Introduction To The Basic Concept Of Modern Physics

During the last years of the Nineteenth Century, the development of new techniques 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.
Several experimental results lead to the birth of the new physics. A brief list
of the most important experiments must contain those performed by H. Hertz
about the photoelectric effect, the measurement of the distribution in frequency of the radiation emitted by an ideal oven (the so-called black body radiation), the measurement of specific heats at low temperatures, which showed
violations of the Dulong–Petit law and contradicted the general applicability
of the equi-partition of energy. Furthermore we have to mention the discovery 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.
From a 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 discovery of the electron and of Rutherford’s studies about atomic structure, 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 statistical theory of matter.
Modern Physics, which is the subject of this notes, is well distinct from
Classical Physics, developed during the XIX century, and from Contemporary
Physics, which was started during the Thirties (of XX century) and deals with
the nature of Fundamental Interactions and 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,account of the main features of Special Relativity, of Quantum 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
follows from their strict interrelations (think for instance of the role played by
special relativity in the hypothesis of de Broglie’s waves or of that of statistical
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 allow a full understanding of the contents and of the consequences of
the new theories.
On the other hand, since the present text is aimed to 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.
That imposes a selection of the topics which are most qualified from the point
of view of the physical content/mathematical formalism ratio.
In the context of special relativity we have given up presenting the covariant formulation of electrodynamics, limiting therefore ourselves to justifying
the conservation of energy and momentum and to developing relativistic kinematics with its quite relevant physical consequences. A mathematical discussion about quadrivectors has been confined to a short appendix.
Regarding Schr¨odinger 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,
we have emphasized its qualitative consequences on energy levels, giving up a
detailed discussion of atomic spectra beyond the simple Bohr model. Therefore
the main analysis has been limited to one-dimensional problems, where we
have examined the origin of discrete energy levels and of band spectra as well
as the tunnel effect. Extensions to more than one dimension have been limited
to very simple cases in which the Schr¨odinger equation is easily separable, like
the three-dimensional harmonic 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
main lines leading to the solution of the three-dimensional motion in a central
potential, hence in particular of the hydrogen atom spectrum.
Going to the last subject, which we have discussed, as usual, on the basis of
Gibbs’ construction of the statistical ensemble and of the related distribution,
we have chosen to consider those cases which are more meaningful from the
point of view of quantum effects, like degenerate gasses, focusing in particular
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 inclusion of significant collections of problems associated with each chapter has been essential. We have possibly tried
to avoid mixing problems with text complements: while moving some relevant topics to the exercise collection may be tempting in order to streamline
the general presentation, it has the bad consequence of leading to excessively
long exercises which dissuade the average student from trying to give an answer before looking at the suggested solution scheme. On the other hand, we
have tried to limit the number of those (however necessary) exercises involving 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 technically 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 divided 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
Schr¨odinger 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