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CRC Press eBooks

2 Quantum Mechanics

Anthony C. Fischer-Cripps

Published Aug 14, 2014• 0 citations
Publisher

Abstract

In 1897, Thomson demonstrated that cathode rays (observed to be emitted from the cathodes of vacuum tubes) were in fact charged particles which he called electrons. Thomson proposed that the atom consisted of a positively charged sphere in which were embedded negatively charged electrons.
DOI
10.1201/b17357-6

Related papers

2 Quantum mechanics

Anthony C. Fischer-Cripps

In 1897, Thomson demonstrated that cathode rays (observed to be emitted from the cathodes of vacuum tubes) were in fact charged particles which he called electrons. Thomson proposed that the atom consisted of a positively charged sphere in which were embedded negatively charged electrons. Rutherford subsequently found in 1911 that the electrons orbited at some distance from a central positively charged nucleus. Rutherford proposed that electrostatic attraction between the nucleus and the electron was balanced by the centrifugal force arising from the orbital motion. However, if this were the case, then the electrons (being accelerated inwards towards the center of rotation) would continuously radiate all their energy as electromagnetic waves and very quickly fall into the nucleus. In 1913, Bohr postulated two important additions to Rutherford’s theory of atomic structure: 1. Electrons can orbit the nucleus in what are called stationary states in which no emission of radiation occurs and in which the angular momentum L is constrained to have values: 2. Electrons can make transitions from one state to another accompanied by the emission or absorption of a single photon of energy E = hf thus leading to absorption and emission spectra.

CRC Press eBooks•Aug 22, 2007•0 citations

Historical Background

M. Srirama Rao

Newtonian mechanics or classical mechanics in its simplest form, known as the laws of mechanics is written in terms of particle trajectories. The trajectory underlies the structure of classical physics and the particle underlies the model of physical reality. Newtonian mechanics has taken such strong roots and everybody believed that everything in this universe can be explained on the basis of these laws. In 1897, the British Physicist J. J. Thompson demonstrated that the beam that leaves the cathode, the so-called cathode rays, consists of a beam of negatively charged discrete particles. Wien equation gives excellent agreement with experiment in the region of short wavelengths and the Rayleigh-Jeans equation appears to be asymptotically correct at long wavelengths. In 1887, Hertz observed photoelectric effect and the first outstanding application of quantum theory was in its explanation by Albert Einstein in 1905.

CRC Press eBooks•Oct 13, 2022•0 citations

Quantum mechanical free electron model

Brian K Tanner

The years of the third decade of the present century were heady times. Old social orders were being swept away, monarchies becoming republics and the United States of America was emerging as a world power. In physics too, revolutions were taking place, none more potent than that which resulted in the emergence of quantum mechanics as a model for the behaviour of sub-atomic particles. It was Newton who first suggested that light was particulate but during the nineteenth century this view had fallen into neglect. Indeed a number of experiments, for example the classic Young's slits experiment, demonstrated quite conclusively that light was a wave motion. How else could interference fringes occur? However, the experiments on the photoelectric effect demonstrated just as conclusively that light energy was carried in packets, or quanta, and that a continuous wave description was not applicable. This ‘wave or particle’ dilemma was not unique to the behaviour of light. J. J. Thompson showed that cathode rays were charged, had a well defined mass and had all the properties expected of a beam of particles. Nevertheless, Davisson and Germer showed that diffraction of electrons could take place, an effect made visually much more dramatic by G. P. Thompson's transmission electron diffraction patterns. Nowadays, electron diffraction is used as a routine analytical tool in all advanced metallurgical and materials science laboratories (Fig. 2.1). Clearly new axioms were required.

Cambridge University Press eBooks•Mar 30, 1995•0 citations

Measuring the de Broglie wavelength of electrons using Thomson e/m tube

Swarnava Mitra

The Thomson e/m experiment laid the foundation of the first atomic particle and for once proved the existence of discrete charged particles, now known as electrons. This work aims at studying the wave nature of electrons by measuring their de Broglie wavelengths. The wavelength was obtained from two different facets, namely, the electric field and the magnetic field, and then it has been shown that the two values are in agreement with each other to an order of 10−11 m.

•Nov 20, 2021•0 citations

Quantum Mechanics Basic Concepts

Lukong Cornelius Fai

Quantum mechanics is the theory of physical phenomena at the atomic and subatomic levels. Historically, quantum mechanics began and took the modern form in the first quarter of the 20th century. The appearance of quantum mechanics was due to the fact that at the end of the last and beginning of our century, there arose a series of experimental facts that could not only get explanations in classical physics but were in contradiction with it: Planetary model [ 1–4 ] : The existence of the planetary atom itself (Rutherford-Bohr model presented by Niels Bohr and Ernest Rutherford in 1913) was in contradiction with classical mechanics and electrodynamics. After the experiments of Rutherford on the planetary model of the atom (the electron revolves about the nucleus), it was necessary to examine the experimental facts on the one hand and theory on the other. From classical mechanics, an electron moving in a definite orbit around the nucleus is constantly accelerated. Knowledge of a classical charged particle shows that when it is accelerated, it emits radiation. From electrodynamics, it is well known that any charged particle with an accelerated motion should emit an electromagnetic wave. So an electron moving around the nucleus would continuously emit radiation and consequently become less energetic. This is because as the electron emits the electromagnetic wave, it loses energy and, consequently, its kinetic energy as well as its velocity reduces. Consequently, from the laws of classical physics and electrodynamics, the radius of the orbit would constantly shrink in size until the electron falls into the atomic nucleus. But this does not take place. Blackbody radiation [ 5–8 ] : Theoretical calculations of blackbody radiation spectrum assume electromagnetic radiation to be a wave and so can exchange energy in any continuous amount and yields a result that does not match with experimental facts. The assumption of Planck shows that electromagnetic radiation of frequency https://www.w3.org/1998/Math/MathML"> ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to exchange energy only in units of https://www.w3.org/1998/Math/MathML"> ℏ ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Planck's calculations agree very well with experimental facts. The Planck constant has the value: 1.1 https://www.w3.org/1998/Math/MathML"> h = 2 π ℏ = 2 π × 1.054 × 10 - 27 e r g - s e c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> Photo-electric effect : The release of electrons from a metal by irradiating it with electromagnetic waves or light is called photo-electric effect [ 9–11 ] . To free an electron from a metal, the light radiation should have a frequency greater than a critical frequency, a characteristic of the metal, while light radiation frequency lower than this critical frequency would not lead to the photo-electric effect. This was shown by Einstein to be consistent with Planck's hypothesis, considering light with frequency https://www.w3.org/1998/Math/MathML"> ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to only exchange energy in the amount https://www.w3.org/1998/Math/MathML"> h ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In order for the electrons to be released from the metal, the following energy is needed: 1.2 https://www.w3.org/1998/Math/MathML"> Ε = h ν = h ν 0 + Τ k i n , Τ k i n = h ν - ν 0 ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> 4 Here, https://www.w3.org/1998/Math/MathML"> h ν 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_7.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> is the binding energy for the given metal and https://www.w3.org/1998/Math/MathML"> Τ k i n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_8.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> is the kinetic energy of the electrons. From the above relation, it is not possible to have any emission of the electrons unless 1.3 https://www.w3.org/1998/Math/MathML"> ν ≥ ν 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_9.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> From the blackbody radiation and photo-electric effect, it is seen that though classically light is a wave, it has a particle nature which is confirmed by the Compton effect. All these can only be explained by quantum mechanics. Compton effect : The scattering of light by an electron can only be explained if a photon of frequency https://www.w3.org/1998/Math/MathML"> ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_10.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> is considered as a particle moving with energy, https://www.w3.org/1998/Math/MathML"> Ε https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_11.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , and momentum, https://www.w3.org/1998/Math/MathML"> p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_12.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : 1.4 https://www.w3.org/1998/Math/MathML"> Ε = h ν , p = p → = Ε c = h ν c = h λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_13.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> Here, https://www.w3.org/1998/Math/MathML"> c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_14.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> and https://www.w3.org/1998/Math/MathML"> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_15.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> are, respectively, the speed and wavelength of light wave. From relativistic mechanics, mass and energy may be considered as a manifestation of the same thing. The energy of a particle can be expressed through its momentum as: 1.5 https://www.w3.org/1998/Math/MathML"> Ε = p 2 c 2 + m 2 c 4 , p → = m v → v → = Ε v → c 2 , m v → = m 1 - v 2 c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_16.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> Here, https://www.w3.org/1998/Math/MathML"> m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_17.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> with https://www.w3.org/1998/Math/MathML"> v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_18.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> are, respectively, the rest mass and velocity of the particle. For the case of the photon, which is a particle moving with the speed of light, https://www.w3.org/1998/Math/MathML"> c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_19.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , if it equals https://www.w3.org/1998/Math/MathML"> 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_20.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , we have https://www.w3.org/1998/Math/MathML"> m v → = 0 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_21.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . The photon momentum has the value 1.6 https://www.w3.org/1998/Math/MathML"> p = Ε c c 2 = Ε c = h λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_22.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> So if we consider a photon as a particle, its rest mass should vanish (see details in references [ 12–14 ]). It is obvious from here that electromagnetic radiation has the property of dualism, i.e., it possesses the corpuscular and wave nature. Davisson - Germer experiment : The notion of the trajectory does not coincide with the experimental fact of the diffraction of an electron. The diffraction of the electron was theoretically foreseen by Louis de Broglie in 1924 (wave character of material particles) and experimentally founded by the American physicist Clinton J. Davisson [ 2 , 4 , 15 ] and the Austrian physicist Erwin Schrödinger (1887–1961) in 1927. Davisson and Germer in 1925 and also the British physicist G. P. Thomson (son of J. J. Thomson, discoverer of the electron) in 1926 showed scattered electrons from crystals and found diffraction patterns consistent with the interference of electrons having the de Broglie wavelength and similar to light interacting with a diffraction grating. G. P. Thomson [ 16 ] found diffraction rings produced by electrons passing through thin foils. Considering an electronic beam incident on a crystal lattice, we observe a diffraction pattern. As diffraction is a wave phenomenon, the electron having a corpuscular nature sometimes has a wave nature. Such observations lead to a general conclusion that all objects have both corpuscular and wave nature. The question on the wavelength associated with the particle was given by Louis de Broglie's hypothesis through the relation: 1.7 https://www.w3.org/1998/Math/MathML"> λ = h p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_23.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> It is instructive to note that experimentation on atomic systems shows that measured quantities achieve quantized values unlike classical mechanics’ predictions, where observables achieve continuous values. 5 Experimentation on dualism : For the dual nature of the particles, sometimes the particles have a corpuscular as well as a wave nature. Classically, particles follow definite trajectories and so do not show interference, whereas waves show interference. These and other experimental facts gave birth to quantum mechanics.

CRC Press eBooks•Apr 19, 2022•0 citations

The problem of the structure of matter

Jorge José Kohanoff

The description of the physical and chemical properties of matter is a central issue that has occupied the minds of scientists since the age of the ancient Greeks. In their route to dissect matter down to what cannot be divided any further, they coined the term atom , the indivisible . Matter became then a collection of atoms. More than twenty centuries had to pass until the development of a more precise concept of atom, thanks, amongst others, to the systematic studies of Mendeleyev and the establishment in 1869 of the periodic table of the elements (Mendeleyev, 1869). The discovery of the electron in 1897 and the first modern model of the atomic structure by Sir Joseph Thomson were soon refined by his student, Sir Ernest Rutherford, who in 1910 showed that an atom was made of a positively charged small nucleus and a number of negatively charged electrons that neutralize the nuclear charge. Much in the spirit of planetary systems, and drawing from the analogy between gravitational and electrostatic interactions, scientists in the beginning of the twentieth century built an image of the atom that consisted of a number Z of electrons – of elementary charge – e – orbiting around the nucleus of charge Ze . A number of experimental observations, though, were incompatible with this idea of orbiting electrons. In particular, according to the successful electromagnetic theory, charged electrons in orbital (radially accelerated) motion should radiate energy, thus decelerating and eventually collapsing onto the nucleus. Clearly, such a picture would imply that matter is essentially unstable, in flagrant contradiction with our everyday experience of the very existence of matter.

Cambridge University Press eBooks•Jun 29, 2006•0 citations
In 1897, Thomson demonstrated that cathode rays (observed to be emitted from the cathodes of vacuum tubes) were in fact charged particles which he called electrons. Thomson proposed that the atom consisted of a positively charged sphere in which were embedded negatively charged electrons. Rutherford subsequently found in 1911 that the electrons orbited at some distance from a central positively charged nucleus. Rutherford proposed that electrostatic attraction between the nucleus and the electron was balanced by the centrifugal force arising from the orbital motion. However, if this were the case, then the electrons (being accelerated inwards towards the center of rotation) would continuously radiate all their energy as electromagnetic waves and very quickly fall into the nucleus. In 1913, Bohr postulated two important additions to Rutherford’s theory of atomic structure: 1. Electrons can orbit the nucleus in what are called stationary states in which no emission of radiation occurs and in which the angular momentum L is constrained to have values: 2. Electrons can make transitions from one state to another accompanied by the emission or absorption of a single photon of energy E = hf thus leading to absorption and emission spectra.
Newtonian mechanics or classical mechanics in its simplest form, known as the laws of mechanics is written in terms of particle trajectories. The trajectory underlies the structure of classical physics and the particle underlies the model of physical reality. Newtonian mechanics has taken such strong roots and everybody believed that everything in this universe can be explained on the basis of these laws. In 1897, the British Physicist J. J. Thompson demonstrated that the beam that leaves the cathode, the so-called cathode rays, consists of a beam of negatively charged discrete particles. Wien equation gives excellent agreement with experiment in the region of short wavelengths and the Rayleigh-Jeans equation appears to be asymptotically correct at long wavelengths. In 1887, Hertz observed photoelectric effect and the first outstanding application of quantum theory was in its explanation by Albert Einstein in 1905.
The years of the third decade of the present century were heady times. Old social orders were being swept away, monarchies becoming republics and the United States of America was emerging as a world power. In physics too, revolutions were taking place, none more potent than that which resulted in the emergence of quantum mechanics as a model for the behaviour of sub-atomic particles. It was Newton who first suggested that light was particulate but during the nineteenth century this view had fallen into neglect. Indeed a number of experiments, for example the classic Young's slits experiment, demonstrated quite conclusively that light was a wave motion. How else could interference fringes occur? However, the experiments on the photoelectric effect demonstrated just as conclusively that light energy was carried in packets, or quanta, and that a continuous wave description was not applicable. This ‘wave or particle’ dilemma was not unique to the behaviour of light. J. J. Thompson showed that cathode rays were charged, had a well defined mass and had all the properties expected of a beam of particles. Nevertheless, Davisson and Germer showed that diffraction of electrons could take place, an effect made visually much more dramatic by G. P. Thompson's transmission electron diffraction patterns. Nowadays, electron diffraction is used as a routine analytical tool in all advanced metallurgical and materials science laboratories (Fig. 2.1). Clearly new axioms were required.
Quantum mechanics is the theory of physical phenomena at the atomic and subatomic levels. Historically, quantum mechanics began and took the modern form in the first quarter of the 20th century. The appearance of quantum mechanics was due to the fact that at the end of the last and beginning of our century, there arose a series of experimental facts that could not only get explanations in classical physics but were in contradiction with it: Planetary model [ 1–4 ] : The existence of the planetary atom itself (Rutherford-Bohr model presented by Niels Bohr and Ernest Rutherford in 1913) was in contradiction with classical mechanics and electrodynamics. After the experiments of Rutherford on the planetary model of the atom (the electron revolves about the nucleus), it was necessary to examine the experimental facts on the one hand and theory on the other. From classical mechanics, an electron moving in a definite orbit around the nucleus is constantly accelerated. Knowledge of a classical charged particle shows that when it is accelerated, it emits radiation. From electrodynamics, it is well known that any charged particle with an accelerated motion should emit an electromagnetic wave. So an electron moving around the nucleus would continuously emit radiation and consequently become less energetic. This is because as the electron emits the electromagnetic wave, it loses energy and, consequently, its kinetic energy as well as its velocity reduces. Consequently, from the laws of classical physics and electrodynamics, the radius of the orbit would constantly shrink in size until the electron falls into the atomic nucleus. But this does not take place. Blackbody radiation [ 5–8 ] : Theoretical calculations of blackbody radiation spectrum assume electromagnetic radiation to be a wave and so can exchange energy in any continuous amount and yields a result that does not match with experimental facts. The assumption of Planck shows that electromagnetic radiation of frequency https://www.w3.org/1998/Math/MathML"> ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to exchange energy only in units of https://www.w3.org/1998/Math/MathML"> ℏ ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Planck's calculations agree very well with experimental facts. The Planck constant has the value: 1.1 https://www.w3.org/1998/Math/MathML"> h = 2 π ℏ = 2 π × 1.054 × 10 - 27 e r g - s e c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> Photo-electric effect : The release of electrons from a metal by irradiating it with electromagnetic waves or light is called photo-electric effect [ 9–11 ] . To free an electron from a metal, the light radiation should have a frequency greater than a critical frequency, a characteristic of the metal, while light radiation frequency lower than this critical frequency would not lead to the photo-electric effect. This was shown by Einstein to be consistent with Planck's hypothesis, considering light with frequency https://www.w3.org/1998/Math/MathML"> ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to only exchange energy in the amount https://www.w3.org/1998/Math/MathML"> h ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In order for the electrons to be released from the metal, the following energy is needed: 1.2 https://www.w3.org/1998/Math/MathML"> Ε = h ν = h ν 0 + Τ k i n , Τ k i n = h ν - ν 0 ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> 4 Here, https://www.w3.org/1998/Math/MathML"> h ν 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_7.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> is the binding energy for the given metal and https://www.w3.org/1998/Math/MathML"> Τ k i n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_8.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> is the kinetic energy of the electrons. From the above relation, it is not possible to have any emission of the electrons unless 1.3 https://www.w3.org/1998/Math/MathML"> ν ≥ ν 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_9.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> From the blackbody radiation and photo-electric effect, it is seen that though classically light is a wave, it has a particle nature which is confirmed by the Compton effect. All these can only be explained by quantum mechanics. Compton effect : The scattering of light by an electron can only be explained if a photon of frequency https://www.w3.org/1998/Math/MathML"> ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_10.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> is considered as a particle moving with energy, https://www.w3.org/1998/Math/MathML"> Ε https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_11.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , and momentum, https://www.w3.org/1998/Math/MathML"> p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_12.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : 1.4 https://www.w3.org/1998/Math/MathML"> Ε = h ν , p = p → = Ε c = h ν c = h λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_13.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> Here, https://www.w3.org/1998/Math/MathML"> c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_14.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> and https://www.w3.org/1998/Math/MathML"> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_15.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> are, respectively, the speed and wavelength of light wave. From relativistic mechanics, mass and energy may be considered as a manifestation of the same thing. The energy of a particle can be expressed through its momentum as: 1.5 https://www.w3.org/1998/Math/MathML"> Ε = p 2 c 2 + m 2 c 4 , p → = m v → v → = Ε v → c 2 , m v → = m 1 - v 2 c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_16.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> Here, https://www.w3.org/1998/Math/MathML"> m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_17.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> with https://www.w3.org/1998/Math/MathML"> v → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_18.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> are, respectively, the rest mass and velocity of the particle. For the case of the photon, which is a particle moving with the speed of light, https://www.w3.org/1998/Math/MathML"> c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_19.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , if it equals https://www.w3.org/1998/Math/MathML"> 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_20.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , we have https://www.w3.org/1998/Math/MathML"> m v → = 0 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_21.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . The photon momentum has the value 1.6 https://www.w3.org/1998/Math/MathML"> p = Ε c c 2 = Ε c = h λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_22.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> So if we consider a photon as a particle, its rest mass should vanish (see details in references [ 12–14 ]). It is obvious from here that electromagnetic radiation has the property of dualism, i.e., it possesses the corpuscular and wave nature. Davisson - Germer experiment : The notion of the trajectory does not coincide with the experimental fact of the diffraction of an electron. The diffraction of the electron was theoretically foreseen by Louis de Broglie in 1924 (wave character of material particles) and experimentally founded by the American physicist Clinton J. Davisson [ 2 , 4 , 15 ] and the Austrian physicist Erwin Schrödinger (1887–1961) in 1927. Davisson and Germer in 1925 and also the British physicist G. P. Thomson (son of J. J. Thomson, discoverer of the electron) in 1926 showed scattered electrons from crystals and found diffraction patterns consistent with the interference of electrons having the de Broglie wavelength and similar to light interacting with a diffraction grating. G. P. Thomson [ 16 ] found diffraction rings produced by electrons passing through thin foils. Considering an electronic beam incident on a crystal lattice, we observe a diffraction pattern. As diffraction is a wave phenomenon, the electron having a corpuscular nature sometimes has a wave nature. Such observations lead to a general conclusion that all objects have both corpuscular and wave nature. The question on the wavelength associated with the particle was given by Louis de Broglie's hypothesis through the relation: 1.7 https://www.w3.org/1998/Math/MathML"> λ = h p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math1_23.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> It is instructive to note that experimentation on atomic systems shows that measured quantities achieve quantized values unlike classical mechanics’ predictions, where observables achieve continuous values. 5 Experimentation on dualism : For the dual nature of the particles, sometimes the particles have a corpuscular as well as a wave nature. Classically, particles follow definite trajectories and so do not show interference, whereas waves show interference. These and other experimental facts gave birth to quantum mechanics.
The description of the physical and chemical properties of matter is a central issue that has occupied the minds of scientists since the age of the ancient Greeks. In their route to dissect matter down to what cannot be divided any further, they coined the term atom , the indivisible . Matter became then a collection of atoms. More than twenty centuries had to pass until the development of a more precise concept of atom, thanks, amongst others, to the systematic studies of Mendeleyev and the establishment in 1869 of the periodic table of the elements (Mendeleyev, 1869). The discovery of the electron in 1897 and the first modern model of the atomic structure by Sir Joseph Thomson were soon refined by his student, Sir Ernest Rutherford, who in 1910 showed that an atom was made of a positively charged small nucleus and a number of negatively charged electrons that neutralize the nuclear charge. Much in the spirit of planetary systems, and drawing from the analogy between gravitational and electrostatic interactions, scientists in the beginning of the twentieth century built an image of the atom that consisted of a number Z of electrons – of elementary charge – e – orbiting around the nucleus of charge Ze . A number of experimental observations, though, were incompatible with this idea of orbiting electrons. In particular, according to the successful electromagnetic theory, charged electrons in orbital (radially accelerated) motion should radiate energy, thus decelerating and eventually collapsing onto the nucleus. Clearly, such a picture would imply that matter is essentially unstable, in flagrant contradiction with our everyday experience of the very existence of matter.