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Chapman & Hall/CRC eBooks

NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS

Vishnu S. Mathur, Surendra Singh

Published Dec 12, 2008• 0 citations
Publisher

Abstract

Classical mechanics, which gives a fairly accurate description of large systems (e.g., solar system) as also of mechanical systems in our every day life, however, breaks down when applied to small (microscopic) systems such as molecules, atoms and nuclei. For example, (1) classical mechanics cannot even explain why the atoms are stable at all. A classical atom with electrons moving in circular or elliptic orbits around the nucleus would continuously radiate energy in the form of electromagnetic radiation because an accelerated charge does radiate energy. As a result the radius of the orbit would become smaller and smaller, resulting in instability of the atom. On the other hand, the atoms are found to be remarkably stable in practice. (2) Another fact of observation that classical mechanics fails to explain is wave particle duality in radiation as well as in material particles. It is well known that light exhibits the phenomena of interference, diffraction and polarization which can be easily understood on the basis of wave aspect of radiation. But light also exhibits the phenomena of photo-electric effect, Compton effect and Raman effect which can only be understood in terms of corpuscular or quantum aspect of radiation. The dual behavior of light, or radiation cannot be consistently understood on the basis of classical concepts alone or explained away by saying that light behaves as wave or particle depending on the kind of experiment we do with it (complementarity). Moreover, a beam of material particles, like electrons and neutrons, demonstrates wave-like properties (e.g., diffraction). A brief outline of phenomena that require quantum mechanics for their understanding follows.

DOI
10.1201/b17176-4

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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. 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Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.

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CRC Press eBooks•Nov 17, 2021•0 citations
Classical mechanics, which gives a fairly accurate description of large systems (e.g., solar system) as also of mechanical systems in our every day life, however, breaks down when applied to small (microscopic) systems such as molecules, atoms and nuclei. For example, (1) classical mechanics cannot even explain why the atoms are stable at all. A classical atom with electrons moving in circular or elliptic orbits around the nucleus would continuously radiate energy in the form of electromagnetic radiation because an accelerated charge does radiate energy. As a result the radius of the orbit would become smaller and smaller, resulting in instability of the atom. On the other hand, the atoms are found to be remarkably stable in practice. (2) Another fact of observation that classical mechanics fails to explain is wave particle duality in radiation as well as in material particles. It is well known that light exhibits the phenomena of interference, diffraction and polarization which can be easily understood on the basis of wave aspect of radiation. But light also exhibits the phenomena of photo-electric effect, Compton effect and Raman effect which can only be understood in terms of corpuscular or quantum aspect of radiation. The dual behavior of light, or radiation cannot be consistently understood on the basis of classical concepts alone or explained away by saying that light behaves as wave or particle depending on the kind of experiment we do with it (complementarity). Moreover, a beam of material particles, like electrons and neutrons, demonstrates wave-like properties (e.g., diffraction). A brief outline of phenomena that require quantum mechanics for their understanding follows.
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.
Because atoms and molecules are small, their behavior can only be understood using quantum mechanics, rather than classical mechanics. Quantum mechanics amounts to studying the behavior of electrons and nuclei and their interactions. We will limit ourselves to slowly moving electrons and nuclei, and thus will not need to discuss relativistic quantum mechanics. Large particles also obey quantum mechanics, but the relevant wavelengths are so small that we don't notice them. Small particles absolutely require the use of quantum mechanics for any meaningful understanding. This treatment is not merely an inconvenience: it leads to useful and interesting phenomena such as the Pauli exclusion principle, the Aufbau principle, Hund's rules, the Heisenberg uncertainty principle, wave-particle duality, the photoelectric effect, Ehrenfest's theorem, and the correspondence principle.
Introduction It is believed that the basic physical building blocks forming the world we live in may be categorized into particles of matter and carriers of force between matter. All known elementary constituents of matter and transmitters of force are quantized. For example, energy, momentum, and angular momentum take on discrete quantized values. The electron is an example of an elementary particle of matter, and the photon is an example of a transmitter of force. Neutrons, protons, and atoms are composite particles made up of elementary particles of matter and transmitters of force. These composite particles are also quantized. Because classical mechanics is unable to explain quantization, we must learn quantum mechanics in order to understand the microscopic properties of atoms – which, for example, make up solids such as crystalline semiconductors. Historically, the laws of quantum mechanics have been established by experiment. The most important early experiments involved light. Long before it was realized that light waves are quantized into particles called photons, key experiments on the wave properties of light were performed. For example, it was established that the color of visible light is associated with different wavelengths of light. Table 2.1 shows the range of wavelengths corresponding to different colors. The connection between optical and electrical phenomena was established by Maxwell in 1864. This extended the concept of light to include the complete electromagnetic spectrum. A great deal of effort was, and continues to be, spent gathering information on the behavior of light.
Quantum mechanics (or simply quantum physics) is the branch of physics that applies to particles and systems of particles, such as atoms and molecules. In that realm the laws of classical mechanics and classical electromagnetism break down, and only quantum mechanics provides the concepts and computational methods that explain observed phenomena. Quantum mechanics uses specially developed tools, such as wave functions, that contain all of the relevant information about particles and systems. is approach is needed to understand the true nature of electromagnetic radiation and how that radiation interacts with matter. e mathematics used in quantum mechanics is highly specialized and often complex. In this chapter we’ll address some of the key questions that require quantum mechanics without presenting the full mathematical formalism. e important questions we’ll address all deal with observable physical phenomena. We’ll discuss how quantum mechanics is used to answer these questions but will omit the advanced mathematical details.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.