roOt of order P(<<l - «2 :::e •• ~

1 This name oriainates from a PrQblom in astroD9Jll)', where a siluilar "Iuation deter

mines the perturbation over long time interVals

• The p~ oftllele theorebllaabe made bJd100sinaa root CIt, 0(1.15), detem1iDiog

the correapondiDa eiaenvoctor ~ • _complotfq~tho,~>with (II-I) vecto.l'J whidt

have to be ortholonal to the first one In order to fonn'a unitary system or axes Asa result of the symmetry proportiea ot the HermitiaD':and:'1Ulitary lIIatrices the coefficients

all I» • • at" list ••• Q".l, are all zero and the matrix A takes the form:

(

<<1 }~ 0 : 0)

o j au t - a",

() ~: : : : ~~

14 VEcrOR SPACES - UNITARY GEOMETRY [eh 1, § 3

1

ap = tX), the vectors which have the property 1 == Ax == (X% form a subspace

of p dimensions in which the direction of the axes is not determined (See the second note on page 12)

3.3 JOINT DIAOONALIZATION OF A SET OF MATRICES

In order that all the matrices of an Hermitian or unitary system can be

reduced at the same time 10 their principal axes, it is necessary and sufficient that they all commute with each other

First we "ill show that this condition is necessary: let A and 8 be two

matrices By a transformation of axes S we reduce them simultaneously to

diagonal form A' and Sf Then obviously they commute with each other:

A'B' = B'A' Hence S-lASS-las = S-lABS == S-l BAS i.e AB == BA

Now we will show that the condition is sufficient: let us suppose that AS =

BA and let us make a transformation of coordinates such that 8 is nal:

diago-We have

If

{J, ~ p", QUe == O

So ~ fiIld that A is a Itep-'WiIe matrlx t of which all the terms are zero,

cxce~ those which are", situated on the main diagonal, or those inside certain

squar~ that share the diagonal; they correspond with the case in which

p, -"ll"e These squares are eaGh related to a subspace \ll in which the

~pal directions of the matrix B are undetermined (the matrix posseasoa a

cirtuiar, spherical, or hyperspherical symmetry)_ One may finaDy choose

these undetermined axes in such a way that A will be completely diagonal

3.4 INVAlUANCE OF A SBCULAll EQUATION

A traDsformatiOll of coordiiudes does not chaDse She form of the secular equation (leIS) Let US carry··out the tra1l1formation of coordiaates 5:

A -+ A' S-lAS;

The rule of multiplication of determiDants gives us

IS-1AS-lII =- IS-11-IA-lII·ISI

== IS-1

1 • lSI • IA- A II == fA -; II (1.16)

Ch 1, I 3, 4] FUNCI10N SPACE 15

3.S TRACE OR SPUR OF A MATRIX

Let A = (aft) be a matrix, the trace is then the sum of all the diagonal elements

,

,= 1

To prove that the trace is invariant we shall write equation (I t S) in the form:

(-l)-+ (_l)'l-l(al1 +a22+ • a,.,.)+ = (-~r+ (_A)',-t Tr A +

Because this equation keeps its form under a transformation of coordinates S" all the coefficients are invariant, in particular the second:

4 Function Space Complete Sets of Orthogonal FDDetlo

Every function of continuous variables and particularly the wave function

in SchrOdinger mecbanics can be represented by vectors in a /UllCtion space

in which the "umber of dimensions is infinite The operators acting on the wave runctions ~l transform these functions into other functions producing

a transformation of this space irlto itself On the other hand quantum ics can also be formulated with the help of certain relations among matri-ces The matrices in this so-called matrix mechanics are matrices with an infinite number of rows and columns They can again be considered as operators that transform a space with an infinite number of dimensions into

mechan-its~f This analogy, brought forward by Hilbert, made it possible for Schradinger and later for Dirac to show the equivalence between wave mechanics and matrix mechanics

There exists between the two spaces just mentioned a difference which appears to be essential: the matrices operate on a space in which the number

of dimensions it denumerably infinite; on the contrary the Dumber of dimen sions in function space is of the order of a continuum We will see, in a moment that this difference is more apparent than real (see § 4.3)

4.1 FUNCTION SPACE

"'(x) defines a function of one variable x The simplest case is that of a discontinuous function where the value is given only for a finite number of

~·alues/or the variable x : Xl, %2, ••• , x., i.c the domaia of the variable x

consists of a set of discrete points The " correspondina values of the function "'1' tit 2 · f • t/J ( "', == ';<xc» can be considered as components of a vector in a sp~ of n dimensions To each different function tp(x) defined

16 VECTOR SPACES - UNITARY GBOMETRY reb 1, § 4 for the same values of x there corresponds a distinct vector which is the

geometrical representation of that particular function

Normally the functions in which we are interested depend on continuous variables So we have to go over to the limit in which the number n of di-mensions of the function space becomes infinite In such a case one may

consider the value ~ of the variable x as an index and say that J/J(e) is the component of a veetor '" along an axis characterized by the index ~4I For

functions of several variable., such as "'(x~ Yt z) the set x, y, z constitutes one single index and the function space shows the same properties as those with

a single variable

In the next section we will reason maiDly by analolY and be satisfied by indicating now and then the mathematical difficulties We will always assume that conditions of convergence are realized

4.2 SCALAR PR.ODUCT'; NORM

In IJ-dimensional space, the scalar product of two vectors % and y has been defined by

fJ

'-1

By anal6gy in the case of two functions of a discontinuous variable, which

are defined for the values Xl' X2' •• II, XII of the variable, we will put:

•

(t/I · tp) = L ",*{Xt)!p(x,,)

t - l

If the functions cp(x) and ';(x) are defined in a continuous domain D

of the variable x the only possibly generalization is

(1/1 • ) = L ,,*(x)cp(x}dx (1.18)

Thia iatearal is the _ar product of two /unctiDNI ';(X) ' and 'P{x) The

wave function in quantum mechanics depends on the coordinates of the particles Xl , Yl, %1, X2' • •• , z,., the domain D is the, finite or infinite con- figuration space in which we will indicate the element by dt == dx1 , dYl •

dZl II dx2 ••• dz p and we have

(t/I • cp) -= Iv ",·.dt (L1Sa)

In particu1arthe norm of the function t/I, which corresponds to the square

of the length of the representing vector can be written

('" · "') J., "," JjJ d'C • ( 1.19)

Ch i, § 4] FUNCflON SPACE 17

We will assume that all the functions which we will consider make the integrals converge, whatever the domain D is, even if it is infinite These

are called square integrable functions

Every function l/I(x) satisfyin, certain broad conditions about contiDuity inside a domain contained between the values x == 1t and x == -1£ can be

developed into a convergent trigonometric series, a series of fi,lntIamental

basis /unctioIU sin nx and cos nx (n = 0, 1 co) If :x is an angle, this

do-main is a circle with unit radius This is the wen-known theorem by Fo~~r

It will be convenient to use complex variables Then the basis functions are

the exponentials ei"~ ;III: exp (inx) where n takes on all inteFrs between

- 00 and + co The possibility of these power series developments is related

to two essential properties of the fundamental functions:

1 III the domain - n, + 1t the basis ftw.ctions multiplied by a convenient

"normalization factor", which happens to be equal to (11 J2i) are

orthogo-nal and normal or orthonormal which means that they satisfy the well·known

and easy-to-check equations

(tp" · £P,.) = - e-u'~ei~dx == 6ma ;

2" _

tp, = (21t)"~*ehU:; ((i", = (2n)-iet m.x

2.j These functions form in this domain a complete or closed system

We will give this proposition the following meaning: in the domain considered

i

we Can develop every function that interests us into a convergent series of basis functions t for instance continuous functions which have a sufficient

number of derivatives

We know that the Fourier development can be extended witho~t trouble

'1 Por reasons or rlsor, matbeDlatJciaD8 Jive in general a more precise de8nJtion or a complote,system: a system of fucdamental fimctions ,,(x) is complete it ODt-caD ftnd for each continuous fUnction .,(x) coefficients PI/ such that

Jim J f-,(x)-EJl.¥.(x>fadx 0, (1 21&)

I it means the absolute square So we deaJ with -an unlimited approximation In the mean

on the domain D or C(Jnv~'gence in tit, lMaR It is Dot necessary that ~)Plltp" is converscnt The ronnulas (1.22) and (1.23)' can result just as wen trorn- tbis detlnition as from our

restricted definition AD the calculations, including those in perturbation theory in

preferred to sacriftce pnerality ip order to simplify the ar~ents

18 VECTOR SPACES - UNITAR,\' GEOMETRY (eh 1, § 4

to an arbitrary number of variables We know also that the limits of the domain of validity of the trigonometric series can be modified by a change of variables We will return to this point in a moment

Spher:Dl fUDcUons, Hermite functions and Laguerre functions which one encounters in quantum theory, possess the same properties They form, each

in their domain (i.e -1, + 1 for the polynomials of Legendre, the surface of

a unit sphere for the spherical harmonics, - co, + CX) for those of Hermite and

There are many other sets of functions which possess these properties Therefore we will consider in a general wayan infinite sequence of functions

1/11 "'2 · • • of a certain number of variables and we will say that they tute a complete system of orthogonal and normal functions inside a certain

consti-domain D, if the following two conditions are fulfilled First we demand that

("'i · ;.) =:= f ",r"'kd't = ~lA:'

~

(1.20)

the integrals being extended over the domain D Second that all the continuous

functions which occur in the applications in physics can be expressed in this

domain by a series development of the form

(1.21)

k being an index which is allowed to talce all integer values between 0 and 00

or ~tween - 00 and + co (as in the case of a series of expon~ntials) If one cotnpares (1.21) with (1.1) and (1.20) with (1.8b) one sees that these equa-tioris can be expressed in the following geometrical language:

~ A complete system of orthogonal functions establishes a unitary system of coordinates which spans completely the function space The p" or the

Fourier coefficients are the components of the vector t/I in this system These

unitary axes establish, by their very definition, a denumerable set Therefore, they permit Feduelion of th~ properties oftbe functional space to those of a

space with an infinite denumerable number of dimensions

If we form the scalar product of "', and", and take (1.20) into account,

we obtain

(1.22)

a wen-known formula from Fourier which allows us to calculate the coeftl-

dents p, It expresses that p, is the ··orthogonal projection" of l/I on the;

Ch 1, § 4, 51 OPEllATORS 19

axis 1/1,- We obtain in the same way the fundamental formula of Parse val

(t/I • ;) = Lfpi"'!fJk,t/llr.d~ = L PiP, e (1.23)

If one can show that this relation is correct ror an arbitrary continuous function, one is sure that the set of 1/1, is complete The choice of orthogonal functions that span the function space depends essentially on the problem that is studied There is an infinite number of possible choices, whether the domain D is finite or infinite The different complete sets can be derive4 from each other by a change of coordinates or unitary transformation (compare

tp(x) == xt/l(x); in the same way the operator d/dx acting on ';(x) gives us

""(x) == (d/dx)t/I(x)

The operators that interest us in quantum mechanics are linear operators •

Any Pnear operator A satisfies the following three conditions:

1 If ex is a number: A(aJ/!) == rJAt/I

2 A(t/I+tp} == Ay,+Alp

3 Both the functions f/I and At/! are normalizable This implies that At/I

is defined in the same domain as '"

By definition, and by analogy with ordinary space, a mapping or tra,ns formatio-nof the function space on itself is a correspondence established among its vectors by a linear operation

Let us span the function space by a complete set of orthogonal axes

'" 1, "'2, · · Let t/I be one of its vectors with the components fJ l' /32, • • , then the development in a series of orthogonal basis functions can be

written as

(1.21)

The projection A makes this vector correspond to a new vector

20 VECTOR SPACES - UNITARY GEOMETRY [eb 1, § S

from which the components p;, in the same coordinate system can be

ex-pressed as a linear function (a series with constant coefficients) of the

coefficients /l" assuming of course that A was linear,

(1.24)

We write:

where At/I.J the function "" transformed by the operator A, can be developed itself in a serieI of fundamental functions '" 1, '" 2' • • • with Fourier coeffi-cients Ga:

The equations (1.25) are infinite in number when one deals with continuous

functions (k == 1,2 ••• ) They are important because they define the matrix (a'l) aa A which represents the linear operator A in the coordinate system

.p, The matrices which we have introduced this way to associate with every lin~ operator and which we will call representation matrices depend

essentially on the fundamental basis functions Their components can be obtalined, according to (1.25), by forming the scalar products

<l/I, · Al/I,) - f l/IiAl/Itd't = t f l/Itl/lJaJtd't = all:; (1.26)

(1.25) and (1.26) are constantly used in quantum theory

The equation (1.25) may still be considered from another point of view:

as a 'CUansformatioD of axes" (compare (1.4a) and (1.2») Particularly if we

span the function space by another complete set of orthogonal functions C{J,

we will have by developing each epic in a series of functions t/I,:

€p" == L , .;,u"' (1.27)

Hence if we introduce a new set of orthogonal functions the linear operators

will undergo a similarity transformation (see (1.7)) Since the transformation matrix U == ("It) conserves the orthogonality of the basis functions it is a

unitary matrix One can verify that this satisfies (1.9)

5.2 BILINEAR FORMS HERMITIAN OPERATORS

In n-dimensional space the scalar product (x -Ay) is a bilinear form of the components of the two vectors x and y (compare (1.10») We will establish the same product (", Alp) supposing that our functions are developed

according to (1.21):

Taking into account (1~25) we obtain

(1/1 • A(p) = f t/I* Aq>d-c = L pil/ljl/!,alk y"d-c == L Pi aUc"lh

(1.28)

which is an infinite bilinear form of Fourier coefficients of the components

of the two functions 1/1 and qJ in the basis system y, ,- Since one defines an

Hermitian operator in an n-dimensional space by the condition

(1/1 'A!f') = (Al/!' q» = L f l/!:atiP'iYIl/!,d-r: = LYka~M~ (1.29)

from which, according to (1.28)

(1.29a) thus, the corresponding matrix is Hermitian too

5 3 REDUCTION OF AN HERMITIAN OPERATOR TO ITS MAIN AXES

Let us reason again by analogy with the n-dimensional space: we want to

find a systenl of basis vectors '" 1 , "'2 to span the function space such that

eacl1 vector A ifr is "parallel" to t/I, or more precisely that the function Ay",

the transforrnation of 1/1, by the operator A, be equal to "', multiplied by a

constant:

(1.30) The functions found in this way are the eigen/unctio1t8 or eigenvectors of

the operator .t\; the numerical values eli its eigenvalues

22 VECTOR SPACES - UNITARY GEOMETRY {Ch~ 1, § S

The solution of this problem for a space with a finite number of dimensions

is relatively simplo ,at least in principle (§ 3) and can be done by purely

algebraic methods In function space, on the contrary, complications and considerable difficulties appear and rigorous discussion of the equation

(1.30)1 has not been completed by the mathematicians

The problem of finding (%, is algebraic, differential, or integral according to whether the operator A is algebraic, differential, or integral The Schr6dinger equation (2.15) is a particular case: The Hamilton operator to which it refers

is a differential operator Discussion of physical problems presented by

quantum mechanics has lead to a better understanding of equation (1.30) since the work of Hilbert

Depending on the nature of the operator A and on the size of the domain

D, which restricts the variables on which the function l/I depends, we are

dealing with two different cases

1 The easiest case arises when the square summable solutions of equation

(1.30) form a denumerable set, corresponding to a discontinuous and

denumerable set of eigenValues a;: ttl' el2, • There may correspond

several ei~nfunctions till to certain constants ex" but this multiplicity is

alwayS-supposed to be finite This is expressed by saying that the spectrum

of eigenvalues of the eigenfunctions is discontinuous or discrete

This case generalizes in a most direct way the results obtained in

n-dime~siona1 space It occurs most often if the domain D is finite (as e.g the case ~f free particles confined to a box), but this condition appears to be neitber necessary nor sufficient It happens for instanc:e in the quantum

the91"Y of a harmonic oscillator that the levels form a discontinuous spectrum

although the size of the domain is infinite"

It can be shown in the case of discrete eigenvalues that the eigenfunctions

.; , of the operator A form a complete set of orthogonal functions which span the function space completely2 In order to prove the orthogonality of

two eiaenfunctions t/I, and J/!" correspondina to two dUI'erent eigenvalues

of an Hermitian operator A, we have

Multiplying the first equation by f/lt and the second by JjI f and 8ubstractiDg

1 One ftDds a buic discussion in 1 v NaUMANN {193l]~

• See, for example, ltlLBD.T and CoURANT (1930] Chap Sand 6 or P lvi Mi.~ a.ad

H FuIlJlACH !19S3] p~ 727 and p~ ilSi>

Ch 1, § S1 OPERATORS 23 and using

we have

(ex, - rl,t)( '" i • VI).) == 0 j~e (t/I i • l/IlIJ == 0,

If several eigenfunctions, for example t/ll' 1/12' "'3, belong to the same

eigenvalue eX we have degeneracy: every linear combination of these three functions is again an eigenfunction beloIlging to the value et:, because the three equations

after multiplication with three arbitrary constants Pi and summation win give

written as a quadratic form in PI according to (1.28)~

(!/I ~ At/!) == L p'taikPk ~

l , l

But if the basis functions are eigenfunctions of the operator A, (1.25) can

be replaced by (1.30) and we obtain

Thus the quadratic form (t/I · At/!) is reduced to a "sum of squaresU

"

2 It happens very often and particularly if the domain D is infinite t}1~t

in addition to the discontinuous spectrum we have a continuous spectrum

of eigenvalues, and it may even happen that the discontinuous spectrum is

not thereat all The theory of atoms with a central force field furnishes an

example: the spectrum of energy levels is discontinuous up to a certain limit

(tlie ionization potential) and becomes continuous above this\> Since the

essential character of the orthogonal functions is that they fonD a

denumer-24 VECTOR SPACES - UNITARY GEOMETRY [eh 1, § 5

able set, obviously, one does not have the right to apply \v:ithout modification

the language of linear vector spaces1

• The development in Fourier series (1.21) is then replaced at least partially (compare (1 32) below) by integrals These integrals can be considered as the limits of series of orthogonal functions This permits us at least to a certain extent to give a sense to the expression of orthogonal functions in the

continuous case It will be sufficient to give two examples Let us consider first functions of a single variable x defined in an arbitrary finite or infinite

domain D and the operator "multiplication by x": (1.30) has now the form

xt/l(x) = exifJ(x),

an equation which has to hold everywhere in the domain D It is obviously

impossible to construct an analytical function which satisfies this condition, but we will imagine a function '" (J, which is everywhere zero except at the point x = eX and we would consider this as an eigenfunction of the preceding equation In order that this has a sense one has, moreover, to demand that the integral over the product of this function and an arbitrary function

f(x) is not zero This makes it necessary that the 0/« becomes infinite for

x = tX We will take this integral equal to f(a):

I f(x)I/I",(x)dx = /(a)

If we permit the existence of such functions we observe that the spectrum

of ~igenvalues and eigenfunctions of the operator x is continuous in the

domain D as the value of r:t, is arbitrary Our function ",«{x) coincides with the

Dirac delta function lJ(x - ex) It is possible to construct analytical expressions

that have this function as their limit 2 •

We have obviously

LI/I",(x)I/Iix)dX = 0,

1 The label i in (1.20) has to be replaced by a parameter E that varies continuously

f '1'; (x)'PI:(X)dT -+ f'P*(X, E)1p(x E')dT

and we will see below that the Kronecker Ott !nust be replaced by a new ki.nd of function

of the variable (E-E') ~

2 For explicit representations see HEtTLER, [1957], section 8

This function ,vas accepted "reluctantly by mathematicians because of its non-rigorous definition It ,vas known, ho\\:cver, that the results obtained via the a-function could alway£ be repeateu without its use Later a French mathematician, L Schwartz, incorpo- rated the underlying ideas into mathematics by defining a g~neraHzed notion of the idea of

a function (LIGHTHILL [1958])

eh t, § 5] OPERAT()R~ 25

SInce one ot trle unctIons oeCOnlCf:~ zero W tell tne Gtier vD.C' IS h erent rom

orthogonal and normal function~ depen.ding on a continuous: index Ct" l"'he

if now Xl = x ?, the hnornl" is i)(O}

The second example will be given \:vitli Fourjer integrahL It \·vill be useful

in the next chapter, 'Ve kn.ow that to ext.end the d01TIain uf validity of the

f'ourier series to an arbitrary interval tZ, + d (0 i.s rcal) it is 5ufficient to

ljJ defined in this domain, having the periodicity 2a and satisfying certain

~oc

tjJ(x) = (2a)-t L Cn exp (inn:.x:/at

(n JS an integer)~ The orthogonal basis functions are the exponentia.ls

(1 ' rv I",)" ""Q} exp ( ' )nnx/ a : ' )

i(d/<L",')s because we have

-i(d/dx) exp (innx/a) = (n1t/a) exp (innxia)~

2a The normalizing factor 1/ J2ii decreases wIlen the domain D extends

I

PvL1v == cff~,i n/a

\Ve obtain

26 VEcrOR SPACES - UNITARY GEOMETRY [eb 1, § 5

and if a -+ 00

t/I{x) == (2n)-+ f:: py exp (ivx)dv, p., == (2tt)-i-f:: ",(x) exp (-illx)dx

(1.32) These are the formulas of Fourier where the index v is a continuous vari· able The series development is replaced by an integral

l"he formulas similar to (1.23) and (1.31) are

and

as one can easily verify

The pree~ding argument is not a proof but the passing to the limit <Xl can easily be justified 1

• As a result we see that the eigenvalues y of the operator

-i(dldx) form a continuous spectrum in the interval - 00, + 00 The corresponding eigenfunctions:

Lim .J -d11/21t exp (ivx)

.4v O

~ve a normalizjng factor that goes to zero if the domain D goes to infinity i.e in\ the limit their "amplitude" is zero

tWe have seen in these two examples the diflculties that will occur if one

d.teDds the idea of eigenfunctions to the case of continuous spectra It is therefore better in physical problems to avoid if possible the continuous spectrum by a limitation of the domain D The theory of black body radiation

is a well-known example of this procedure: Conowina Rayleish and Jeans one supposes the radiation enclosed in a recta.nplar box with perfectly

~waUs The ampli~de .; of the waves is zero at the boundaries and therefore may be developed ill a Fourier series and it is not necessary to use

an integral expression

A similar procedure is used in solid state physics where the Born-von

~ periodic boundary conditions enable us to use a Fourier series also

lti the general case in which we rulve a juxtaposition of a continuous

spes-trum and a discontinuous spectrum of eigenfunctions the last, although

1 See the standard book~ on c~~d(:Ultu (e.g WElf,STER, [1955] p 153.)

Ch 1, § S1 OPERATORS 27

consisting of an infinite number of functions, does not form a complete set The combination of the two spectra, however, does form a complote system The development of an arbitrary function 1 consists then of a sum of a series corresponding to the discontinuous spectrum, and of an integral providing the continuous spectrum

Comparative table of the properties of function space and If-dimensional vector space

"-II Reduction to main axes I Ax, = C{.Xi

E~genvalues ! X' A}f ,= :E !XiX~ XI

(1.28) (!.ll) I (If' • Alp} '.~ (A1p' rp), (1.29)

CHAPTER 2

'rHE PRINCIPLES OF QUANTUM MECHANICS

1 Waves 1.1 CLASSICAL WAVES

In classical physics wave motion in a continuum is described by the

three-dimensional wave equation,

(2.1)

A possible elementary solution of this equation is given by

",(x, y, z, t) == A exp {-2niV (t- IXX+P;+'VZ)} (2.2)

where A is the amplitude, 11 is the frequency, c the velocity of propagation, and CI., P and y the direction cosines of the normal to the plane wavefront The wavelength A = c/v which we prefer to replace by the wave vector k,

which has the components

k" = 21tfXV/C; k, == 2ttpv/c; k z = 2TCYVjC (2.3) The magnitude k is 21t times the number of waves per em; the direction of Ie

is the direction of the wavefront

Each complete solution of the wave equation (in a box or in infinite ~e) can be obtained by superposition of elementary solutions The function that satisfies (2.1),

29

30 THE PRINCIPLES OF QUANTUM MECHANICS [eh 2, t 1

is substantially different from zero over a localized region the function is

usually called a wave packet

If we take together all waves of the same frequency, that is of the same

absolute k-value, but of different directions (2.2a) becomes

GO

Y, = L: a.t(x, y, z) exp {-21riVlt} (2.4)

t=O

The coefficients of (2.2a) and (2.4) are generally complex because the

different spectral components usually have different phases The relative phases will determine whether we will have constructive or destructive inter-ference, one of the most striking properties of waves

The product a: Qt(x, y, z) is the intensity of a particular spectral

compo-nent Vic of the wave at a certain point x, y, z On the other hand, if we look for

the average value < "' '" > of l/I* '" over a time T long compared to the periods 1/'1k, we have:

<1/1*1/1> = lim ! L: fTa:ase2'rl (Vr-v.)t dt L: a; a,(x y, z) (2.5)

T-+~ T ",Jo ,

or,the average intensity <"' "'> at a point (x, y, z) is the sum over all the

intensities of the different spectral components

The expression (2.5) is obtained by integrating separately the terms in

which ", ~ v" and those in which 1', = V •• The first set of integrals are

oscillat-ingffunctions of T with a constant amplitude After multiplicatibn with

T- i the product goes to zero for T -+ 00 The second integral is proportional

toT hence it is unnecessary to take the limit of the product

1.2 QUANTUM MECHANICAL WAVES

Early in the development of quantum mechanics a postulate was

intro-duced that the energy was proportional to a frequency

It was successively rea1ized that the nature of the wave was not always

electromagnetic and an additional postulate

(2.8)