The unknowns that remain in quantum mechanics are not the mysteries that people often think of. The real mysteries of quantum mechanics are annexed to foundational knowledge of quantum physics that, given all indications, just appeared into existence at our measurement level, or second layer of quantum physics. These miracles of knowledge include Planck's energy equation https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν , Born's rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , elements of the Schrödinger equation, and Dirac's notation. In Quantum Clear we elucidate the physics leading to https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , but that's it. And it remains certain that those who introduced this wonderful knowledge did not elaborate further except for the explanation of Schrödinger who linked mathematical discovery to ‘like a great gift…’. Ward, on the other hand, described the arrival of knowledge with ‘ no way I can trace the source. ’
The unknowns that remain in quantum mechanics are not the mysteries that people often think of. The real mysteries of quantum mechanics are annexed to foundational knowledge of quantum physics that, given all indications, just appeared into existence at our measurement level, or second layer of quantum physics. These miracles of knowledge include Planck's energy equation https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν , Born's rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , elements of the Schrödinger equation, and Dirac's notation. In Quantum Clear we elucidate the physics leading to https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , but that's it. And it remains certain that those who introduced this wonderful knowledge did not elaborate further except for the explanation of Schrödinger who linked mathematical discovery to ‘like a great gift…’. Ward, on the other hand, described the arrival of knowledge with ‘ no way I can trace the source. ’
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.
In quantum mechanics, the physics of probability amplitudes, that exists in the imaginary and complex mathematical realm, is brought to the measurable realm by the iconic Born rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 that convert the superposition probability amplitude https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 into a measurable interferometric probability distribution. When Born first introduced his rule, in 1926, he did so in the absence of a physical derivation. In fact, Born never offered a derivation for his most crucial rule. In this chapter a transparent interferometric derivation of Born's rule is given. Furthermore, it is indicated that far from a ‘collapse’ the https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition, for interferometric configurations, clearly involves a mathematical expansion . The https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition for quantum entanglement is also discussed.
Quantum mechanics is the newest science and should therefore be the most precise, but unfortunately, it's quite the opposite. If you don't understand quantum mechanics, ask yourself if I'm right, because others don't either. The field of experimental quantum mechanics is so vast and confusing that its theoretical practitioners will always have to adopt new explanations from their own context to conform to the facts of quantum mechanics. Quantum mechanics was created in the early 20th century to establish mathematical and physical laws or rules for subatomic objects subject to an external potential field in order to explain so-called quantum systems. Studying the spatiotemporal evolution of a quantum system, defined as a microscopic subatomic system of low energy density subjected to a potential field, external or internal, is a rather complex subject. However, in 1927, the well-known Schrödinger PDE, valid for infinite free space subject to Dirichlet boundary conditions, was introduced. Schrödinger's PDE must be complemented by the Bohr/Copenhagen interpretation of the wave function Ψ involving the rules of quantum systems of instantaneous entanglement and superposition. We all know that the Schrödinger equation is incomplete because it lives and operates in the incomplete space R^4 and therefore the wave function Ψ is incomplete in itself and cannot be defined as scalar, a vector, nor a tensor. The complex wave function Ψ has never been properly defined. Moreover, the Schrödinger PDE is not Lorentz invariant and is not compatible with the special theory of relativity, and is obviously even less so with the general theory of relativity. However, quantum mechanics has come a long way, both in theory and practice, since 1927, nearly a century ago. This long journey has added even more illogical and confusing properties to the Ψ wave function, such as causality and wave function collapse. Many attempts have been made over the last century to reform the Schrödinger equation, the most notable being to combine it with the general theory of relativity, but all these attempts have been in vain. The breakthrough came in 2020-2024 [2,3,4], when the author of this article claimed to replace the classical Schrödinger equation Ψ with the PDE for its square Ψ^2, which is logical and physical. Ψ^2 is needed to express the quantum energy density flux (U(x,y,z,t) = Ψ^2 (x,y,z,t)) and we therefore propose a new Schrödinger equation for Ψ^2 which should have the form of the energy density diffusion PDE such as that of thermal conduction. Equation 3 should be supplemented by the advanced artificial intelligence proposed by the author [5,6]. Comparing Schrödinger's classical PDE of 1927 with the PDE proposed for Ψ^2 in 2020 shows that we currently have two different or distinct theories of quantum mechanics! Which one is more theoretically true and more practically useful? This is the purpose of this article.
If we include all three spatial dimensions, the time-dependent wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , y , z , t ) depends on four coordinates. A wave function of this form determines the probability density https://www.w3.org/1998/Math/MathML" display="inline"> ρ t ( x , y , z ) = | Ψ ( x , y , z , t ) | 2 for measurements of the spatial position https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) . The product https://www.w3.org/1998/Math/MathML" display="inline"> | Ψ ( x , y , z , t ) | 2 d x d y d z therefore equals the probability of finding a particle in a small cubic volume with side lengths https://www.w3.org/1998/Math/MathML" display="inline"> ( d x , d y , d z ) around https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) at time t . The normalization condition for this probability density is given by 4.1 https://www.w3.org/1998/Math/MathML" display="block"> ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ | Ψ ( x , y , z , t ) | 2 d x d y d z = 1 for all t .
For a century, quantum theorists have been reading the mathematical entrails of quantum mechanics (QM) to divine the nature of quantum reality. But to little avail. In this paper a different approach is taken, namely to identify and explain the basic intuitive ideas involved in QM. This does not tell one how those basic `gears' all mesh together in the beautiful mathematics of QM. But this does give one some intuitive (\textit{anschaulich})) ideas about the quantum reality described in the seemingly hard-to-interpret mathematical framework.
Heisenberg's uncertainty principle is one of the unique and iconic features of quantum mechanics that distinctly separates this physics, based on indeterminacy, from any and all classical deterministic theories. Besides its succinct utter beauty, https://www.w3.org/1998/Math/MathML" display="inline"> Δ x Δ p ≈ h , as articulated by Feynman ‘it protects quantum mechanics.’ In addition to its foundational importance and ‘protection’ function, the uncertainty principle plays crucial experimental roles in the measurement of coherent emission linewidth https://www.w3.org/1998/Math/MathML" display="inline"> Δ ν and in determining the nonlocality of the photon via measurements of coherence length https://www.w3.org/1998/Math/MathML" display="inline"> Δ x . Physically, it can be shown that https://www.w3.org/1998/Math/MathML" display="inline"> Δ x Δ p ≈ h has its origin in the physics of measurable interferometric quantum probability distributions. This insight teaches us that, unlike probability amplitudes, https://www.w3.org/1998/Math/MathML" display="inline"> Δ x Δ p ≈ h firmly resides in the measurable realm.