linus pauling introduction to quantum mechanics MCGraw hill

After introductory chapters on classical mechanics and the equation and its physical interpretation on a postulatory basis,and have then given in great detail the solution of the wave eq

00

Osmania Universitq

This book should bereturned on or before theBarkedbelow

Associate Professor of Chemistry, Harvard University

INTERNATIONAL STUDENTEDITION

'

NEWYORKAND LONDON

KOGAKUSHA COMPANY, LTD

QUANTUM MECHANICS

INTERNATIONALSTUDENT EDITION

Exclusive rights by Kogakusha Co., Ltd for manufacture and export

from Japan. This book cannot be re-exported from the country to

COPYRIGHT, 1935, BY THEMCGRAW-HILL BOOK COMPANY, INC.

fill rights reserved This book, or

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TOSHO INSATSU PRINTING CO , LTD TOKYO, JAPAN

In writing thisbook wehave attemptedto produceatextbook

of practical quantum mechanics for the chemist, the mental physicist, and the beginning student of theoreticalphysics The book is not intended to provide a critical discus-sion of quantum mechanics, nor even to presen^ia thoroughsurvey of the subject We hope that it does gb^ela, lucid and

experi-easily understandable introduction to a limjAdVpori*<m >

quantum-mechanical theorv; n*w*iv. *hat i^WJ< - - Via 1

to provide for the reader a means of equipping himself with a

practical grasp of this subject, so that he can apply quantum

mechanics to most of the chemical and physical problems which

may confronthim

The book is particularly designed for study by men withoutextensive previous experience with advanced mathematics, such

as chemists interested in the subject because of its chemical

applications We have assumed on the part of the reader, in

addition to elementary mathematics through the calculus, only

some knowledge of complex quantities, ordinary differential

equations, and the technique of partial differentiation It

may be desirable that a book written for the reader not adept

at mathematics be richer in equations than one intended for

the mathematician; for the mathematician can follow a sketchyderivation with ease, whereas if the less adept reader is to be

led safely through the usually straightforward but sometimesrather complicated derivations of quantum mechanics a firm

guiding hand must be kept on him Quantum mechanics is

essentially mathematical in character, and an understanding

ofthe subjectwithoutathoroughknowledgeofthemathematical

obtained The student not thoroughly trained in the theory

of partial differential equations and orthogonal functions must

learnsomethingofthese subjects ashestudiesquantum

mechan-ics. In order that he may do so, and that he may follow the

discussions given without danger of being deflected from thecourseof theargument byinability tocarry throughsomeminor

step, we have avoided the temptation to condense the various

discussions into shorter andperhaps more elegant forms.

After introductory chapters on classical mechanics and the

equation and its physical interpretation on a postulatory basis,and have then given in great detail the solution of the wave

equation for important systems (harmonic oscillator, hydrogenatom) and thediscussion ofthe wavefunctions andtheirproper-

ties, omitting none of the mathematical steps except the most

l^JT!^?577 A, similarly detailed treatment has been given

in the discussion di pertin1optionShruor^,the variation method,the structure of simple molecules, and, in general, 'iu ,

important section of the book

In order to limit the size of the book, we have omitted from

discussion such advanced topics as transformation theory and

general quantum mechanics (aside from brief mention in the

last chapter), the Dirac theory of the electron, quantization

of the electromagnetic field, etc. We have also omitted severalsubjects which are ordinarily considered as part of elementary

quantum mechanics, but which are of minor importance to thechemist, such as the Zeeman effect and magnetic interactions ingeneral, the dispersion of light and allied phenomena, and

most of the theoryof aperiodic processes

Theauthors are severallyindebtedtoProfessorA Sommerfeldand Professors E U Condon and H P Robertson for their

own introduction to quantum mechanics The constant advice

of Professor R C Tolman is gratefully acknowledged, as well

as the aid of Professor P M. Morse, Dr L E Sutton, Dr

G W. Wheland, Dr L 0 Brockway, Dr J. Sherman, Dr S.

PREFACE

CHAPTER I

SURVEY OF CLASSICAL MECHANICS

SECTION

1. Newton's Equationsof Motionin theLagntngianForm 2

la. TheThree-dimensional Isotropic HarmonicOscillator . 4

Ib Generalized Coordinates \. A . 6

Ic. TheInvariance of the Equations of Motionin the Lagraib-gianForm 7

Id. An Example: The Isotropic Harmonic Oscillator in Polar Coordinates 9

le. TheConservationof Angular Momentum 11

2. TheEquationsof Motionin the HamiltonianForm 14

2a Generalized Momenta 14

26. The Hamiltonian Functionand Equations 16

2c. TheHamiltonian Function and theEnergy 16

2d. A General Example 17

3. TheEmissionand Absorptionof Radiation . 21

4. Summary ofChapter I 23

CHAPTER II THE OLD QUANTUM THEORY 5. TheOriginof the Old Quantum Theory 25

5a. ThePostulates ofBohr 26

56. The Wilson-Sommerfeld Rulesof Quantization 28

5c Selection Rules. The CorrespondencePrinciple 29

6. The Quantizationof SimpleSystems 30

6a. The Harmonic Oscillator. Degenerate States 30

66. TheRigidRotator 31

6c. TheOscillatingand Rotating Diatomic Molecule 32

6d. TheParticle in aBox 33

6e Diffractionbya Crystal Lattice 34

7. TheHydrogenAtom 36

7a Solution of theEquationsofMotion 36

76 Application of theQuantumRules. TheEnergyLevels 39

7c Description of the Orbits 43

7d. Spatial Quantization 45

8. TheDecline of theOld QuantumTheory 47

vi CONTENTS

CHAPTER IIITHE SCHRODINGER WAVE EQUATION WITH THE

Charac-teristicEnergyValues 58

9d. The Complex ConjugateWaveFunction ty*(x,t) 63

lOc. Further Physical Interpretation. Average Values of

11. The Harmonic Oscillator in WaveMechanics . 67

116. TheWaveFunctionsfor the Harmonic Oscillatorandtheir

12. TheWaveEquationfor aSystem of Point Particles 84

126. TheAmplitudeEquation 86

12c. TheComplex ConjugateWave Functionty*(xi ZAT, t) 88

14. TheParticle in a Box 95

15. The Three-dimensional Harmonic Oscillator in Cartesian

18a. The Separation of the Wave Equation. The

19a. TheLegendre FunctionsorLegendre Polynomials 126

20. TheLaguerre Polynomialsand AssociatedLaguerre Functions . 129

216. TheNormalState of the HydrogenAtom 139

CHAPTER VI

PERTURBATION THEORY

23a. ASimple Example: T>/> P^^tujj^ejjJH^mjinnin QsillRf.nr 160

24a. AnExample: Applicationofa Perturbationto aHydrogen

25a. An Example:TheStark Effect of thePlane Rotator . 177

CHAPTER VII

THE VARIATION METHOD AND OTHER APPROXIMATE

METHODS

266. AnExample fThe NormalState of

the_Helium Atom . 184

27d. ApproximationbytheUseof DifferenceEquations 202

via CONTENTS

CHAPTER VIII

THE SPINNING ELECTRON AND THE PAULI EXCLUSION

29. The Helium Atom The PauliExclusionPrinciple 210

29c. TheAccurateTreatmentof theNormal Helium Atom 221

29e. The Polarizability oftheNormal HeliumAtom 226

CHAPTER IX

MANY-ELECTRON ATOMS

30 Slater's Treatmentof Complex Atoms 230

30a. Exchange Degeneracy 230

30d. Evaluationof Integrals *

239

30e. Empirical Evaluationof Integrals Applications . 244

31 a. The Lithium Atom and Three-electron Ions 247

32. The Methodof the Self-consistent Field 250

336. The Thomas-Fermi StatisticalAtom 257

CHAPTER X

THE ROTATION AND VIBRATION OF MOLECULES

35. The RotationandVibration ofDiatomic Molecules 263

356. TheNatureof the ElectronicEnergyFunction 266

35 ASimple PotentialFunctionforDiatomic Molecules . 267

35d. A MoreAccurate Treatment The MorseFunction 271

36a. TheRotationof Symmetrical-top Molecules 275

36b. TheRotationofUnsymmetrical-top Molecules 280

CONTENTS ix

37a. NormalCoordinatesin Classical Mechanics 282

376. NormalCoordinates inQuantum Mechanics 288

CHAPTER XI

PERTURBATION THEORY INVOLVING THE TIME, THE

RESONANCE PHENOMENON

39. TheTreatmentof aTime-dependent Perturbation bythe Methoa

40. TheEmission andAbsorptionof Radiation.V 299

Moleculev-The Franck-Condon Principle 309

CHAPTER XII

THE STRUCTURE OF SIMPLE MOLECULES

42. TheHydrogenMolecule-ion 327

42d. ExcitedStates of theHydrogen Molecule-ion 340

43c. TheTreatmentofJames andCoolidge 349

43d. ComparisonwithExperiment 351

43/ Oscillation and Rotation of the Molecule. Ortho and

INTRODUCTION TO QUANTUM

MECHANICS

CHAPTER I

Thesubject of quantummechanics constitutesthemost recentstep in the very old search for the general laws_goyrning themotion of matter For a longtime investigators confined their

stage it was properly considered a branch of physics. Sincethe development of atomic theory there has been a change of

emphasis Itwas recognized that theolderlaws are not correct

when applied to atoms and electrons, without considerable

modification Moreover, the success which has been obtained

hadtheresult ofdepriving physicsof sole claimuponthem, since

it is now realized that the combining power of atoms and, in

explicable in terms of the laws governing the motions of the

electrons andnuclei composing them

Although it is the modern theory of quantum mechanics in

which we are primarily interested because of its applicatiqns_to

chemical jjroblems, it is desirable for us first to discuss briefly

the background of classical mechanics from which it was

devel-oped By so doing we not only follow to a certain extent the

historical development, but we also introducein a more familiar

formmany conceptswhichareretainedinthelater theory We

student is advised to consider the exercises of the first fewchapters carefully and to retain for later reference the results

which aresecured

2 SURVEY OF CLASSICAL MECHANICS [1-1

In the first chapterno attempt willbe made togive anyparts

of atomic and molecular problems With this restriction, we

bodies, non-conservative systems, non-holonomic systems,

sys-tems involving impact, etc. Moreover, no use is made of

Hamilton'sprinciple or oftheHamilton-Jacobipartialdifferential

equation By thus limiting the subjects to be discussed, it is

possible to give in a short chapter a thorough treatment of

1. NEWTON'S EQUATIONS OF MOTION IN THE LAGRANGIAN

FORM

for the three Cartesian coordinates of the iih particle withmass Wi, Newton's equations for npoint particles are

mtx =

i =

1, 2, - -

whereX,-, F;, Ztare the three components oftheforce acting on

that

Byintroducingcertain familiar definitionswechange Equation

1-1 into a form which will be more useful later. We define as

T =

Ifwelimit ourselves toacertainclassofsystems,called

conserva-tivesystems, it ispossible to defineanotherquantity, the potential

energy V, which is a function of the coordinates x\y\z\

of all the suchthat the force

1-1] NEWTON'S EQUATIONS OF MOTION 3

on each particle are equal to partial derivatives of the potentialenergy with respect to the coordinates of the particle (withnegative sign); that is,

It ispossible to finda functionVwhichwillexpressinthismanner

forces ofthe types usually designatedasmechanical,electrostatic,

and gravitational. Since other types of forces (such as

electro-magnetic) for which such a potential-energy function cannot

be set up are not important in chemical applications, we shall

not considerthem in detail.

With these definitions, Newton's equationsbecome

There are three such equations for every particle, as before.

These results are definitely restricted to Cartesian coordinates;but by introducing a new function, the Lagrangianfunction L,

defined for Newtonian systems as the difference of the kinetic

and potential energy,

L =L(XI, i/i, 21,

, xn, yn , z n , Xif

, z n ) =

-T

-F, (1-6)

wecanthrowthe equationsofmotionintoa form whichweshalllater prove to be valid in any system of coordinates (Sec. Ic).

In Cartesian coordinates T7 is a function of the velocities

xi, , z n only, and for the systems to which our treatment

is restricted F is a function of the coordinates only; hence theequations of motion given in Equation 1-5 on introduction of

SURVEY OF CLASSICAL MECHANICS P-la

In the following paragraphs a simple dynamical system is

discussedbythe useof these equations

la. The Three-dimensional Isotropic Harmonic Oscillator

form, we choose a system which has played a very importantpart in the development of quantum theory This is thelwrn,wiwscj$M&r, aparticleboundto anequilibriumpositionby

a force which increases in magnitude linearly with its distance

r from the point. In the three-dimensional isotropic harmonic

,

represent-ing a force of magnitude kr acting in a negative direction; i.e.,

from the position of the particle to the origin, k is called the

force constant or Hookas-law constant Using Cartesian

I-la] NEWTON'S EQUATIONS OF MOTION

given case. The quantity VQ is related to the constant of the

restoring forceby the equation

the motion It is seen that the particle may be described as

carrying out independent harmonic oscillations along the x, y,

and zaxes, with different amplitudesXQ, 2/o, andZQ anddifferent

phase angles 6 X , dy , and d z respectively

The energyofthe system isthe sum ofthe kinetic energyand

the potential energy, and isthus equal to

Onevaluation,it isfoundtobe independent ofthetime,with thevalue 27r2mj/2)(:r2) + 2/o + zl) determined by the amplitudes of

6 SURVEY OF CLASSICAL MECHANICS [I-lb

along this axis as describedby Equation 1-14 Itstotalenergy

is givenbythe expression 2T2

mj>gzJ.

Ib Generalized Coordinates Instead of Cartesian

coordi-nates xij 2/1,2i, ,xn, 7/n, n, it is frequentlymore convenient

touse some other-set of coordinatesto specify the configuration

of the system. For example, the isotropic spatial harmonic

polarcoordinates; again, thetreatment of asystem composedof

two attracting particles in space, which will be considered

rectangularcoordinates.

Ifwe choose, anyset of 3n coordinates, which we shall alwaysassume to be independent and at the same time sufficient in

the system, then there will in general exist 3n equations, called

the equations of transformation, relating the new coordinates

(1-18)

There is such a set of three equations for each particle i. The

functions/t , g iy hi may be functions of any or all of the3n new

coordinates q^ so that these new variables do not necessarily

in the case of two particlesthe six new coordinates may be the

three Cartesian coordinates of the center of mass together

with the polar coordinates of one particle referred to the other

particle as origin.

As is known from the theory of partial differentiation, it is

possible to transform derivatives from one set of independent

variables to another, an example of this process being

dq2 dt "*" +

dq*n ~3T' IThis sameequation can be put in the much more compact form

3n

I-lc] NEWTON'S EQUATIONS OF MOTION 7This gives the relation between any Cartesian component ofvelocityandthetimederivatives ofthenewcoordinates Similarrelations, of course, hold for t/ and 2< for any particle. The

quantities g/, byanalogy withxif are called generalizedvelocities,

even though they do not necessarily have the dimensions of

length dividedby time (forexample, <?/maybe an angle).Since partial derivatives transform in just the same manner,

Since Q, isgiven by an expression in terms of V and <//which is

analogoustothatfortheforceXiintermsof V and#,,it iscalled

proof we shall apply a transformation of coordinates to

Equa-tions 1-5,using themethodsofthe previous section

SURVEY OF CLASSICAL MECHANICS [I-lc

From Equation l-19b we obtain directly

Furthermore, becausethe order of differentiation is immaterial,

d/dXi\ _ i d /dXi\

Byintroducing Equations 1-26 and 1-25 in 1-24 and using the

result in Equation 1-23, we get

Finally,the introductionoftheLagrangianfunctionL = T Vy

withVa functionofthe coordinatesonly,givesthemore compact

form

I-ld] NEWTON'S EQUATIONS OF MOTION 9

(It is important to note that L must be expressed as a function

ofthe coordinates and theirfirsttime-derivatives.)

Since the above derivation could be carried out for any value

of ,/, there are 3n such equations, one for each coordinate g/.

They are called the equations of motion in the Lagrangianform and are of great importance. The method by which they were

derived shows that they are independent of the coordinatesystem

Wehaveso far ratherlimitedthe types ofsystems considered,but Lagrange's equations are much more general than we have

indicatedandby a properchoice ofthefunctionLnearlyall

dynami-cal problems can be treated with their use. These equations aretherefore frequently chosen as the fundamental postulates of

Coordinates Theexample which wehave treatedin Section lacan equally well be solved by the use of polar coordinates r,

correspond-ing to Equation 1-18 are

Withthe useofthesewefind forthekineticandpotential energies

ofthe isotropicharmonic oscillatorthe followingexpressions:

10 SURVEY OF CLASSICAL MECHANICS I-td

plane containing the origin. This conclusion enables us tosimplify the problem by making a change of variables Let usintroduce new polar coordinates r, #', x such that at the time

t = the plane determined by the vectorsrand v,theposition

andvelocityvectorsoftheparticle att = 0,isnormaltothenew

coordinatesiftwoparameters # andv? , determining theposition

oftheaxisz'intermsof theold coordinates, aregiven (Fig. 1-2)

FIG 1-1. The relation of polar

coor-dinates r, t? ,and <p to Cartesian axes.

Fio 1-2. Therotation of axes.

In terms of the new coordinates, the Lagrangian function L

and the equations of motion have the same formas previously,

because the first choice of axis direction was quite arbitrary.

However, since the coordinates have been chosen so that theplaneofthe motionisthex'y'plane,the angle#'is alwaysequal

to a constant, v/2 Inserting this valueof & in Equation 1-33and writingit interms of x instead of <?, we obtain

Me] NEWTON'S EQUATIONS OF MOTION 11

or, usingEquation 1-37,

Multiplication by r and integration with respect to the time

We havethus obtained the dependence of r on the time, and

by integratingEquation 1-37 we could obtain xas a functionof

the time, completing the solution Elimination of the timebetween these two results would give the equation of the orbit,which is an ellipsewith center atthe origin. It isseen that theconstant v again occurs asthefrequency ofthe motion

le The Conservation of Angular Momentum The example

principle of wide applicability, the principle of the conservation

momentum.

12 SURVEY OF CLASSICAL MECHANICS [I-le

Equation1-37showsthatwhenxisthe angular velocity oftheparticle about afixed axis z'and r isthe distanceof the particle

xisa constantofthe motion.1

This quantity is called the angular momentum of the particleabout theaxisz'.

It is not necessaryto choosean axisnormaltothe planeofthemotion, as z' in this example, in order to apply the theorem

ThusEquation 1-33, written forarbitrary directionz, is at once

integrable to

theleft side of thisequationisthe angularmomentumabout the

axisz.2 It isseen to beequalto a constant, p^.

In orderto apply the principle, it is essential that the axis ofreference bea fixedaxis. Thusthe angle dof polar coordinateshas associated with it an angular momentum p# = mr2

$ about

an axis in the xy plane, but the principle of conservation of

angular momentum cannot be applied directly to this quantitybecause the axis is not, in general, fixed but varies with <?. Asimple relation involving p* connects the angular momenta

1Thephrase aconstant ofthe motion is oftenusedin referring to a constant

2Thisis sometimesreferred to as thecomponent of angularmomentum

I-le] NEWTON'S EQUATIONS OF MOTION 13

px and p<p about different fixed axes, one of which, px , relates

to the axisnormalto the plane ofthe motion This is

Pxdx = Pd$ + Pvdv, (1-41)

an equation easily derived by considering Figure 1-3 The

sides of the small triangle have the lengths rsinM>, rdx, and

rd& Since they form a right triangle, these distances areconnected bythe relation

= rL

sm*

which gives, onintroductionof the angularvelocities x,<f>, and #

andmultiplication by m/dt>

Equation 1-41 follows from this and the definitions of px , p*,

Conservation of angular momentum may be applied to more

general systems than the one described here It is at onceevident thatwe have not used the specialformof the potential-energy expression except for the fact that it is independent ofdirection, since this function enters into the r equation only

Therefore the above results are true for a particle moving in

anyspherically symmetricpotential field.

Furthermore, we can extend the theorem to a collection of

point particles interacting with each other in any desired way

but influenced by external forces only through a sphericallysymmetric potential function If we describe such a system by

using the polar coordinates of each particle, the Lagrangianfunctionis

n

5/Wi(^? + ?*&* + r2sin2

Instead of <pi, ^2 , , <p n , we now introduce new angular

coordinates a, 0,

, * given bythelinearequations

<Pi = oi + 6i0 + + kiK,

14 SURVEY OF CLASSICAL MECHANICS [I-2a

ent a is an angle about the axis z such that if a is increased

byAa, holding /?, , K constant, the effect isto increaseeach

<pibyAa,or,inother words,torotatethewhole systemof particles

aboutzwithoutchangingtheirmutualpositions. Byhypothesisthe value of V is not changed by such a rotation, so that V is

independentof a. Wethereforeobtain the equation

ddL^dL^ddT^

Moreover, from Equation 1-42 we derivethe relation

t-1 1=1

Hence, calling the distance rlsin t>t of the ith particlefrom the

zaxis pi, we obtainthe equation

This is the more general expression of the principle of the servation of angular momentum which we were seeking. Insuch a system of many particles with mutual interactions, as,

con-for example, an atom consisting of a number of electronsand a

nucleus, the individual particles do not in general conserveangularmomentum but the aggregate does

The potential-energy function V need be only cylindrically

symmetric about the axis z for the above proof to apply,

sincetheessentialfeaturewastheindependenceof Vonthe angle

a about z. However, in that case z is restricted to a particular

direction in space, whereas if V is spherically symmetric thetheorem holdsforany choiceof axis.

Angular momenta transform likevectors, the directions of the

vectorsbeing the directions of the axes about whichthe angular

momenta are determined It is customary to take the sense

of the vectorssuch as to correspondtothe right-hand screwrule.

3. THE EQUATIONS OF MOTION IN THE HAMILTONIAN FORM

2a Generalized Momenta. In Cartesian coordinates the

momentum related to the directionx mx since V

I-2a] EQUATIONS OF MOTION IN HAMILTONIAN FORM 15

restricted tobea functionofthe coordinatesonly, can bewritten

Reference to Equation 1-31, which gives the expression for the

kinetic energyin polar coordinates, shows that

Likewise, in the case of a number of particles, the angular

momentum conjugate to the coordinate a is

dT dL

this to other coordinate systems, the generalized momentum pk

conjugate tothe coordinate qk isdefined as

so that Equations 2-5 and 2-6 form a set of 6n first-order

dif-ferential equations equivalent to the 3n second-order equations

ofEquation 1-29

being in general a function of both the qy

s and g's, the

u<lk

definition of pk given by Equation 2-5 provides 3n relations

between the variables #*, (fc, and p*, permitting the elimination

ofthe 3n velocities q^ so that the system can nowbe described

interms the3ncoordinates andthe3n momenta

16 SURVEY OF CLASSICAL MECHANICS (I-2c

in this way bethrown into an especially simple form, involving

a functionHofthe pk's andqk's called theHamiltonianfunction.

2b The Hamiltonian Function and Equations. For servativesystems.1

con-we shall showthat the functionHis thetotal

energy (kineticplus potential) ofthe system, expressed interms

of the pk's and g^'s. In order to have a definition which holds

for more general systems, we introduce H bythe relation

3n

Althoughthis definitioninvolvesthevelocities</*,H maybemade

afunction ofthe coordinates and momentaonly, byeliminating

the velocities through the use of Equation 2-5 From the

definitionwe obtain for thetotal differential of H ths equation

whence, if // is regarded as a function of the qk's and p/t's, we

obtain the equations

2c. The Hamiltonian Function and the 'Energy Let us

con-sider the time dependence of H for a conservative system Wehave

onthetimet. Wehaverestrictedourdiscussion to conservativesystemsby

V

1-81 THE EMISSION AND ABSORPTION OF RADIATION 21

coordinate.

3. THE EMISSION AND ABSORPTION OF RADIATION

permitthecompletediscussion ofthe emissionand absorptionof

electromagnetic radiation by a system of electrically charged

results of this discussion It is found that these results are not

in agreement with experiments involving atoms and molecules;

it was, indeed, just this disagreement which was the principal

factor in leading to the development of the Bohr theory of the

time, when an apparently satisfactory theoretical treatment of

dynamical systems composedof electrons and nucleiisprovided

by the quantum mechanics, the problem of the.emission and

absorptionof radiation stilllacks asatisfactory solution, despitethe concentration of attention onitbythemostable theoreticalphysicists It will be shownin a subsequent-chapter 'however,

that, despite ourlack of a satisfactory conceptionof thenature

of electromagnetic radiation, equations similar to the classical

equations of this section can be formulated which represent

correctly the emission and absorption of radiation by'atomicsystemstowithin the limitsof error of experiment

Accordingtotheclassicaltheory therate ofemissionofradiantenergyby anaccelerated particle of electric charge e is

is converted into radiant energy, v is the acceleration of the

Let us first consider a system of a special type, in which a

particle of charge e carries out simple harmonic oscillation

with frequency valong the x axis, according tothe equation

x = x cosZirvt. (3-2)Differentiating this expression, assuming that x* is independent

ofthe time,we obtainfor theaccelerationthe value

* -4irVz

22 SURVEY OF CLASSICAL MECHANICS The averagerate of emission ofradiantenergy bysucha system

[I-is consequently

dE

inasmuch as the average valuecos2

2irvtover a cycle is "one-half.

As a result of the emission of energy, the amplitude z of themotion will decrease with time; if the fractional change in

energyduring acycle of themotion issmall, however, this

equa-tion retainsits validity

Theradiationemittedbysucha system hasthefrequencyvoftheemitting system It is plane-polarized, the plane of theelectric

vector being the planewhichincludesthexaxisandthedirection

ofpropagation ofthelight.

In case that the particle carriesout harmonicoscillationsalong

all three axes x, y, and 2, with'frequencies vxy vv , and v z and

amplitudes (at a given time) x, 2/0, andZQ,respectively,thetotalrate of emission of radiant energy will be given as the sum of

three termssimilarto the Tightside ofEquation 3-4, one givingthe rate of emission of energy as light of frequency v x , one of

vy , andone of v

If themotionof theparticle isnot simple harmonic, it can berepresented by a Fourier series or Fourier integral as a sum or

integral of harmonic terms similar to that of Equation 3-2;

be emittedatarate givenbyEquation3-4, thecoefficient ofthe

Fourierterm being introducedin place of x .

The emission of light by a system composed of several

inter-acting electrically charged particles is conveniently discussed in

the following way A Fourier analysis is first made of themotionof thesystemina givenstateto resolve itintoharmonicterms For a given term, corresponding to a given frequency

ofmotion v,thecoefficientresultingfromtheanalysis (whichisafunction of the coordinates of the particles) is expanded as a

, z n are the coordinates of the particles relative to some

origin(suchasthe centerofmass)andX = c/visthewave length

of the radiation with frequency v. The term of zero degree inthis expansion is zero, inasmuch as the electric charge of thesystem does not change with time The term of first degree

addition to the harmonic function ofthe time, only

1-4] SUMMARY OF CHAPTER I 23

a function of the coordinates The aggregate of these

first-degreetermsinthe coordinateswiththeirassociatedtimefactors,

summed over all frequency values occurring in the originalFourier analysis, represents a dynamical quantity ktiown as the

(3-5)

in which r denotes the vector fromtheorigin tothe position of

approximation the radiation emitted by a system of severalparticles can be discussed by making a Fourier analysis of the

in this representation of P, there will be emitted radiation of

frequency v at a rate given by an equation similar to Equation

3-4, with exo replaced by the Fourier coefficient inthe

electric-momentexpansion. Theemissionofradiationbythismechanism

is usually called dipole emission, the radiation itself sometimesbeing described as dipole radiation

The quadratic terms in the expansions in powers of x\/\

, Zn/X form a quantity Q called the quadrupole moment

ofthe system,andhigherpowersformhighermoments Therate

ofemissionofradiantenergyasaresultof thechangeof

quadru-pole and higher moments of an atom or molecule is usually

negligibly small in comparison with the rate of dipole emission,

andin consequencedipoleradiation aloneisordinarily discussed

dipole radiation is zero and the presence of veryweak radiation

can bedetected,the processof quadrupoleemissionis important

The purpose of this survey of classical mechanics is twofold:

first,toindicatethepathwherebythemoregeneral formulations

andofHamilton, have been developed fromtheoriginalequations

quantum-mechanical methods

In carrying out thefirstpurpose, we havediscussed Newton's

inCartesian coordinatesandthenaltered theirformby

24 SURVEY OF CLASSICAL MECHANICS [1-4

the introduction of the kinetic and potential energies. By

definingtheLagrangianfunctionforthespecial case ofNewtonian

systemsandintroducingitintothe equationsofmotion,Newton'sequations were then thrown intotheLagrangian form Follow-ing an introductory discussion of generalized coordinates, the

proof of the general validity of the equations of motion in theLagrangian form for any system of coordinates has been given;

and it has also been pointed out that the Lagrangian form

ofthe equationsofmotion, althoughwe havederiveditfromtheequations of Newton, is really more widely applicable thanNewton's postulates, because by makingasuitable choiceoftheLagrangian function a very wide range of problems can betreatedinthis way

In the second section there has been derived a thirdform for

the equations of motion, the Hamiltonian form, following theintroductionofthe conceptofgeneralizedmomenta, andtherela-

tion betweenthe Hamiltonian function andthe energyhasbeen

discussed

In Section 3 a very brief discussion of the classical theory of

the radiation of energy from accelerated charged particles hasbeen given, in order to have a foundation for later discussions

radiation

Finally, severalexamples (whichare latersolvedbythe useof

treated by themethods discussed in this chapter

General References onClassical Mechanics

W.D MACMILLAN:"TheoreticalMechanics StaticsandtheDynamics

of aParticle," McGraw-Hill Book Company, Inc., NewYork, 1932.

S L.LONEY: "Dynamicsof a Particleand of RigidBodies," Cambridge

J.H JEANS:"TheoreticalMechanics,"Ginn and Company,Boston, 1907.

Cambridge, 1928.

Chemistry," Chemical Catalog Company, Inc., New York, 1927, Chap.II,

TheElements of Classical Mechanics

W.E. BYEBLY:"GeneralizedCoordinates," Ginnand Company, Boston,

CHAPTER II

THE OLD QUANTUM THEORY

Theold quantum theorywas bornin 1900, when MaxPlanck1

announced his theoretical derivation of th^jdistrihutionjaw for,

black-body radiation which he had previously formulated fromempirical considerations He showed that the results of experi-

ment on the distribution of energy with frequency of radiation

in equilibrium with matter at a given temperature can beaccounted for by postulating that the vibrating particles of

matter (considered to act as harmonic oscillators) do not emit

or absorb lightcontinuously but instead onlyin discrete

quanti-ties ofmagnitudehv proportional to the frequency vofthelight.

The constant of proportionality, h> is a new constant of nature;

it iscalledPlanck'sconstantandhas themagnitude6.547 X 10~27

erg sec Its dimensions (energy X time) are those of the old

dynamical quantity called action;they aresuch that the product

ofh andfrequency v (with dimensionssec"1

) has the dimensions

of energy Thedimensionsofhare alsothoseofangular

momen-tum,and we shallsee laterthatjust ashvisaquantumofradiantenergy of frequency v, so is h/2w a natural unit or quantum of

angularmomentum.

The developmentofthequantumtheorywasatfirstslow Itwas not until 1905 that Einstein2

suggested that the quantity

of radiant energy hv was sent out in the process of emission of

portion of radiant energy Einstein also discussed the

and the heat capacities of solid bodies in terms of"the quantum

theory When light fallsona metalplate, electrons areemittedfromit. The maximum speedof thesephotoelectrons, however,

1M PLANCK, Ann.d Phys (4) 4, 553 (1901).

2 A.EINSTEIN, Ann.d Phys (4) 17, 132 (1905).

26 THE OLD QUANTUM THEORY n-6a

is not dependent on the intensity of the light, as would beexpected from classical electromagnetic theory, but only on its

frequency; Einstein pointed out thatthis isto be expectedfromthequantumtheory, theprocessof photoelectricemission involv-ing the conversionoftheenergyhvofone photonintothekinetic

energy of a photoelectron (plus the energy required to remove

the electron from the metal). Similarly, Einstein's law of

photochemical equivalence states that one molecule may beactivated to chemical reactionbythe absorptionof one photon.

The third application, to the heat capacities of solid bodies,

Planck's postulate regarding the emission and absorption ofradiation in quanta hv suggested that a dynamical system such

as an atom oscillating about an equilibrium position with

fre-quency VQ might not be able to oscillate with arbitrary energy,but only with energy values which differ from one another by

integral multiples of hvQ From this assumption and a simpleextension of the principles of statistical mechanics it can be

should not remain constant with decreasing temperature, butshould at some low temperature fall off rapidly toward zero.

This prediction of Einstein, supported by the earlier

experi-mental work of Dewar on diamond, was immediately verified

bytheexperimentsofNernst and Euckenonvarious substances;

and quantitative agreement between theory and experiment for

simplecrystalswasachievedthrough Debye'sbrilliantrefinement

ofthe theory.1

6a. The Postulates of Bohr The quantum theory had

developed to this stage before it became possible to apply it

to the hydrogen atom; for it was not until 1911 that thereoccurred the discovery by Rutherford of the nuclear constitu-tion of the atom its composition from a small heavy posi-tively charged nucleus and one or more extranuclear electrons

Attempts weremadeimmediatelytoapplythequantumtheorytothe hydrogen atom The successful effort of Bohr2

in 1913,despiteitssimplicity, maywell be consideredthe greatest singlestepinthedevelopment ofthetheory ofatomic structure

1 P.DEBYB, Ann.d Phya (4) 39,789(1912); seealsoM BORNandT.VON

H-5a] THE ORIGIN OF THE OLD QUANTUM THEORY 27

Itwas clearlyevident that the lawsofclassicalmechanicalandelectromagnetic theory could not apply to the Rutherfordhydrogen atom. According to classical theory the electron

inahydrogen atom,attracted towardthe nucleusby an

inverse-square Coulomb force, would describe an elliptical or circularorbit about it, similarto that of the earth about the sun ^The

agc_eleratioiLDf the chargedjgartjcles would lead to the emission

ofthe electron inits orbit, and tomultiplesof this as overtones.

With the emission of energy, the radius of the orbit_would

the emittedlightshouldshowawide rangeof frequencies.) This

isnotatallwhatisobserved the radiation emittedbyhydrogen

and, moreover, these frequencies are not related to one another

byintegral factors, asovertones, butinsteadshow an interesting

additiverelation,expressedintheRitzcombination principk,and

in addition a still more striking relation involving the squares

of integers, discovered by Balmer Furthermore, the existence

of stable non-radiating atoms was not to be understood on the

basis of classical theory, for a system consisting of electrons

revolving about atomic nucleiwouldbeexpected to emitradiantenergyuntil the electronshad falleninto the nuclei.

Bohr, no doubt inspired by the work of Einstein mentionedabove, formulatedthetwo followingpostulates, whichtoa greatextentretain their validity inthe quantum mechanics

I. The Existence ofStationary States. An atomic system can

definitevalueoftheenergyWofthe system;andtransitionfromone stationary state to another is accompanied by the emission

or absorption as radiant energy, or the transfer to or fromanothersystem, of an amount of energy equal to the difference

in energyof thetwo states.

II. TheBohrFrequency Rule Thefrequencyofthe radiationemitted bya systemontransitionfromaninitialstate ofenergy

from the state of energy Wi to that of energy TF2 ) is given by

the equation1

1Thisrelationwassuggestedbythe Ritz combination principle,whichit

28 THE OLD QUANTUM THEORY [H-6b

Wt - Wi

Bohrinaddition gave amethod ofdetermining the quantized

states of motion the stationary states of the hydrogen atom

angular momentum of circular orbits to integral multiples ofthe quantum h/2ir, though leading to satisfactory energy

inthe next section

Problem 5-1. Consider an electron moving in a circular orbit about a

/r2, the total energy is equal to one-half the

/r. Evaluatetheenergyof the stationary states for

which theangularmomentum equals nh/27rj with n 1, 2, 3,

1915 W. Wilson and A Sommerfeld discovered independently1

a powerful method of quantization, which was soon applied,

especially by Sommerfeld and his coworkers, in the discussion

lines of frequencies v\ and vi occur in the spectrum of agiven atom it is

This led directly to the idea that a set ofnumbers, called term values, can

can be expressed as differences of pairs ofterm values. Term values are

. Thenormalstate

W

Themodernstudent, towhomtheBohrfrequencyrulehasbecome

common-place, mightconsider that this rule is clearly evident in theworkofPlanck

mechanicalfrequenciesof theharmonicoscillator (the onlysystemdiscussed)

andthefrequencyoftheradiationabsorbedandemittedbythisquantizedsystem delayed recognition of the fact that a fundamental violation of

W WILSON,Phil. Mag.29, 795 (1915); A. SOMMERFELD, Ann.d Phys.

H-5c] THE ORIGIN OF THE OLD QUANTUM THEORY 29

of the fine structure of the spectra of hydrogen and ionized

helium, their Zeeman and Stark effects, and many other nomena The first step of their method consists in solving the

therefore making use of the coordinates #!,, <?3n and thecanonicallyconjugatemomentapi, ,p3n astheindependent

variables The assumption is then introduced that only those

following conditions aresatisfied:

tfpkdqk nk h, k =1, 2,

, 3n; nk =an integer. (5-2)

These integrals, which are called action integrals, can be latedonlyfor conditionally periodic systems; that is,for systems

calcu-forwhichcoordinates can befound eachofwhich goesthrougha

cycle as a function of the time, independently of the others

The definite integral indicated by the symbol $ is taken overone cycle of the motion Sometimes the coordinates can bechosen inseveral different ways, in which case the shapesof thequantized orbitsdepend onthe choiceof coordinate systems,butthe energyvalues do not

determination oftheenergylevels of certain specificproblems in

Sections 6 and 7.

5c. Selection Rules The Correspondence Principle. The

cal-culating the intensities of spectral lines emitted or absorbed by

a system, that is, the probabilities of transition from one

sta-tionary state to another with the emission or absorption of aphoton Qualitative informationwas provided, however, by an

auxiliary postulate, known as Bohr's correspondence principle,which correlated the quantum-theory transition probabilities

withtheintensity ofthelight of various frequencieswhich wouldhave been radiatedbythe system accordingto classical electro-

magnetic theory In particular, if no light of frequency

cor-responding to a given transition would have been emitted

place The results of such considerations were expressed inselection rules.

For example, the energy values nhvQ of a harmonic oscillator

in the following are such as

30 THE OLD QUANTUM THEORY [H-6a

permit the emission or absorption of light of frequencies whichare arbitrary multiples (n2 UI)VQ of the fundamental fre-

the fundamental frequency v

, with no overtones, as discussed

inSection3;consequently,inaccordance withthe correspondence

principle, it was assumed that the selection rule An = 1 was

valid, the quantizedoscillatorbeing thusrestrictedtotransitions

to the adjacent stationary states

8. THE QUANTIZATION OF SIMPLE SYSTEMS

a particle of mass ra bound to the equilibrium position x =

by a restoring force kx= 4ir*mvlx and constrained to move

along the xaxistheclassicalmotionconsists inaharmonic

oscilla-tionwith frequency v

, as describedbythe equation

Thus we see thatthe energy levels allowed by the oldquantum

theory are integral multiples of hvo, as indicated in Figure 6-1

Theselection ruleAn = 1 permitsthe emissionandabsorption

A particle bound to an equilibrium position in a plane by

restoring forces with different force constants in the x and y

directions, corresponding to thepotentialfunction

m(^2 + vV),