We show that a simplified force between the electron and the nucleus solves Quantum Mechanics in the simplest way possible. We update and rewrite Bohr's postulates, we show the simple relation between the electric and magnetic force, we show how light is emitted, and find a new term for the fine structure constant which is the simplest term there is. Simplified Orbital Mechanics results in a new atomic model which allow mathematically the observed electron cloud. All calculations can be found in this google spreadsheet: https://docs.google.com/spreadsheets/d/1jE6hjJNPcY7h98DkkGVEJiO9x3LK-_ZK6553cd2b_M8/edit#gid=498321884
We show that a simplified force between the electron and the nucleus solves Quantum Mechanics in the simplest way possible. We update and rewrite Bohr's postulates, we show the simple relation between the electric and magnetic force, we show how light is emitted, and find a new term for the fine structure constant which is the simplest term there is. Simplified Orbital Mechanics results in a new atomic model which allow mathematically the observed electron cloud. All calculations can be found in this google spreadsheet: https://docs.google.com/spreadsheets/d/1jE6hjJNPcY7h98DkkGVEJiO9x3LK-_ZK6553cd2b_M8/edit#gid=498321884
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
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.
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 process of observation disturbs the thing being observed and that disturbance might possibly affect other observations. Heisenberg showed that simultaneous measurements of an electron's position and its momentum interfere with each other and thus induce essential uncertainties into those measurements. Quantum mechanics represents the behavior of electrons around nuclei in terms of mathematical constructs called wave functions, or orbitals, which contain all possible information about the electron. The square of the wave function gives the probability of finding the electron in a given region of space. For mathematical reasons, physicists and chemists almost always use the abstract concept of wavefunctions (orbitals), rather than that of electron probability clouds, to describe electrons. The distribution of probability within a probability cloud is not uniform. For some, the greatest probability is near, or even at, the nucleus
Most textbooks say that quantum mechanics came into being when Max Planck, in a desperate attempt to construct a theory that explained experimental data, explored the radical idea of energy being radiated or absorbed by all bodies in discrete packets (or quanta). The energy content of a quantum of radiation is proportional to the frequency of the radiation emitted or absorbed, as in , and the proportionality constant bears the name of Planck. Planck himself was rather uncomfortable with the idea of energy packets, and spent a considerable amount of time in later years trying to “restore” the classical picture of continuous energy exchange between bodies and surroundings. However, in 1905 Einstein was able to use the idea of the quantum to explain the mechanism of the photoelectric effect, which cannot be explained in any other way! This established the reality of the quantum and so, it may be said that Einstein, rather than Planck, is the father of quantum mechanics. Within a few years, Neils Bohr constructed his model of the hydrogen atom in which the electron can only occupy certain special orbits around the nucleus, and any jump between these orbits is accompanied by energy being absorbed or emitted in the form of quanta. The nearly exact agreement of Bohr’s predictions based on the model and observed hydrogen atomic spectra was among the first impressive accomplishments of quantum mechanics.
Niels Bohr’s model of the hydrogen and hydrogen-like atoms, in which the electron is allowed to revolve in certain circular orbits around the nucleus, was extended by Arnold Sommerfeld to allow for the possibility of elliptical orbits. Each elliptical orbit was constrained by two quantization conditions specified by a positive integer nφ and a non-negative integer nr. If, in the presence of the attractive Coulomb force between the electron and the nucleus, the dynamics of the atom were taken to be governed by the Newtonian laws of motion, the energy Ɛ of the two-particle system was found to be a function of the positive integer n = nr + nφ. The atomic energy levels thus exhibited an n-fold degeneracy, which would be lifted in the presence of external electric or magnetic fields, thus accounting for the fine-structure of the spectral lines emitted by the atom—phenomena observed, respectively, in the Stark effect and the Zeeman effect. Sommerfeld also examined the behavior of his elliptical orbits by allowing the revolving electron to obey Einstein’s relativistic dynamics. In this case, the energy-level degeneracies were lifted (even in the absence of external perturbations), and the elliptical orbits were found to precess around the nucleus—similarly to the precession of, say, the orbit of the planet Mercury around the Sun. What is astonishing about Sommerfeld’s relativistic expression of the energy Ɛ as a function of nφ and nr is that it is precisely the same expression as obtained for hydrogen and hydrogen-like atoms from Dirac’s equation, which came more than a decade later. Recall that the spin of the electron was not yet discovered when Sommerfeld developed his model, and that in Dirac’s fully-relativistic quantum mechanics, the spectral fine-structure arises from the coupling of the electron’s spin to its orbital motion. The present paper provides a streamlined yet detailed account of Sommerfeld’s elliptical orbits in hopes of bringing about a deeper appreciation for the miraculous coincidence of Sommerfeld’s result with that derived from Dirac’s equation.