The Book of Physics: Volume 2

By Simone Malacrida

In this book, the great history of physics discoveries is traced, starting from the scientific revolution of Galileo and Newton to the physics of today and the near future.
The understanding of physics is approached both from a theoretical point of view, expounding the definitions of each particular field and the assumptions underlying each theory, and on a practical level, going on to solve more than 350 exercises related to physics problems of all sorts.
The approach to physics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science.
The entire book is divided into five distinct sections: classical physics, the scientific revolutions that took place in the early twentieth century, physics of the microcosm, physics of the macrocosm, and finally current problems that are the starting point for the physics of the future.
The paper stands as an all-encompassing work concerning physics, leaving out no aspect of the many facets it can take on.

• Language: English
• Publisher: Simone Malacrida
• Release date: Dec 27, 2022
• ISBN: 9798201732578

Simone Malacrida

Simone Malacrida (1977) Ha lavorato nel settore della ricerca (ottica e nanotecnologie) e, in seguito, in quello industriale-impiantistico, in particolare nel Power, nell'Oil&Gas e nelle infrastrutture. E' interessato a problematiche finanziarie ed energetiche. Ha pubblicato un primo ciclo di 21 libri principali (10 divulgativi e didattici e 11 romanzi) + 91 manuali didattici derivati. Un secondo ciclo, sempre di 21 libri, è in corso di elaborazione e sviluppo.

Book preview

The Book of Physics: Volume 2

SIMONE MALACRIDA

In this book, the great history of physics discoveries is traced, starting from the scientific revolution of Galileo and Newton to the physics of today and the near future.

The understanding of physics is approached both from a theoretical point of view, expounding the definitions of each particular field and the assumptions underlying each theory, and on a practical level, going on to solve more than 350 exercises related to physics problems of all sorts.

The approach to physics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science.

The entire book is divided into five distinct sections: classical physics, the scientific revolutions that took place in the early twentieth century, physics of the microcosm, physics of the macrocosm, and finally current problems that are the starting point for the physics of the future.

The paper stands as an all-encompassing work concerning physics, leaving out no aspect of the many facets it can take on.

ANALYTICAL INDEX

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PART TWO: THE REVOLUTIONS OF THE EARLY TWENTIETH CENTURY

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13 – QUANTUM PHYSICS

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14 – SPECIAL RELAVITY THEORY

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PART THREE: PHYSICS OF MICROCOSM

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15 – PHYSICS OF MATTER

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16 – CHEMICAL PHYSICS

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17 – QUANTUM FIELD THEORY

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18 – NUCLEAR PHYSICS

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19 – PHYSICS OF PARTICLES AND INTERACTIONS

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PART FOUR: PHYSICS OF MACROCOSM

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20 – THEORY OF GENERAL RELATIVITY

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21 - ASTRONOMY

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22 - ASTROPHYSICS

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23 - COSMOLOGY

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24 – PHYSICS OF BLACK HOLES

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PART FIVE: TODAY'S PROBLEMS AND TOMORROW'S PHYSICS

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25 – ATTEMPTS AT UNIFICATION

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26 – THE THEORY OF EVERYTHING

PARTE TWO: THE REVOLUTIONS OF THE EARLY TWENTIETH CENTURY

13

QUANTUM PHYSICS

The first revolutionary theory that we are going to explain concerns quantum physics which will be inextricably linked to the microcosm.

This theory will explain many of the phenomena that had triggered the crisis of classical physics, introducing new scientific horizons.

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Planck's solution for the black body spectrum

One of the main discrepancies that led to the overcoming of classical physics was the explanation of the black body spectrum.

According to the known scheme, the energy could assume any possible value and, therefore, the statistical distribution of the energy followed the well-known Boltzmann law, derived from classical thermodynamics:

This led to a distribution of the blackbody spectrum which was known as the Rayleigh-Jeans formula:

in full agreement with the experimental data for the infrared region, but not for the ultraviolet region, as already mentioned in the previous paragraph.

In 1900, Planck hypothesized that energy could not assume any possible continuous value, but only some discrete data from the following expression:

where n is a positive integer and h a constant defined as Planck's constant.

In doing so, the statistical distribution of the energy (averaged on the discrete sums and not on the continuous integrals) becomes:

and the spectral distribution of the black body took on a new form, in full agreement with the experimental data, also in the ultraviolet region.

The logical passage planned by Planck was of extraordinary importance.

For the first time it was admitted that energy, or any physical entity, was a discrete quantity and could not assume any value.

Planck introduced the concept of discrete energy to match the theory with the experimental data regarding the black body spectrum and called these allowed values of energy as quanta. From then on, the resulting theory assumed the term quantum physics and the adjective quantum was used as a qualifier of each part of this theory.

The spectrum of the black body was therefore explained in this new vision, but all the other problems were not and, moreover, there was no overarching theory that provided for all these empirical results.

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Einstein's solution for the photoelectric effect

In 1905 (remarkably the same year as the publication of the special theory of relativity), Einstein proposed a solution to explain the phenomenology of the photoelectric effect.

Einstein accepted Planck's hypothesis and applied it to the photoelectric effect.

The energy of an electromagnetic wave depended only on the frequency.

The photoelectric effect described by Hertz's experiments found an easy explanation if one accepted the hypothesis of a quantized energy dependent only on the frequency of the electromagnetic wave.

This is why below a certain frequency, there was no emission of electrons, since there was not enough energy to stimulate this emission and this also explained why the energy of the emitted electrons was proportional to the frequency.

Einstein called the quanta of light, and of electromagnetic waves in general, with the name of photons.

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The Bohr model

Planck's hypothesis had somehow explained the two inconsistencies relating to the black body spectrum and the photoelectric effect.

The question of the stability of matter and of giving a general explanation for why energy was a discrete and non-continuous quantity remained open.

In 1913 Bohr proposed a first model of atom which followed the rules of quantum physics, but had to introduce postulates to explain the stability of matter.

Inspired by Rutherford's experiments, he understood that the negatively charged electron revolved around a positively charged atomic nucleus, and introduced some variations to the previous atomic model.

First of all, he also quantized the angular momentum of an electron revolving around the nucleus by introducing a direct dependence with Planck's constant, as he did years earlier for energy (the quantization rules were later extended and completed by Sommerfeld in 1916).

In doing so, we began to understand how quantization was a much more widespread process than Planck's relation implied.

Later, he postulated that an electron revolved around the nucleus in predefined (quantized) orbits without emitting electromagnetic radiation (all to explain the stability of the atom).

The emission of electromagnetic radiation occurs only when the electron jumps from one orbit to another and the emitted (or absorbed) energy respects both the Planck relation and the principle of conservation of energy.

The radii of the stable orbits are also quantized and related to the principal quantum number and atomic number as follows:

The second fraction is exactly the radius of the fundamental level of hydrogen, the simplest atom of all being formed by a single electron and a single proton.

The energy of these stable orbits was given by

which for n=1 corresponds exactly to the energy of the first bound state of hydrogen.

Bohr's atom represents the first systematic attempt to reconcile the new quantum theory with what has been experimentally found in other disciplines, such as electromagnetism and chemistry, but it had the defect of having to postulate certain assumptions to explain the stability of matter and was not in agreement with what was measured for atoms other than that of hydrogen.

Furthermore, the dualism between wave and particle, which had become so evident since the publication of Maxwell's equations, was not explained.

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New discoveries: Compton effect

A further step towards a new general theory was obtained in 1920 with the explanation of the Compton effect.

Considering the X-rays scattered by the electrons and combining Planck's energy equation with that of Einstein's energy for special relativity explained the experimental evidence that the wavelength variation depended on the angle of incidence according to the following formula:

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De Broglie's solution for the wave-particle duality

Just the comparison between two energy equations, that of quantum physics and that of special relativity, led to the last piece necessary to overcome those problems mentioned previously.

In 1924, De Broglie set one of those milestones intended to completely subvert concepts hitherto considered separate.

Starting from these four equations (the first is the energy equation according to special relativity, the second is Planck's relation, the third is the definition of speed of light according to Maxwell's equations, the fourth is the definition of momentum ):

obtained with simple mathematical steps the following relationship:

This relation connects a wave quantity, such as wavelength, with a material quantity, such as momentum, by saying that their product is equal to a constant.

De Broglie intuited that this relationship was the fundamental basis for overcoming the eternal dualism between the wave nature and the corpuscular nature of physical entities, simply stating that each of them is, at the same time, both wave and particle and posing that dualism not as a problem , but as a new frontier.

Through this relationship, the wavelength of the electron was calculated, which therefore was not only a particle, but also a wave.

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Young and the two fissures

The English scientist Young had carried out, back in 1801, experiments on light to demonstrate its wave nature.

Scientists understood how this experimental apparatus could be useful for confirming, or not, the wave-particle dualism.

Take a weak source of light and a photographic plate.

Between them, place an opaque barrier with two parallel slits.

Construct a similar experimental setup in which the weak source of light is replaced by a weak source of electrons.

If the sources emit one photon (or one electron) at a time, the plate is impressed with single points of light, so the photons and electrons behave like particles.

If, on the other hand, the number of photons (or electrons) emitted is increased, the plate shows the classic interference fringes typical of the corpuscular nature.

Furthermore, and this is the most shocking aspect, although photons and electrons behave like particles if emitted individually, it is not possible to determine which of the two slits they passed through.

The dual nature is present in an intrinsic way, i.e. it is not possible to separate a single behavior of this dualism.

Unbeknownst to him (the photographic plates of the nineteenth century were in fact insensitive to weak beams of light), Young had devised an experiment that could have resolved the wave-particle dualism a good 125 years earlier!

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Quantum mechanics according to Schrodinger

All these experimental and theoretical evidences, which followed one another over twenty years, needed a general explanation that included them all, just as in the 19th century Maxwell's equations incorporated the experiences of Volta, Ampére, Oersted and Faraday.

It was De Broglie's report that gave the final impetus to the quantum arguments.

In 1926, with four different articles, Schrodinger showed that De Broglie's wave mechanics satisfied Bohr's quantization rules and following the parallelism between optics and mechanics (i.e. between wave and corpuscular nature) he established a new equation which became the basis of mechanics quantum.

Newton's mechanics became an approximation of quantum mechanics for large energies and for much larger spatial scales than the wavelength established by De Broglie's relation.

The new equation derived naturally from Newton's mechanics by simply applying the De Broglie relation and the following correspondence rules (let's consider the one-dimensional case, at least for now):

Where is it:

Instead of continuous quantities such as E and p, discrete operators were introduced, in full accordance with the quantization procedure.

Schrodinger's equation thus assumes this general form (for multidimensional cases, just think of dependencies also on the y and z coordinates):

This equation reveals multiple aspects that explain almost all the new properties of quantum mechanics.

The solutions of this equation are wave functions, a name given by Schrodinger himself to recall the basis of wave mechanics.

1) First of all, a generic potential V(x) appears in this equation.

Depending on the shape of this potential (step, hole, harmonic oscillator and so on) there are different solutions to this equation.

2) Secondly, there are strong similarities between this equation and what is derived from Maxwell's equations, under appropriate rewrites. Therefore, simple correspondences can be made and a sort of parallel numerical calculation can be established, always keeping in mind the large basic differences (continuous quantities on the one hand, discrete quantities on the other).

3) The third observation concerns the time factor which is a pure phase factor. This observation, together with the fact that the second side of the equation is itself a complex number, makes for a huge difference from Maxwell's equations.

In the case where the wave functions can be expressed in this form

Schrodinger's equation takes on a simplified form, relating to stationary states:

which is an equation with eigenvalues, given by energy, while u(x) are the eigenfunctions.

Schrodinger's equation is therefore an energy equation.

Energy can only assume predefined values, in other words this equation provides for the quantization of energy and this is a first result in favor of it.

We will see shortly how the predictions coincide with the experimental verifications.

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The probabilistic view

Before continuing, a necessary clarification should be made.

To the question what does the wave function represent?, quantum mechanics can only give this answer the solution of Schrodinger's equation.

Put another way, there is no correspondence between the wave function and an observable physical quantity.

By itself, the wave function does not represent anything.

This will be one of the philosophical problems that we will explain at the end of this chapter.

The real novelty of quantum mechanics, however, was given by the fact that the square module of the wave function represents the probability of finding the wave/particle in a given place at a given time.

The evolution from a deterministic to a probabilistic mechanics shed new light on physics itself.

Atomic physics, the basis of all other sectors given that the atom is the constituent basis of matter, foresaw that it is not possible to state with certainty where a given particle is, but only to establish its probability.

The probabilistic interpretation of Schrodinger's equation was given only one year after 1926, by Born.

With this clarification and by studying Schrodinger's equations as the potentials V(x) varied, the knowledge of classical physics was extended, reaching new scientific horizons.

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The innovations compared to the classical mechanics

A first point was the prediction that the wave function could also extend to areas that, instead, classical physics considered forbidden.

In the case of the potential step, for example, quantum mechanics predicts that the wave/particle can overcome the step even if the associated energy is lower, which is impossible for classical physics.

This effect, known as the tunnel effect, underlies much of the way modern computers work, such as computers and cell phones. Indeed, quantum physics was the forerunner of many fields such as solid state physics, matter, semiconductors and nanotechnology.

Likewise, within the classically permitted area, there are particular points in which the probability of finding the wave/particle is null.

A second point is the verification that the energy can only assume discrete values below some thresholds, for example the aforementioned potential step, while it becomes continuous spectrum above them.

A third point is given by the zero-point energy.

From Schrodinger's equation we can see how the lowest energy solution is never zero, but a multiple of ½ hf which is precisely defined as zero point energy, ie the minimum possible. Planck's equation must therefore be modified in this sense (with n positive integer):

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The solutions

Considering Schrodinger's equation in spherical coordinates and carrying out the solution for the radial part, we find as solutions the functions u(r) given by the known Laguerre polynomials, the first of which is the following:

where the subscript 10 refers to the two discrete numbers used to identify this polynomial.

The first subscript is precisely n, the principal quantum number already introduced by Bohr, while the second subscript l accounts for the shape (spherical if it is equal to zero, as in this example) and can vary only for positive integers less than no.

In essence, the first Laguerre polynomial as expressed above is the radial part of the wave function referred to the ground state of the hydrogen atom.

Rereading it in a probabilistic key, the square module of this function is the probability of finding the electron in the hydrogen atom.

It is clearly seen how the probability is zero near the atomic nucleus (r=0) while the probability of finding the electron somewhere is equal to the certain event given that, for a suitable A, the following relationship holds:

Quantum mechanics therefore explains why the electron does not fall towards the atomic nucleus under the force of attraction of Lorentz and also predicts that there are no fixed orbits, given that classical determinism is not applicable, in favor of quantum probabilism.

The name given to these areas of probability of finding the electron is that of orbital.

The quantum number l therefore gives the shape of the orbitals based on the probability of finding or not the electron in that specific area.

For the first bound state of hydrogen, it is easy to verify that the maximum probability of finding the electron occurs precisely in the case of the Bohr radius and that, at that radius, the bond energy is in the stable state, i.e. the least energy principle.

Unlike wave mechanics, Schrodinger's equation explains very well even the most complex atoms and not just hydrogen.

Also, with the definition of the orbitals, comes an easy theoretical understanding of the properties of the periodic table and the octet rule.

Atomic physics described by quantum mechanics encompasses a good part of the experiments of physical-chemistry and physics of matter, in particular atomic and molecular spectra, especially after what we are going to say shortly.

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Operator evolution and uncertainty principles

Quantum mechanics can also be expressed in an operator form, recalling the Hamilton relations of classical mechanics and applying them to the quantum case.

Newton's laws could be written in this elegant way:

with p and q continuous observables (momentum and position) also called canonical operators, while H was the (continuous) Hamiltonian function defined as:

A fundamental achievement of classical mechanics was the commutation of the canonical operators; in other words qp-pq=0.

Applying the correspondence rules mentioned for energy and momentum, in quantum mechanics the canonical operators were associated with the discrete operators as follows:

While the Hamiltonian function took the form of a discrete operator called Hamiltonian:

With this symbolism, the general Schrodinger equation and that for stationary states simply become the following:

In quantum mechanics, the canonical operators do not commute. In fact, there is this relationship:

Which is a direct consequence of (and which also explains) Heisenberg's uncertainty principle, enunciated only a few years after 1926.

In particular, Heisenberg stated that any physical quantity not commuting with another one underwent the following inequality:

where [A,B] is the commutator defined as AB-BA while it is a generic discrete operator and the symbolism is that of the bra-ket used by Dirac (which we will find again shortly in this description).

In the case of canonical operators, this inequality reduces to the well-known formulation of the uncertainty principle:

This inequality states that it is not possible to determine, with absolute precision and at the same time, the position and velocity of a particular particle.

If we wanted to know the position of an electron through an experiment with suitable meters, we could not say anything about its speed and vice versa.

This affirmation, absolutely valid, loses its meaning in the macroscopic world, when the distances are much higher than the wavelength, but it is of fundamental importance in the atomic and nuclear world.

Furthermore, two new concepts in physics were introduced.

The first is that of indeterminism.

Not only did quantum mechanics bring physics from an absolute vision to a probabilistic one, but a further perturbation was introduced given by the indeterminacy of the physical variables.

This also caused disruptive effects on a philosophical level, in the same way as relativity had precisely relativized previously absolute concepts, such as space and time.

However, the real focal point was given by the very concept of measurement and the role of the observer.

It was clear that the experiment itself was going to change the state of a physical quantity (hereafter called observable) and that nothing could be said about the value of that observable before and after the experiment.

Thus arose a very evident discrepancy between physical reality and observed reality and the measurement itself was a way of revealing the observables.

This physical and philosophical problem of quantum mechanics still remains open.

Two other observables that do not commute are energy and time, for which:

There is therefore a limit to the minimum value of energy spacing and this minimum coincides precisely with the zero point energy.

Similarly, time pulses cannot be discerned below this quantum limit and this can be found, for example, in lasers.

To explain the atomic spectra it was necessary to resort to the quantization of the angular momentum, introducing a new quantum number which can assume integer values from –la +l.

Furthermore, quantum mechanics foresaw a new quantity linked to the total momentum, which was given the name of spin which was in no way comparable to the classic angular momentum.

The spin explained many practical findings, including the octet rule and the occupation of electronic levels, and also accounted for further differences in atomic spectra.

Associated with spin was the introduction of the last quantum number.

The quantization rules of operator quantum mechanics are therefore the following, with the relative eigenvalues, eigenfunctions and quantum numbers and generalize Sommerfeld's rules:

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University-level exercises

Exercise 1

Consider the family of states:

And the one-dimensional Hamiltonian: