Chapter 1 Review of Classical Physics

8/2/2019 Chapter 1 Review of Classical Physics

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1.1 Review of Classical Physics 3

particles of which it is composed, collections of atoms in molecules andsolids, and, on a cosmic scale, the origin and evolution of the universe. Ourdiscussion of modern physics in this text touches on each of these fields.We begin with the relativity theory, exploring its assumptions, implications,and experimental verification. After reviewing the experiments that sig-naled the inadequacy of classical concepts of particles and waves, we discussthe success of the quantum theory (or wave mechanics, as it is sometimesknown) in resolving those failures. A complete discussion of wave mechan-ics requires mathematical proficiency beyond the level of this text; thereforeweundertake only a superficial introduction to the techniques and applica-tions of wave mechanics. The remainder of the text deals with applicationsof these principles, first to the study of the structure and properties of theatom, next to the study of groups of atoms, both small groups in moleculesand large groups in solids, and then to the study of the structure andproperties of the atomic nucleus and the elementary particles. Finally, wetum from the microscopic to the cosmic and discuss the applications ofmodern physics to the understanding of some problems of astrophysicsand cosmology.As you undertake this study, keep in mind that the details of the storyof modern physics have been written only during this century, and thatmany of the discoveries have been made during our lifetimes. This meansthat the story of modern physics is not yet complete and will continue toevolve. Often we find that each time we look deeper or refine our techniqueswe learn something new and must revise our theories to account for thenew discoveries. It is this process of refinement and revision that advancesthe frontiers of science.

1.1 REVIEW OF CLASSICAL PHYSICSAlthough we will find many areas in which modern physics differs radicallyfrom classical physics, we frequently find the need to refer to concepts ofclassical physics. Here is a brief review of some of the concepts of classicalphysics that we may need.

MechanicsA particle of mass m moving with velocity v has a kinetic energy defined by

(Ll)and a linear momentum p defined by

p = mv (1.2)In terms of the linear momentum, the kinetic energy can be written

2K =P__ (1.3)2m


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4 Chapter 1 Introduction

When one particle collides with another, we analyze the collision byapplying two fundamental conservation laws:Conservation laws I. Conservation of Energy. The total energy of an isolated system (onwhich no net external forces act) remains constant. This means (in thecase of a collision) that the total energy of the two particles beforethe collision is equal to the total energy of the two particles afterthe collision.

II. Conservation of Linear Momentum. The total linear momentum ofan isolated system remains constant. For the collision, the total linearmomentum of the two particles before the collision is equal to the totallinear momentum of the two particles after the collision. Since linearmomentum is a vector, application of this law usually gives us twoequations, one for the x components and another for the y components.These two conservation laws are of the most basic importance to under-standing and analyzing a wide variety of problems in classical physics.Problems 1-3, 15, and 16 at the end of this chapter review the use ofthese laws.The importance of these conservation laws is both so great and so funda-mental that, even though in Chapter 2 we learn that the special theory ofrelativity modifies Equations 1.1, 1.2, and 1.3, the laws of conservation ofenergy and linear momentum remain valid.Another application of the principle of conservation of energy occurswhen a particle moves subject to an external force F. Corresponding tothat external force there is often a potential energy U, defined such that(for one-dimensional motion)

L=rxp

xFIGURE 1.1 A particle of massm, located with respect to the origino by position vector r and movingwith linear momentum p, has angularmomentum L about O.

F= _dUd x (1.4)

The total energy E is the sum of the kinetic and potential energies:E=K+U (1.5)

As the particle moves, K and U may change, but E remains constant. (InChapter 2, we find that the special theory of relativity gives us a newdefinition of total energy.)When a particle moving with linear momentum p is at a displacementr from the origin 0, its angular momentum L about the point 0 is defined(see Figure 1.1) by

L=rXp (1.6)There is a conservation law for angular momentum, just like that for linearmomentum. In practice this has many important applications. For example,when a charged particle moves near, and is deflected by, another chargedparticle, the total angular momentum of the system (the two particles)remains constant if no net external torques act on the system. If the secondparticle is so much heavier than the first that its motion is unchanged bythe influence of the first particle, the angular momentum of the first particleremains constant (because the second particle acquires no angular momen-tum). A similar situation occurs when a comet moves in the gravitational

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field of the Sun. As it approaches the Sun, r decreases, and so p mustincrease if L is to remain constant; the comet therefore accelerates as itapproaches the Sun.

Electricity and MagnetismThe electrostatic force (Coulomb force) between two charged particles qland qz has magnitude

F= _1_qlq24771: :0 r2

The direction of F is along the line joining the particles (Figure 1.2). Inthe SI system of units, the constant 114771: :0 has the value

(1.7)

_1_ = 8.988 X 109N'm2/C24771: :0

The corresponding potential energy isU=_1_Qlq2

4771: :0 rIn all equations derived from Equation 1.7 or 1.8 as starting points, thequantity 114771: :0 must appear. In some texts and reference books, you mayfindelectrostatic quantities in which this constant does not appear. In suchcases,the centimeter-gram-second (cgs) system has probably been used, inwhich the constant 114771: :0 is defined to be 1. You should always be verycareful in making comparisons of electrostatic quantities from differentreferences and check that the units are identical.A magnetic field B can be produced by an electric current i.For example,the magnitude of the magnetic field at the center of a circular current loopof radius r is (see Figure 1.3a)

(1.8)

B = /L oi2rThe SI unit for magnetic field is the tesla (T), which is equivalent to anewton per ampere-meter. The constant /L o is

(1.9)

/L o = 477 X 10-7 N'S2/C2Be sure to remember that i is in the direction of the conventional (positive)current, opposite to the actual direction of travel of the negatively chargedelectrons that typically produce the current in metallic wires. The directionof B is chosen according to the right-hand rule: if you hold the wire in theright hand with the thumb pointing in the direction of the current, thefingers point in the direction of the magnetic field.It isoften convenient to define the magnetic moment p of a current loop:

I p l = iA (1.10)where A is the geometrical area enclosed by the loop. The direction of pis perpendicular to the plane of the loop, according to the right-hand rule.

1.1 Review of Classical Physics 5

~+FIGURE 1.2 Two charged parti-cles experience equal and oppositeelectrostatic forces along the linejoining their centers. If the chargeshave the same sign (both positive orboth negative), the force is repulsive;if the signs are different, the forceis attractive.

Magnetic moment


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(a)

(b )

When a current loop is placed in a uniform external magnetic field Bext'there is a torque T on the loop that tends to line up p. with Bext:

T = P. x Bext (1.11)

FIGURE 1.3 (a ) A circular cur-rent loop produces a magnetic fieldB at its center. (b ) A current loopwith magnetic moment I. L in an exter-nal magnetic field Bex!.The field ex-erts a torque on the loop that willtend to rotate it so that I .L lines upwith Bex!.

Figure 1.3b shows a current loop in an external magnetic field. Anotherway to interpret this is to assign a potential energy to the magnetic momentp. in the external field Bext:

u = -p. . Bext (1.12)When the field Bex! is applied, p. rotates so that its energy tends to aminimum value, which occurs when p. and Bex! are parallel.Itis important for us to understand the properties of magnetic moments,

because we will find that particles such as electrons or protons have magneticmoments. Although we don't imagine these particles to be tiny currentloops, their magnetic moments do obey Equations 1.11 and 1.12.A particularly important aspect of electromagnetism is electromagnetic

waves. In Chapter 3 we discuss some properties of these waves in moredetail; in particular, interference and diffraction are fundamental to ourdiscussions in this text. Electromagnetic waves travel in free space withspeed c (the speed of light), which is related to the electromagnetic constantseo and J L o

(1.13)The waves have a frequency II and wavelength A , which are related by

c = A ll (1.14)The wavelengths range from the very short (nuclear gamma rays) to thevery long (radio waves). Figure 1.4 shows the electromagnetic spectrumwith the conventional names assigned to the different ranges of wavelengths.

Kinetic Theory of MatterThe mean kinetic energy of the molecules of an ideal gas at temperatureTis

K = !kT (1.15)

Broadcast

Wavelength(m)10-2 10-4 10-6 10-8 10-10 10-12

Visiblellight ~1012

Frequency(Hz)FIGURE 1.4 The electromagnetic spectrum. The boundaries of the regions are not sharply defined.


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1.2 Units and Dimensions 7

where k is the Boltzmann constantk = 1.381 X 10-23 J /K

TheSI unit of temperature is the kelvin (K), not "degree kelvin." Be carefulnot to confuse the symbol K for kelvin with the symbol K for kinetic energy;also be careful about confusing the Boltzmann constant k with the wavenumber k = 2 7 T / A .For rough estimates, the quantity kT is often used as a measure of themean kinetic energy per particle. For example, at room temperature(20C= 293 K), the mean kinetic energy per particle is about 4 X 10-21 J,while in the interior of a star where T ~ 107 K, the mean energy is about1 0- 16 J.

1.2 UNITS AND DIMENSIONSNearly all of the physical constants and variables we will be using haveboth units and dimensions. The dimensions of a constant or variable tellus something about the kind of constant or variable it is; a quantity thatin one frame of reference has the dimension of length, for example, willhave the dimension of length in every frame of reference, although itsmagnitude and the units in which we express it may vary. We often referto mass, length, and time as the fundamental dimensions.Inworking problems, it isalways necessary to be sure that your equationsare dimensionally consistent; if, for example, you have an equation thatcontains the term (v + m) where v = velocity and m = mass, it is a surebet that you have made a mistake somewhere-two quantities can neverbeadded unless they have the same dimensions. [However, if your equationcontains the term (av + m), where a is a constant, it may be dimensionallycorrect if a has the proper dimensions.]Checking your results for dimensional consistency is a good habit toacquire. It is one of the simplest techniques to apply and can almost alwaysbe done quickly by inspection. Ifyour results check for dimensional consis-tency, it doesn't guarantee that they are correct; the reverse, however, istrue-lack of dimensional consistency always indicates an error.It is sometimes possible for a quantity to have units, but no dimensions!Suppose your watch runs slow and loses 6.0 seconds each day. Its rate ofloss is therefore R = 6.0 s/day. R is a dimensionless quantity-it hasdimensions of tit-but it has units, and its value changes as its units change;wecould also express R as 0.10 min/day or 0.25 s/hour, or even in a unitlessform as 6.9 X 10-5, which gives the fractional loss of time in any interval.As another example, all conversion factors such as 25.4 mm/inch or 1000g/kg are dimensionless.We generally use SI units to express physical quantities. However, inmodern physics our calculations often yield results that turn out to be verysmallor very large when expressed in SI units. For example, typical energiesassociated with atomic or nuclear processes may be 10-19 to 10-12 J, andtypical sizes of atomic and nuclear systems range from 10-10 to 10-15 m.


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Although there is no reason that we could not use these SI units in ourstudy of modern physics, we frequently give way to convenience and usedifferent units.

LengthThe SI unit for length is the meter (m), but we will need lengths muchsmaller than the meter for atomic and nuclear systems. We use the followingunits for lengths:micrometer = /Lm = 10-6 mnanometer = nm = 10-9 mfemtometer = fm = 10-15 m

Wavelengths are conveniently measured in nanometers-visible light haswavelengths in the range 400 to 700 nm. Atomic sizes are typically 0.1 nm,and nuclear sizes are about 1 to 10 fm. (The unit fm is sometimes knownas the fermi in honor of Enrico Fermi, who was a pioneer experimentaland theoretical nuclear physicist.)

EnergyThe SI unit of energy is the joule (J), which is too large to be of convenienceThe electron-volt for atomic and nuclear physics. A much more convenient unit is the electron-volt (eV), defined as the energy gained or lost by a particle of charge equalin magnitude to that ofthe electron in moving through a potential differenceof one volt. (The volt is an SI unit, so we have not gone too far astray!)With the electronic charge of 1.602 X 10-19 C and with 1 V = 1 J/C, wehave the equivalence

1 eV = 1.602 X 10-19 JSome convenient multiples of the electron-volt arekeY = kilo electron-volt = 103 eVMeV = mega electron-volt = 106 eVGeV = giga electron-volt = 109 eV

(In some older works you may find reference to the BeV, for billion elec-tron-volts; this is a source of confusion, for in the United States a billionis 109 while in Europe a billion is 1012.)

Electric ChargeThe standard unit of electric charge is the coulomb (C), and the basicquantity of charge, that of the electron, is e = 1.602 X 10-19 C. Veryfrequently we wish to find the potential energy of two basic charges sepa-rated by typical atomic or nuclear dimensions, and we wish to have the

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1.2 Units and Dimensions 9

result expressed in electron-volts. Here is a convenient way of doing this.Let us find the potential energy of two electrons at a separation of r =1.00 nm:

1 e2U=--41700 rThe quantity e2/417oo can be expressed in a convenient form:

4::0 = (8.988 X 109 N~~2) (1.602 X 10-19 C )2=2.307 X 10-28 N .m?

1 109nm=2.307 X 10-28 Jm 1.602 X 10-19 J/eY ~= 1.440 eV nm

With this useful combination of constants it becomes very easy to calculateelectrostatic potential energies. For two unit charges separated by 1.00 nm,

1e2 e21 1U =--- =--- = 1.440 eYnm..,......,..".----41700 r 41700r 1.00 nm= 1.44eY

For calculations using typical nuclear sizes, the femtometer is a more conve-nient unit of distance, so

e2 1m 1015fm1MeY4_- = 1.440 eY nm -109 -- 106 Y1700 nm m e

= 1.440 MeYfmIt is remarkable (and convenient to remember) that the quantity e2/417oohas the same value of 1.440 whether we use typical atomic energies andsizes (eYnm) or typical nuclear energies and sizes (MeYfm).

MassThe kilogram (kg) is the basic SI unit of mass, but it is too large a unit tobe particularly useful for work in atomic and nuclear physics. Anotherdifficulty, as we discuss in Chapter 2, is that we are frequently interestedin using Einstein's equation E = me? to express the mass of a particle interms of its rest energy. Using c2, a very large number, to do this conversionis inconvenient and can lead to mistakes. We avoid this problem by keepingthe factor of c2 with the mass units, and remembering that m = E/c2 Forexample, you will often find in tables of the masses of the elementaryparticles that an electron has a mass of 0.511 MeY/c2 Although this maylook like an energy unit, it is really a measure of mass-the factor of c2does the conversion. What is important is that we can keep the factor ofc2 without bothering to put in its numerical value. You can check this result


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by converting MeV to joules and putting in the numerical value of c2, andyou should obtain 9.11 x 10-31 kg, the mass of the electron. (To avoid round-off errors, use four significant figures for the quantities in this calculation andthen round off the final result to three significant figures.)Atomic mass unit Another mass unit that we find convenient to use is the unified atomicmass unit, u. This is most convenient in calculating atomic and nuclearbinding energies. The atomic mass unit is defined so that the mass of anatom of the most abundant isotope of carbon (12e) is exactly equal to 12u. All other atomic masses are measured relative to this value. In AppendixB you will find a table of atomic masses.Another useful mass unit based on 12Cis the molar mass. One mole ofany substance is the quantity of that substance that contains the samenumber of elementary entities (atoms, molecules, or whatever) as 0.012 kgof 12C.This number is called the Avogadro constant and has the value

N A = 6.022 X 1023 per moleThat is, one mole of 12Ccontains N A atoms and has a molar mass of(exactly) 0.012 kg, one mole of nitrogen gas contains N A molecules of N2and has a molar mass of 0.028 kg, and one mole of water contains NAmolecules of H20 and has a molar mass of 0.018 kg.


The Speed of LightOne of the fundamental constants of nature is the speed of light, c, whichyou will be using quite frequently throughout your study of modern physics.Its value is

c = 2.998 X 108 mlsIt will often be convenient for us to measure speeds by comparing themwith the speed of light; in Chapter 2 you will find many examples ofproblems in which speeds are given as a fraction of c, for example, v =0.6c. Fortunately, many of the equations of special relativity involve not vbut vic, and thus it is often not necessary to convert 0.6c to a numericalspeed in meters per second.

Planck's ConstantAnother of the fundamental constants of nature is Planck's constant, h,with a value of

h = 6.626 X 10-34 JsPlanck's constant obviously has dimensions of energy X time, but with alittle bit of manipulation, you can show that it also has dimensions oflinear momentum X displacement and also of angular momentum. Planck'sconstant will appear in many applications when we begin our study of


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1.3 Significant Figures 11

quantum physics, and each different way of expressing its dimensions repre-sents an important application.We have already mentioned our desire to use electron-volts rather thanjoules for energy, and so it is useful to express Planck's constant using eV:h = 4.136 X 10-15 ev-s

We also encounter the product he in many calculations. In our units ofconvenience, you should be able to derive its value:he = 1240 eVvnm

= 1240MeYfmIt is interesting to note that he and e2 /4rr8o have the same dimensions,and we have in fact calculated both quantities in the same units, eYnm.The ratio of these two quantities is a pure number, independent of thesystem of units we have chosen, and we will learn that this ratio is offundamental importance in atomic physics. The dimensionless constant a,called the fine structure constant, is actually Z m times the ratio: Fine structure constant

a = Zn e2 /4rr8ohe= 2 rr 1.440 eY 'nm1240 eVvnm= 0.007297 (1.16)

Thisnumber is often expressed as a = 1/137.0.

1.3 SIGNIFICANT FIGURESThe number of significant figures in a quantity tells us something about itsprecision-a quantity that is measured to eight significant figures is knownmore precisely than one that is measured to four significant figures. Incalculations, the number of significant figures in the result is determinedby the number of significant figures in the input quantities by applyingtwo rules:1. When adding or subtracting, the least significant digit of the numberbeing added or subtracted determines the least significant digit of thesum or difference. The number of significant figures does not matter.2. When multiplying or dividing, count the number of significant figuresin the quantities being multiplied or divided. The number of significantfigures in the product or quotient is determined by the factor with thesmallest number of significant figures. The location of the least significantdigit does not matter.


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Here are some examples that illustrate the use of these rules.

E X A M P L E 1 . 1Find the mass difference between a proton and a neutron. Express theresult in u and in MeV/c2 S O L U T I O NThe proton and neutron masses are (to seven significant figures):

m = 1.008665 ump = 1.007276 u

The difference is1.008665 u - 1.007276 u = 0.001389 u

In all cases, the last digit (indicated in boldface) is the least significant. Inaddition or subtraction, we do not count the number of significant figures.It is of no importance for the subtraction that each of the two masses hasseven significant figures while the difference has only four.The conversion factor from u to MeV/c2 is

1 u = 931.50 MeV/c2The mass difference is therefore

0.001389 u X 931.501~e VIc2= 1.294MeVIc2

The mass difference in u has four significant figures, the conversion factorhas five significant figures, and the product can therefore have only foursignificant figures, according to the second rule given above.

E X A M P L E 1 . 2A proton and an electron can combine to form an atom of hydrogen. Findthe total mass of a proton and an electron.S O L U T I O NThe values of the masses are

mp = 1.007276 ume =5.4858 X 10-4 u

The combined mass is1.007276 u + 0.00054858 u = 1.007825 uNotice that the position of the least significant digit (the 6 in mp) determinesthe position of the least significant digit in the sum.


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1.4 Theory, Experiment, Law 13

E X A M P L E 1 . 3The value of he was given in the last section as 1240 eYnm. Compute thevalue of he to four significant figures and determine if the zero in the lastdigit is significant.~ H U T I O NThe values given previously for hand e contain four significant figures. Toavoidround-off errors, we use the values of the constants to five significantfigures, and then round off the final result to four figures.

h = 6.6261 X 10-34 J. se = 2.9979 X lO B m/s

1 eY = 1.6022 X 10-19 JTherefore,

h _ (6.6261 X 1O-34Js)(2.9979 X lO B m/s)(109nm/m)e - 1.6022 X 10-19 J/eY= 1239.8eYnm

Rounding off to four figures, we obtain he = 1240 eY.nm, so the zero issignificant. (Of course, it is good practice not to leave zeros on the end ofnumbers such as this, where we don't know whether or not they are signifi-cant. It would be better to express this result as 1.240 X 103 eYnm.)

Proper attention to significant figures is a matter of habit, and the sooneryou form this good habit the less trouble you will have in expressing theresults of calculations. Simply because your calculator display shows eightdigits does not make them all significant, and only the significant onesshould be written down as the answer to a problem.

1.4 THEORY, EXPERIMENT, LAWWhen you first began to study science, perhaps in your elementary or highschool years, you may have learned about the "scientific method," whichwas supposed to be a sort of procedure by which scientific progress wasachieved. The basic idea of the "scientific method" was that, on reflectingoversome particular aspect of nature, the scientist would invent a hypothesisor theory,which would then be tested by experiment and if successful wouldbe elevated to the status of law. This procedure is meant to emphasize theimportance of doing experiments as a way of testing hypotheses and re-jectingthose that do not pass the tests. For example, the ancient Greeks hadsomerather definite ideas about the motion of objects, such as projectiles, in


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the Earth's gravity. Yet they tested none of these by experiment, so con-vinced were they that the power of logical deduction alone could be usedto discover the hidden and mysterious lawsof nature and that once logichadbeen applied to understanding a problem, no experiments were necessary. Iftheory and experiment were to disagree, they would argue, then there mustbe something wrong with the experiment! This dominance of analysis andfaithwas sopervasive that it was2000years before Galileo, usingan inclinedplane and a crude timer (equipment surely within the abilities of the earlyGreeks to construct), discovered the laws of motion, which were laterorganized and analyzed by Newton.In the case of modern physics, none of the fundamental concepts is

obvious from reason alone. Only by doing often difficult and necessarilyprecise experiments do we learn about these unexpected and fascinatingeffects. We therefore emphasize the experiments that have been done tostudy such modern physics topics as relativity and quantum physics. Theseexperiments have been done to unprecedented levels of precision-of theorder of one part in 106 or better-and it can certainly be concluded thatmodern physics has been tested far better in the twentieth century thanclassical physics was tested in all of the preceding centuries.Nevertheless, there is a persistent and often perplexing problem associ-

ated with modem physics, one which stems directly from your previousacquaintance with the "scientific method." This concerns the use of theword "theory," as in "theory of relativity" or "quantum theory," or even"atomic theory" or "theory of evolution." There are two contrasting andconflicting definitions of the word "theory" in the dictionary:1. A hypothesis or guess.2. An organized body of facts or explanations.The "scientific method" refers to the first kind of "theory," while whenwe speak of the "theory of relativity" we refer to the second kind. Yetthere isoften confusion between the two definitions, and therefore relativityand quantum physics are sometimes incorrectly regarded asmerely hypoth-eses, on which evidence is still being gathered, in the hope of somedaysubmitting that evidence to some sort of international (or intergalactic)tribunal, which in tum might elevate the "theory" into a "law." Thus the"theory of relativity" might someday become the "law of relativity," likethe "law of gravity." Nothing could be further from the truth!The theory of relativity and the quantum theory, like the atomic theory

or the theory of evolution, are truly "organized bodies of facts and explana-tions" and not "hypotheses." There is no question of these "theories"becoming "laws"-the "facts" (experiments, observations) ofrelativity andquantum physics, like those of atomism or evolution, are accepted byvirtually all scientists today; whether one calls them theories or laws ismerely a question of semantics and has nothing to do with their scientificmerits. Like all scientific principles, they will continue to develop andchange asnew discoveries are made; that is the essence ofscientificprogress,and it must be remembered that the search for ultimate truths or eternallaws is not a goal of science.


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Suggestions for Further Reading 15

Still, there are two remaining questions that you may find particularlybothersome as you study modern physics. First is the "how" of thesetheories. The experimental evidence that forms the basis of modern physicsisalmost always indirect-no one has ever "seen" a quantum or a pi mesonor even a nucleus, and no one has ever traveled at nearly the speed of lightand "seen" the effects of relativity; similarly, no one has ever "seen" singleatoms joining to form compounds or "seen" one species evolving intoanother. Yet the experimental evidence for all of these processes is socompelling that no one who approaches them in the spirit of free and openinquiry can doubt them:The second vexing question concerns the "why" of these theories. WhydoesNature behave according to Einstein's relativity, rather than accordingto Galileo's? Why do particles sometimes behave as waves, and wavessometimes as particles? Why do atoms join to form compounds? Why dohigher forms of life evolve from lower forms? Although scientists canprovide extremely precise answers to the "how" of these theories, theycannot provide the answers to the "why," not because their powers ofobservation or experimental abilities are limited, but rather because thequestions are outside the realm of experimentation. These questions areof extreme importance, and as potential practitioners of pure or appliedscienceyou should be aware of them and spend some time thinking aboutthem. If answers to these questions are to be found at all, they will befound not in the field of science, but in the fields of philosophy or theology.As you begin to study the facts ot'modern science, you should keep theseadditional questions in mind and perhaps seek your instructor's opinionsconcerning them. Although such speculations are an exciting intellectualendeavor in their own right, they will not be discussed in this text.

SUGGESTIONS FOR FURTHER READINGIfyouneed to review some topics from classical physics, here are some introductorytexts that provide the necessary background:H. C. Ohanian, Physics, 2nd ed. (New York, Norton, 1989).R.Resnick, D. Halliday, and K. S. Krane, Physics, 4th ed. (New York, Wiley, 1992).R. A. Serway, Physics for Scientists and Engineers, 3rd ed. (Philadelphia, Saun-ders, 1990).

P . A.Tipler, Physics for Scientists and Engineers, 3rd ed. (New York, Worth, 1991).H. D. Young, University Physics, 8th ed. (Reading, MA, Addison-Wesley, 1992).Sometimes it helps your understanding to read about a subject from a slightlydifferent perspective or written in a slightly different way. Here are some othermodern physics books at about the same level as this text:A. Beiser, Concepts of Modern Physics, 4th ed. (New York, McGraw-Hill, 1987).H. C . Ohanian, Modern Physics (Englewood Cliffs, NJ, Prentice-Hall, 1987).


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S. T. Thornton and A. Rex, Modern Physics for Scientists and Engineers (Philadel-phia, Saunders, 1993).P. A. Tipler, Elementary Modern Physics (New York, Worth, 1992).

Some more advanced (but still undergraduate level) texts:J. J. Brehm and W. J. Mullin, Introduction to the Structure of Matter (New York,Wiley, 1989).

R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei,and Particles, 2nd ed. (New York, Wiley, 1985).R. B. Leighton, Principles of Modern Physics (New York, McGraw-Hill, 1959).Some descriptive, historical, philosophical, and nonmathematical texts which givegood background material and are great fun to read:A. Baker, Modern Physics and Anti-Physics (Reading, Addison-Wesley, 1970).F. Capra, The Tao of Physics (Berkeley, Shambhala Publications, 1975).G. Gamow, Thirty Years that Shook Physics (New York, Doubleday, 1966).R. March, Physics for Poets (New York, McGraw-Hill, 1978).E. Segre, From X-Rays to Quarks: Modern Physicists and their Discoveries (SanFrancisco, W. H. Freeman, 1980).G. L. Trigg, Landmark Experiments in Twentieth Century Physics (New York,Crane, Russak, 1975).F. A. Wolf, Taking the Quantum Leap (San Francisco, Harper & Row, 1981).

G. Zukav, The Dancing Wu Li Masters, An Overview of the New Physics (NewYork, Morrow, 1979).

Gamow, Segre, and Trigg have contributed directly to the development of modernphysics and their books are written from a perspective that only those who werepart of that development can offer. The books by Capra and Zukav draw interestingparallels between modern physics (especially quantum theory and particle physics)and oriental philosophy.

QUESTIONS1. Under what conditions can you apply the law of conservation of energy?

Conservation of linear momentum? Conservation of angular momentum?2. Which of the conserved quantities are scalars and which are vectors? Is there

a difference in how we apply conservation laws for scalar and vector quantities?3. What other conserved quantities (besides energy, linear momentum, and angu-

lar momentum) can you name?4. What is the difference between potential and potential energy? Do they have

different dimensions? Different units?5. In Section 1.1 we defined the electric force between two charges and the

magnetic field of a current. Use these quantities to define the electric field ofa single Charge and the magnetic force on a moving electric charge.6. Other than from the ranges of wavelengths shown in Figure 1.4, can you think

of a way to distinguish radio waves from infrared waves? Visible from infrared?


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Problems 17

That is, could you design a radio that could be tuned to infrared waves? Couldliving beings "see" in the infrared region?

7. What is the difference between dimensions and units?8. Can a quantity have units but no dimensions? Dimensions but no units?9. In the past century a macroscopic standard for time (the rotation of the Earth)

has been replaced with a microscopic standard (the frequency of light emittedby an atom). Similarly, a macroscopic standard for length (a metal bar oflength one meter) has been replaced with a microscopic standard (based onthe basic unit of time and the speed of light). Might it be possible to replacethe current macroscopic standard of mass (the standard kilogram) with amicroscopic standard? If so, how would we connect the macroscopic measure-ment with the microscopic standard?

10. Consider the equation y =X-I with X = 0.98. By strictly following the rulefor significant figures, we obtain y = 1.0. What is wrong with expressing theresult in this way? How can we modify the rules for significant figures toinclude calculations such as this one?

PROBLEMS1 . An atom of mass m moving in the x direction with speed v collides elastically

with an atom of mass 3m at rest. After the collision the first atom moves inthe y direction. Find the direction of motion of the second atom and the speedsof both atoms (in terms of v) after the collision.

2. An atom of mass m moves in the positive x direction with speed v. It collideswith and sticks to an atom of mass 2m moving in the positive y directionwith speed 2 v 1 3. Find the resultant speed and direction of motion of thecombination, and find the kinetic energy lost in this inelastic collision.

3. An atom of beryllium (m = = 8.00 u) splits into two atoms of helium (m = =4.00 u) with the release of 92.0 keV of energy. If the original beryllium atom isat rest, find the kinetic energy, speed, and momentum of the two helium atoms.

4. Express the following speeds as a fraction of the speed of light: (a) a typicalautomobile speed (100 km/h ) ; (b) the speed of sound (330 m /s); ( c) the escapevelocity of a rocket from the Earth's surface (11 km/s) ; (d) the orbital speedof the Earth about the Sun (Earth-Sun distance = 1.5 X 108 km).

5. Show that Planck's constant h has dimensions of linear momentum X dis-placement.

6. Starting from Coulomb's law, show that e2/41TBo has dimensions of energyX distance.

7. (a) Starting from Newton's universal law of gravitation, show that Gm2 hasdimensions of energy X distance. (b) Evaluate Gm2 in units of eVvnm usingthe proton mass. (c) Evaluate the ratio Gm2/ (e2/41TBo) . Is this a pure number?What is its significance?

8. Use the Avogadro constant to find the mass in kilograms equivalent to oneatomic mass unit (u).

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16. Suppose the beryllium atom of Problem 3 moves in the positive x directionand has kinetic energy 60.0 keY. One helium atom is found to move at anangle of 30 with respect to the x axis. Find the direction of motion of thesecond helium atom and find the velocity of each helium atom. Work thisproblem in two ways as you did the previous problem. (Hint: Consider onehelium to be emitted with velocity components Vx and Vy in the beryllium restframe. What is the relationship between Vx and vy? How do v< and Vy changewhen we move in the x direction at speed v?)

17. It is sometimes convenient to represent the behavior of a particle on a graph+-+-+--+--+--+--+--+-+-+-+-+-+ .. x showing its position on one axis and its linear momentum on another (Figure1.5 is an example for a particle confined to move in one dimension). Thecoordinate space defined by this system is called ph as e s pace. Draw the phasespace graphs representing: (a) a particle at rest; (b) a particle moving atconstant positive speed; (c) a particle originally at rest at x = 0 moving witha constant positive acceleration; (d) a particle in simple harmonic motionabout a point xo on the positive x axis; (e) a ball thrown upward to a heightx = h.


Px

FIGURE 1.5 Problem 17.

9. In the Bohr model of the hydrogen atom, the electron moves in a circularorbit of radius h2eo / 1Tmee2 with angular momentum h / 21T . Find the speed ofthe electron as a fraction of the speed of light. Express your result in termsof the fine structure constant, Equation 1.16.10. A gas cylinder contains argon atoms (m=40.0 u). The temperature isincreasedfrom 293 K (20C) to 373 K (100C). (a) What is the change in the averagekinetic energy per atom? (b) The container is resting on a table in tbe Earth's

gravity. Find the change in the height of the container that produces the samechange in the average energy per atom found in part (a).11. The Bohr magneton /L B is equal to ehl anm., (a) Show that /L B can be expressedin units of joules per tesla. (b) Show tbat /L B has the same dimensions as amagnetic moment (current X area).12. The orbital radius of an electron in an atom of hydrogen is h2eo / 1Tmee2

(a) Show that this quantity has the dimension of length. (b) Compute its valueto four significant figures. (Hint: Multiply numerator and denominator bycommon factors to allow use of some of the combinations of constants com-puted in this chapter.)13. A fundamental unit frequently encountered is the Compton wavelength ofthe electron, h/m.c. (a) Sbow that this quantity has the dimension of length.

(b) Compute the value of h/m.c to four significant figures.14. A helium nucleus consists of two protons and two neutrons, and has a massof 4.001506 u. Find the mass difference between a helium nucleus and itsconstituents. Express your result in u and in MeV/c2.15. Suppose the beryllium atom of Problem 3 were not at rest, but instead movedin the positive x direction and had a kinetic energy of 40.0 keV. One of thehelium atoms is found to be moving in the positive x direction. Find thedirection of motion of the second helium, and find the velocity of each of thetwo helium atoms. Solve this problem in two different ways: (a) By directapplication of conservation of momentum and energy. (b) By applying theresults of Problem 3 to a frame of reference moving with the original berylliumatom and then switching to the reference frame in which the beryllium ismoving.

c

Chapter 1 Review of Classical Physics

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