Table Of ContentPreface
DuringthelastyearsoftheNineteenthCentury,thedevelopmentofnewtech-
niques and the refinement of measuring apparatuses provided an abundance
of new data whose interpretation implied deep changes in the formulation of
physical laws and in the development of new phenomenology.
Severalexperimentalresultsleadtothebirthofthenewphysics.Abrieflist
ofthemostimportantexperimentsmustcontainthoseperformedbyH.Hertz
about the photoelectric effect, the measurement of the distribution in fre-
quency of the radiationemitted by an ideal oven(the so-calledblack body ra-
diation),themeasurementofspecificheatsatlowtemperatures,whichshowed
violations of the Dulong–Petit law and contradicted the general applicability
of the equi-partition of energy. Furthermore we have to mention the discov-
ery of the electron by J. J. Thomson in 1897, A. Michelson and E. Morley’s
experiments in 1887, showing that the speed of light is independent of the
reference frame, and the detection of line spectra in atomic radiation.
Froma theoretical point of view, one of the main themes pushing for new
physics was the failure in identifying the ether, i.e. the medium propagating
electromagnetic waves, and the consequent Einstein–Lorentz interpretation
of the Galilean relativity principle, which states the equivalence among all
reference frames having a linear uniform motion with respect to fixed stars.
In the light of the electromagnetic interpretation of radiation, of the dis-
coveryofthe electronandofRutherford’sstudiesaboutatomicstructure,the
anomaly in black body radiation and the particular line structure of atomic
spectra lead to the formulation of quantum theory, to the birth of atomic
physics and, strictly related to that, to the quantum formulation of the sta-
tistical theory of matter.
Modern Physics, which is the subject of this notes, is well distinct from
Classical Physics, developedduringtheXIXcentury,andfromContemporary
Physics,whichwasstartedduringtheThirties(ofXXcentury)anddealswith
the nature ofFundamental Interactionsand with the physics of matter under
extreme conditions. The aim of this introduction to Modern Physics is that
of presenting a quantitative, even if necessarily also synthetic and schematic,
VI Preface
accountofthe mainfeatures ofSpecial Relativity, ofQuantum Physics and of
its application to the Statistical Theory of Matter. In usual textbooks these
three subjects are presented together only at an introductory and descriptive
level, while analytic presentations can be found in distinct volumes, also in
view of examining quite complex technical aspects. This state of things can
be problematic from the educational point of view.
Indeed, while the need for presenting the three topics together clearly
followsfromtheirstrictinterrelations(thinkforinstanceoftheroleplayedby
specialrelativityinthehypothesisof deBroglie’swavesorofthatofstatistical
physics in the hypothesis of energy quantization), it is also clear that this
unitary presentation must necessarily be supplied with enough analytic tools
so as to allowa full understanding of the contents and of the consequences of
the new theories.
Onthe otherhand, since the presenttextis aimedto be introductory,the
obvious constraints on its length and on its prerequisites must be properly
taken into account: it is not possible to write an introductory encyclopaedia.
Thatimposesaselectionofthetopicswhicharemostqualifiedfromthepoint
of view of the physical content/mathematical formalism ratio.
Inthe contextofspecialrelativitywehavegivenuppresentingthe covari-
ant formulation of electrodynamics, limiting therefore ourselves to justifying
theconservationofenergyandmomentumandtodevelopingrelativistickine-
matics with its quite relevant physical consequences. A mathematical discus-
sion about quadrivectors has been confined to a short appendix.
Regarding Schro¨dinger quantum mechanics, after presenting with some
care the origin of the wave equation and the nature of the wave function
together with its main implications, like Heisenberg’s Uncertainty Principle,
wehaveemphasizedits qualitativeconsequencesonenergylevels,givingupa
detaileddiscussionofatomicspectrabeyondthesimpleBohrmodel.Therefore
the main analysis has been limited to one-dimensional problems, where we
haveexaminedthe originofdiscreteenergylevelsandofbandspectra aswell
asthetunneleffect.Extensionstomorethanonedimensionhavebeenlimited
toverysimplecasesinwhichtheSchro¨dingerequationiseasilyseparable,like
the three-dimensionalharmonic oscillator and the cubic well with completely
reflecting walls, which are however among the most useful systems for their
applications to statistical physics. In a brief appendix we have sketched the
mainlinesleadingtothesolutionofthethree-dimensionalmotioninacentral
potential, hence in particular of the hydrogen atom spectrum.
Goingtothelastsubject,whichwehavediscussed,asusual,onthebasisof
Gibbs’constructionofthestatisticalensembleandoftherelateddistribution,
we have chosen to consider those cases which are more meaningful from the
pointofviewofquantumeffects,likedegenerategasses,focusinginparticular
on distribution laws and on the equation of state, confining the presentation
of entropy to a brief appendix.
In order to accomplish the aim of writing a text which is introductory
and analytic at the same time, the inclusionof significantcollections of prob-
Preface VII
lems associated with each chapter has been essential. We have possibly tried
to avoid mixing problems with text complements: while moving some rele-
vant topics to the exercise collection may be tempting in order to streamline
the generalpresentation, it has the bad consequence of leading to excessively
long exercises which dissuade the average student from trying to give an an-
swer before looking at the suggested solution scheme. On the other hand, we
have tried to limit the number of those (however necessary) exercises involv-
ing a mere analysis of the order of magnitudes of the physical effects under
consideration. The resulting picture, regarding problems, should consist of a
sufficiently wide series of applications of the theory, being simple but techni-
cally non-trivial at the same time: we hope that the reader will feel that this
result has been achieved.
Going to the chapter organization, the one about Special Relativity is di-
vided in two sections, dealing respectively with Lorentz transformations and
with relativistic kinematics. The chapter on Wave Mechanics is made up
of eight sections, going from an analysis of the photo-electric effect to the
Schro¨dinger equation and from the potential barrier to the analysis of band
spectra. Finally, the chapter on the Statistical Theory of Matter includes a
first part dedicated to Gibbs distribution and to the equation of state, and
a second part dedicated to the Grand Canonical distribution and to perfect
quantum gasses.
Genova, Carlo Maria Becchi
April 2007 Massimo D’Elia
VIII Preface
Suggestion for Introductory Reading
• K.Krane: Modern Physics, 2nd edn (John Wiley,New York 1996)
Physical Constants
• speed of light in vacuum: c=2.998 108 m/s
• Planck’s constant: h=6.626 10−34 J s =4.136 10−15 eV s
• ¯h=1.055 10−34 J s =6.582 10−16 eV s
• Boltzmann’s constant : k =1.381 10−23 J/◦K =8.617 10−5 eV/◦K
• electron charge magnitude: e=1.602 10−19 C
• electron mass: me =9.109 10−31 Kg=0.5110 MeV/c2
• proton mass: mp =1.673 10−27 Kg=0.9383 GeV/c2
• permittivity of free space: (cid:1)0 =8.854 10−12 F / m
Contents
1 Introduction to Special Relativity ......................... 1
1.1 Michelson–Morley Experiment and Lorentz Transformations .. 2
1.2 Relativistic Kinematics................................... 8
Problems ................................................... 16
2 Introduction to Quantum Physics.......................... 29
2.1 The Photoelectric Effect.................................. 29
2.2 Bohr’s Quantum Theory ................................. 34
2.3 De Broglie’s Interpretation ............................... 36
2.4 Schro¨dinger’s Equation................................... 42
2.4.1 The Uncertainty Principle .......................... 46
2.4.2 The Speed of Waves ............................... 48
2.4.3 The Collective Interpretation of de Broglie’s Waves .... 49
2.5 The Potential Barrier .................................... 49
2.5.1 Mathematical Interlude: Differential Equations
with Discontinuous Coefficients ..................... 51
2.5.2 The Square Barrier ................................ 53
2.6 Quantum Wells and Energy Levels......................... 60
2.7 The Harmonic Oscillator ................................. 66
2.8 Periodic Potentials and Band spectra ...................... 71
Problems ................................................... 77
3 Introduction to the Statistical Theory of Matter........... 93
3.1 Thermal Equilibrium by Gibbs’ Method .................... 97
3.1.1 Einstein’s Crystal .................................100
3.1.2 The Particle in a Box with Reflecting Walls...........102
3.2 The Pressure and the Equation of State ....................103
3.3 A Three Level System....................................105
3.4 The Grand Canonical Ensemble and the Perfect Quantum Gas108
3.4.1 The Perfect Fermionic Gas .........................110
3.4.2 The Perfect Bosonic Gas ...........................118
X Contents
3.4.3 The Photonic Gas and the Black Body Radiation......121
Problems ...................................................124
A Quadrivectors .............................................133
B The Schro¨dinger Equation in a Central Potential ..........137
C Thermodynamics and Entropy.............................147
Index..........................................................151
1
Introduction to Special Relativity
Maxwell equations in vacuum space describe the propagation of electromag-
√
netic signals with speed c ≡ 1/ µ0(cid:1)0. Since, according to Galilean relativity
principle, velocities must be added like vectors when going from one inertial
referenceframetoanother,thevectorcorrespondingtothevelocityofalumi-
noussignalinoneinertialreferenceframeOcanbeaddedtothevelocityofO
with respectto a new inertialframeO(cid:2) to obtainthe velocityofthe luminous
signalasmeasuredinO(cid:2).Foragenericvalueoftherelativevelocity,thespeed
of the signal in O(cid:2) will be different, implying that, if Maxwell equations are
valid in O, they are not valid in a generic inertial reference frame O(cid:2).
In the Nineteenth Century the most natural solution to this paradox
seemed that based, in analogy with the propagation of elastic waves, on the
assumption that electromagnetic waves correspond to deformations of an ex-
tremely rigid and rare medium, which was named ether. However that led to
the problem of finding the reference system at rest with ether.
Taking into account that Earth rotates along its orbit with a velocity
which is about 10−4 times the speed of light, an experiment able to reveal
the possible change of velocity of the Earth with respect to the ether in two
differentperiodsoftheyearwouldrequireaprecisionofatleastonepartover
ten thousand. We will show how A. Michelson and E. Morley were able to
reach that precision by using light interference.
Anotheraspectofthesameproblemcomesoutwhenconsideringtheforce
exchanged between two charged particles at rest with respect to each other.
From the point of view of an observer at rest with the particles, the force is
given by Coulomb law, which is repulsive it the charges have equal sign. An
observerinamovingreferenceframe mustinsteadalsoconsiderthe magnetic
fieldproducedbyeachparticle,whichactsonmovingelectricchargesaccord-
ingtoLorentz’sforcelaw.Ifthevelocityoftheparticlesisorthogonaltotheir
relativedistance,itcanbeeasilycheckedthatthe Lorentzforceisoppositeto
theCoulombone,thusreducingtheelectrostaticforcebyafactor(1−v2/c2).
Even if small, the difference leads to different accelerations in the two refer-
ence frames, in contrast with Galilean relativity principle. According to this
2 1 Introduction to Special Relativity
analysisCoulomblawshouldbe validin no inertialreference systembutthat
at rest with ether. However in this case violations are not easily detectable:
for instance, in the case of two electrons accelerated through a potential gap
equal to 104 V, one would need a precision of the order of v2/c2 (cid:3)4 10−4 in
order to reveal the effect, and such precisions are not easily attained in the
measurementofa force.For this reasonitwasmuchmoreconvenientto mea-
sure the motion of Earth with respect to the ether by studying interference
effects related to variations in the speed of light.
1.1 Michelson–Morley Experiment
and Lorentz Transformations
The experimental analysis was done by Michelson and Morley who used a
two-arminterferometer similar to what reported in Fig. 1.1. The light source
L generates a beam which is split into two parts by a half-silvered mirror S.
Thetwobeamstraveluptotheendofthearms1and2oftheinterferometer,
where they are reflected back to S: there they recombine and interfere along
thetractconnectingtotheobserverinO.Theobserverdetectsthephaseshift,
which can be easily shown to be proportional to the difference ∆T between
the times needed by the two beams to go along their paths: if the two arms
havethesamelengthl andlightmoveswiththesamevelocitycalongthetwo
directions, then ∆T =0 and constructive interference is observed in O.
Fig. 1.1. A sketch of Michelson-Morley interferometer
1.1 Michelson–Morley Experiment and Lorentz Transformations 3
Ifhowevertheinterferometerismov-
ing with respect to ether with a ve-
locity v, which we assume for sim-
plicity to be parallel to the second
arm,then the path of the firstbeam
will be seen from the reference sys-
tem of the ether as reported in the
nearbyfigureandthe timeT needed
to make the path will be given by
Pitagora’s theorem:
c2T2 =v2T2+4l2 (1.1)
from which we infer
2l/c
T = (cid:1) . (1.2)
1−v2/c2
If we instead consider the second beam, we have a time t1 needed to make
half-path and a time t2 to go back, which are given respectively by
l l
t1 = c−v , t2 = c+v (1.3)
so that the total time needed by the second beam is
2l/c T
T(cid:2) =t1+t2 = 1−v2/c2 = (cid:1)1−v2/c2 (1.4)
and for small values of v/c one has
T v2 lv2
∆T ≡T(cid:2)−T (cid:3) (cid:3) ; (1.5)
2c2 c3
thisresultshowsthattheexperimentalapparatusisinprincipleabletoreveal
the motion of the laboratory with respect to ether.
If we assume to be able to reveal time differences δT as small as 1/20 of
the typical oscillationperiod of visible light (hence phase differences as small
as 2π/20), i.e. δT ∼5 10−17 s,(cid:1)and we take l =2m, so that l/c(cid:3)0.6 10−8s,
we obtain a sensitivity δv/c = δT c/l ∼ 10−4, showing that we are able to
reveal velocities with respect to ether as small as 3 104m/s, which roughly
corresponds to the orbital speed of Earth. If we compare the outcome of two
suchexperimentsseparatedbyanintervalof6months,correspondingtoEarth
velocities differing by approximately105m/s, we should be able to revealthe
motion of Earth with respect to ether. The experiment, repeated in several
differenttimesoftheyear,clearlyshowed,togetherwithothercomplementary
observations, that ether does not exist.
Starting from that observation, Einstein deduced that Galilean transfor-
mation laws between inertial reference frames:
