Difference between revisions of "Quantum Mechanics"

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Full Title or Meme

This page focuses on the first development of the Quantum Mechanics of the Eventful Universe as developed by Werner Heisenberg and his contemporaries in the late 1920's.

Mathematically, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic.

The Universe is mystic to man, and must ever remain so; for he cannot transcend the limits of his consciousness, his knowledge being only knowledge of its changes.^[1]

Context

Werner Heisenberg was not happy with the state of Quantum Mechanics as articulated by the Copenhagen school of Niels Bohr and so went off to an isolated island in the North Sea (for his Hay Fever) to think through a better solution focused on the observed events. Shortly after this Edwin Schrödinger developed another model based on the flow of a quantum from one (potentially observable) event to another known as the wave equation. He later showed that these two models were consistent with each other in spite of the different goals of the models, the wave equation dealt with the flow of the particle and Quantum Mechanics dealt with the event that was observed when the electron was measured. The best description of the history of this time is given by Max Born's Noble Lecture.^[2]

Mathematical Basis for Theories

We’re spoiled by how accurate modern mathematical theories are. When you study quantum field theories, for instance, you have the Particle Model. For the photon, which carries the electromagnetic force, it’s close enough. It’s not literally a little ball, but we have machines that can detect a local bit of energy. The detector makes a click, and that’s a photon. ... I perceive a new humbleness in the community. After having made these excursions to the very rich and exotic frameworks of loops, strings and other extended objects, where we got stuck in one way or another, we’re starting to rediscover the beauty of quantum field theory as originally put forward. As one example, Causal Dynamical Triangulation is part of the trend of going back to basics.^[3]

Energy

During the centuries after Newton and Leibniz introduced the calculus and hew ways of evaluating Force and analysis of physical effects in their world, a new understanding of energy and heat flow took the mathematics used to solve physics in a new direction. First of all energy was described as the sum of kinetic energy of partials in motion, plus the potential energy which is typically gravitation. (H = T + V) Rather than starting with F = ma, which works so well in the classical Lagrangian model, Hamilton focused on the multi-body problem and formulated a mathematical model based on position (x or q) and momentum (p = mv). The two partial differential equations to replace the force equation are symmetrical based on the Hamiltonian equations based on this total energy.

∂H/∂p = dx/dt = v
∂H/∂x = dp/dt = ma (= F)

These equation are more natural in Quantum Mechanics where the particle seems to have some blurriness that can be viewed as a result of its wave-like nature. In fact the particle can now be viewed as a wavelet which comes with some lack of clarity as to its exact position. Once it is decided to use energy as the core principle, most of the rest of the weird results of Quantum Mechanics come out of the Hamiltonian model naturally. The certainty as to position and velocity that worked in Newton's model are no longer possible with the Hamiltonian model. Also the existence of the photon that has not rest mass, but does carry momentum can be accommodated by the Hamiltonian. The downside to the adoption of the Hamiltonian is the immense mathematical complexity that arises from its use.

Quantum of Energy

While the quantum nature of light (radiation of energy) is well-known, sound is radiating energy as well. In 2023 researchers reported that sound quanta (phonons) have been created and shown to exhibit similar behavior, like splitting into two Entangled components.^[4] This may be a clue that any transmission of energy must be quantized and exhibit quantum behavior.

Quantum of Action

The quantum of action is a fundamental physical constant that appears in quantum mechanics and is used to describe the behavior of particles and waves at the atomic and subatomic level. It is also known as Planck’s constant (h) and has a value of approximately 6.62607015 × 10 −34 joule second. When the values are multiples of a constant least amount, that amount is referred to as a quantum of the observable2. Thus Planck’s constant h is the quantum of action, and ℏ (i.e., h/2π) is the quantum of angular momentum, or spin.^[5]

The Feynman path integral is the most successful application of the practice of using a formula for the minimization of the action of a particle to predict the path of that particle. As far as physicists can tell, it precisely predicts the behavior of any quantum system — an electron, a light ray or even a black hole. The path integral has racked up so many successes that many physicists believe it to be a direct window into the heart of reality.^[6] This formula is uses together with the sum of all possible paths to determine physical reality, or so most physicists believe. The only unknown is which possible paths need to be included in this process. It certainly has been a success.^[7] Physicists have even managed to estimate the path integral for the strong force, the extraordinarily complex interaction that holds together particles in atomic nuclei. They used two main hacks to do this. First, they made time an imaginary number, a strange trick that turns amplitudes into real numbers. Then they approximated the infinite space-time continuum as a finite grid. Practitioners of this “lattice” quantum field theory approach can use the path integral to calculate properties of protons and other particles that feel the strong force, overcoming rickety mathematics to get solid answers that match experiments.

Fourier transforms

Fourier created these long before their wide applicability was known. It can be used in the original quantum solution to the emission of light by a black-body which led Planck to employ quanta to light radiation.^[8] As Heisenberg thought through his solution as a set of oscillators, he reported that "The idea suggest itself that one should write down the mechanical laws not as equations for the positions and velocities of the electrons, but as equations for the frequencies and amplitudes of the Fourier expansion." When Fourier Analysis is used to formulate Quantum Mechanics the uncertainty relationship is a foregone conclusion.^[9] Since Fourier transforms were used because of the assumption that the light quanta were the result of either classical or virtual oscillators, it seems that Quantum Mechanics needs to be formulated better with other assumptions.

Spectroscopy

It was discovered that each element was likely to generate "spectral" lines only in very specific patterns of discrete frequencies when it was heated to luminescence. The primary goal for the emerging Quantum Mechanics was a cogent explanation of these lines. In other words it was the common search for causes that has driven much of human yearning for knowledge.

The first good quality spectra of hydrogen atom was recorded in 1853 by Anders Ångstrom. In 1885, Balmer recognized a pattern in the spectra and gave the Blamer formula: λ = B m² /( m²−2² ). Later, Rydberg generalized this formula. The first successful theoretical explanation of the Rydberg formula was given by Bohr–Sommerfeld model, which is now known as ‘old quantum mechanics’. Although the old quantum mechanics was able to explain a lot about the hydrogen atom spectra, including splitting of spectral lines in presence of electric field (Stark effect), it was still not able to explain quite a few experimental Observations such as, splitting of spectral lines in presence of magnetic field (anomalous Zemman effect), presence of hyper-fine spectral lines structure and hydrogen atom in presence of crossed electric and magnetic field, etc. Old quantum mechanics was also inadequate to explain different intensities of spectral lines in the atomic spectra.

Observation

The role of the observer in physics was first introduced in Eistein's special theory of relativity to enable different observers to measure distances in different inertial frames while still agreeing on a constant speed of propagation of light.

The success of Quantum Mechanics was demonstrated when Pauli was able^[10] to derive the Balmer Spectrographic lines from Heisenberg and Born's work. The Balmer Lines had been observed first. The Copenhagen interpretation was that physics was about explaining observable and not trying to determine what sub-atomic reality might actually entail. Heisenberg when defending the Copenhagen interpretation went so far in 1955 to declare "we cannot and should not replace these concepts by any others."^[11](page 197) The term "Observation" is unfortunate in that is seems to imply that some human must be the observer. In this page any event that resolves any physical aspect of a quantum particle must be viewed as an observation, whether or not a human was involved. Thus Schrödinger's cat paradox will mean that the impact of the particle on the detector is observation enough whether a human observed the event or not. It might just be that the "Collapse of the Wave Function" is indistinguishable from an Observation.

Bell's Theorem

Bell described a method in 1964 to show whether a "hidden variable" as describe by Bohm^[8] could determine the actual path of a a quantum particle. Subsequent tests showed that such variable could not exist and that the probability methods of Quantum Mechanics were actually all that could be said about the path of a particle.

Falsifiability

The philosopher Popper was the first to describe a means to determine if a theory could be accepted as truth. The theory must come with a description of reality that could (at least theoretically) be proven false. If such a test was not possible, the theory could never be accepted to be true. Such unfalsifiable theories were just metaphysics.

Problems

The Bohr Model

In 1920 there was a model of a quantum atom that has electrons spinning around a nucleus, that fad just been discovered by Rutherford, a New Zealander working in Canada and England. Some success was obtained in determining the differences between electron "orbits" as a light photon of a specific energy was emitted and measured whenever an electron "fell" from one orbit to another at a lower energy. Unfortunately, that Bohr model of electron orbits is unable to predict the fine details of the simplest atom, Hydrogen, and Bohr, in the 1920's, was adamantly opposed to the concept of light quanta.

The Particle Model

Ever since Newton developed his theory of gravitation a fully mechanistic view of moving bodies had led physics to believe that physical laws were deterministic, that is, that if all of the positions and velocities of the physical objects in the universe could be know that the entire past and future could also be known. But if we consider a photon to be a particle, then when it is sent through the two-slit experiment, we cannot know with certainty where it will land on the detection screen. Many physicists, including Einstein and Bohr rebelled against any such interpretation. Einstein by insisting on certainty and Bohr insisting that light could not be composted of quanta.

This model is focus on actual Observations of discontinuous events (an action plus a transformation) when particle interact. The distinction between the particle and wave models is like a Fourier transform: it can describe a wave in tine, or in an analysis which has no time component to it.

Probabilities

It seems that the current understanding of Quantum Mechanics only creates probabilities of the outcome of any measurement of a object ow atomic size or lower. Here is what John von Neumann said. "It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics—the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible."^[12]

It was a known fact that a probability must be in the range of 0 (will not happen) to 1 (must happen). This definition did not work with quantum mechanics. We will see below that Heisenberg allowed negative numbers that in effect corresponded to the adjustments needed to make the particles behave like waves with constructive and destructive interference.

Oscillators

One theory was that if an atom could only emit light of a predetermined frequency, then there could be real, or virtual oscillators in the atom tuned to the frequencies that were emitted.

Heisenberg was trying to understand the spectrum of the hydrogen atom. Classically, with certain approximations, one expects the spectrum for high energy hydrogen atom to be harmonic. This fact can be represented by Fourier expansion of position as follows,

x(t) = Σ an exp (−inωt)  sum from n=0 to ∞

Here the equation can be related to the intensities of the corresponding harmonics. In the high quantum number limit, the hydrogen atom does exhibit a harmonic spectrum

The Wave Model

The Wave Model was very good at predicting the distribution of light photons (and electrons) impacting on a fixed screen when they were diffracted, as is clear in the "two slit" experiments from the beginning of Quantum Mechanics. In the wave model and is many of its predecessors, the interesting part of physics was in the descriptions of the states of the reality. In the Quantum Mechanics of the Schrödinger equation its was the state of the wave as it propagated through space. Every formula was a linear equations of differential equations. Space was continuous and the core of physics follows the calculus invented by the same Issac Newton that established the laws of gravity.

The Wave Model is not testable by directed Observations. It exists only in theory and does not describe what happens when an Observation occurs.

Is it Logical?

The Law of excluded middle can be violated by some quantum operations. So in that sense we can say the Aristotle's Laws of Thought (aka Boolean logic) were not up the the challenge of describing how Quantum Mechanics worked. Which is not to say that Quantum Mechanics is not logical, just that the logic needs to fully articulated as it is not the same as classical logic, just as Quantum Mechanics is not the same as classical mechanics.

Composition

One the the conundrums that Heisenberg tased was the the composition of two event, such as an electron falling from one level to a lower level and from there onto an even lower level. Heisenberg used multiplication of compose the two events into a new state. He deduced that put together elements corresponding to the same initial and final states, summing over all possible intermediaries. This realization gave hie the key by which he could devise a multiplication rule that was both manageable and sensible.^[11] (page113)

Accepted Solutions

At its core, Quantum mechanics is simply a mathematical model for predicting the statistical behavior of microscopic particles by measuring experiments we use to explore those behaviors which have been spectacularly successful: in terms of power and precision, beyond any theory we have ever had.^[13] But still, it is just a model and so false, even though it is extremely useful.^[14] The following is a list of the mathematical concepts that have been testing against experimental results and not been contradicted. From time to come the ideas have been tested against the common sense of the world around us, but generally that common sense has been overridden by the experiments. Some common sense ideas from classical physics haven been retained, like the existence of particles that have a deterministic position and velocity in the space that we occupy here on earth. See the wiki page Particle Model for the case that particles are a myth, which would make the position of a myth a suspect concept. The concepts of energy and momentum, on the other hand, seem to be born out by experimentation.

Indistinguishable Particle

Boltzmann statistics were based on a Hamiltonian where every particle was known and tracked. Planck determined that his quanta required indistinguishable particle statistics. in 1924 S. N. Bose published a new treatment of the statistics of indistinguishable particles in 1924 after Einstein had it translated into German.^[15] This was the beginning of a trend to use sophisticated mathematical structures to define Quantum Mechanics without recourse to classical physical theory^[16] that was completed by von Neumann as the fundamental, rigorous basis for Quantum Mechanics.^[12]

• More details on the statistics that apply to classical and quantum "particles" can be found on the page Statistical Physics in this wiki.
• More details.on the features of an Identical Particle are to found on that page in this wiki.

Quantum Numbers

At first there was one quantum number that Bohr assigned to the electron "orbits" as they were added to the atom. Afterwards more numbers were added, which Bohr was not happy about. These are the final four:

1. n = principal, this is Bohr's "orbit" (in a neutral atom the highest number is the atomic number of the element, that is, the number of protons in the nucleus.)
2. l = azimuthal
3. m = magnetic
4. s = spin

These names are historical for what the physics at the time thought they they represented. Today they should be considered to be just labels with no connection to the ordinary uses of those names.

Heisenberg's Solution

Heisenberg realized that the classical view expressed in the Hamiltonian was not working for Quantum Mechanics so he took the first step away from the classical view by expressing both position and momentum as dependent on 2 factors. He went off by himself on an island in the North Sea to working out a way to combine transitions of electrons in the Hydrogen atom between any two energy levels. For example, a jump of two levels would be the result of the first jump to the next lower level times the jump on down the next level.^[17]

While his original solution created a new mathematics for multiplication, it was realized by Born that this type of multiplication was already known as matrix multiplication and so Heisenberg's solution^[18] was recast as matrix mechanics, which Born included in his second paper on Quantum Mechanics. The following will be presented in the matrix formalism that has come to be standard as it was described by Dirac. When Heisenberg left for^[11]

Eigenvalues

In 1926 Erwin Schrödinger published a series of 4 papers that changed the idea of the quantum number.^[19] Whereas Bohr created the quantization number as a natural integer as input the the theory, Schrödinger found the quantum number as a result of formulas of Quantum Mechanics. On other words, he discovered a way to turn the continuous mathematical models back into descrete results. Nature itself is composed only of descrete objects and descrete events. Continuity is an illusion that helps our limited intellect to make models of reality that we can comprehend.

Schrödinger Wave Equations

The origin for the quantum wave theories was de Broglie's extending the light quanta (photon) waves to electrons and eventually all other quantum "particles". His breakthrough was to select the wavelength for the electron as λ = h/p, Planck's constant divided by the momentum.^[20] This relates the wavelength, a wave-like attribute, with the momentum, a particle-like attribute. While that makes sense for the photon, the application to the electron was revolutionary.

Transformation Theory

Transformation Theory is what von Neumann^[12] page 2 and 6 called the work of Dirac and Jordan in which they make possible a grasp of physical problems which is exceptionally simple mathematically

Hilbert space

Von Neumann worked in Göttingen with Hilbert during the 1930's and developed a fully formed mathematical theory for Quantum Mechanics that pretty much survive to this day,^[12]

Sum of all Paths

Feynman Least Action Thesis proposes that to get a full potential of where one individual "particle" might go we need to calculate the probability of all possible paths and crate a weighted sum of all of these possibilities to determine the probability of an event where the "particle" will eventually interact with another particle creating an observable event.

While it was Feynman that brought this concept to fruition, Schrödinger described the basic idea back at the beginning. Following is a quote from his paper. ^[21]

If we like paradoxes, we may say that the system exists, as it were, simultaneously in all the positions kinematically imaginable.

Physical Reality

The essential problem with the Hamiltonian as a basis for Quantum Mechanics is that there is no point where the 'particle" actually is. The zero point energy of any physical system is not quiet. Not only can we not know where each particle is, but the particle is constantly "jiggling around". We now have some ideas about the physical reality of each particle in local enforcement of interest. Researchers from 2020 have been discovering that the these base level energies and excitons above that level can (and must) be measured to have a workable Quantum Computer.^[22]

De Broglie's Caution

The originator of the wave nature of matter, Louis De Broglie, was troubled by the direction of Quantum Mechanics and offer this caution in this book "The Revolution in Physics"^[23]

The old mechanics, corresponding to the approximation of geometrical optics - all the images and conceptions that we used in it - all these must be abandoned when we pass beyond the limits of this approximation. We cannot therefore avail ourselves, at least without precautions, of the notions of a position, of a velocity, and of a trajectory of a [particle]. [Any] system of postulates ... have to satisfy the essential condition of leading us back to the conceptions and results of the old mechanics whenever the ψ-wave obeys the laws of geometrical optics. ... The interpretations of the new mechanics is of a probabilistic nature.

Entanglement

When two particles are Entangled, they share a quantum state that is described by a wave function. When one of the particles is measured, it collapses the wave function of both particles, which means that the other particle’s wave function is also collapsed. This means that the other particle’s state is determined by the measurement of the first particle. The order of measurements does not matter because both measurements will collapse the wave function of both particles and determine their states.^[24]

Despite its spookiness, entanglement is a core feature of quantum physics. When any two objects interact in quantum mechanics, they generally become entangled and will stay entangled so long as they remain isolated from the rest of the world—no matter how far apart they may travel. In experiments, physicists have maintained entanglement between particles more than 1,000 kilometers apart and even between particles on the ground and others sent to orbiting satellites. In principle, two entangled particles could sustain their connection on opposite sides of the galaxy or the universe. Distance simply does not seem to matter for entanglement, a puzzle that has troubled many physicists for decades. If space is emergent, entanglement’s ability to persist over large distances might not be terribly mysterious—after all, distance is a construct. Leonard Susskind says. “The continuity and the connectivity of space owes its existence to quantum-mechanical entanglement.”

Coherence

there is a difference between coherence and quantum entanglement. In quantum mechanics, coherence refers to the property of a quantum system to maintain a well-defined phase relationship between its states 1. On the other hand, entanglement is a property of two or more quantum systems that are correlated in such a way that the state of one system cannot be described independently of the state of the other system.^[25]

In other words, coherence is a property of a single quantum system, while entanglement is a property of multiple quantum systems. Coherent electrons have the same quantum status, so they have the same spin, while entangled electrons have opposite (antiparallel) spin, and they always act like a couple.^[26]

Specific Uses

In keeping with the purposes of this wiki the application of Quantum Mechanics to computer and communications applications. Click on the names below for more information.

• The Quantum Computing Threat describes how existing private keys can be exposed.
• The Quantum Information Theory describes how quantum effects can be used create provably secure communication channels.

Alternate Models

• The physical layer of IEEE 802 LANs has some interesting behaviors that might help explain why Quantum Mechanics is statistical. Of the three data link layers proposed to IEEE 802 at the start in 1980 there are two that exemplify the difference been statistical and deterministic physical layers in networking. Ethernet (statistical) and Token Ring (deterministic) are two types of physical layer technologies. Ethernet is the most widely used LAN technology today. It is a wired network technology that uses coaxial cables, twisted pair cables, or fiber optic cables to transmit data between computers. Token Ring is another type of LAN technology that was popular in the 1980s and early 1990s. It uses a ring topology to transmit data between computers. While both were operating in the 10 to 16 megabit range at the start, Ethernet's flexibility allowed it to grow to 10 or more gigabits per second while Token Ring's rigidity limited its growth. Token Ring's reliability gave it some advantage in the start that is not available with statistical approaches, but the speed (and cost) advantage of statistic won out in the end. That fact might help understand why Quantum Mechanics is statistical.
• In the current model fluctuation of space-time are a result of the statistical nature of Quantum Mechanics, but more physics are beginning to think that it is the other way around, that the fluctuations are the basic cause and the statistical nature of Quantum Mechanics is the result of them.

References

1. ↑ George Henry Lewes, Problems of Life and Mind. First Series: The Foundations of a Creed Vol. 2. Boston: Osgood. section 1 (1875) https://www.google.com/books/edition/Problems_of_Life_and_Mind_The_principles/0J8RAAAAYAAJ?hl=en&gbpv=1
2. ↑ Max Born. The Statistical Interpretation of Quantum Mechanics, (1954-12-11) https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf
3. ↑ Renate Loll quoted Charlie Wood, The Physicist Who Glues Together Universes Quanta (2023-05-25) https://www.quantamagazine.org/renate-loll-blends-universes-to-unlock-quantum-gravity-20230525/
4. ↑ University of Chicago, Researchers 'split' phonons in step toward new type of quantum computer University of Chicago (2023-06-11) https://phys.org/news/2023-06-phonons-quantum.amp
5. ↑ https://www.britannica.com/science/quantum
6. ↑ Charlie Wood, How Our Reality May Be a Sum of All Possible Realities https://www.quantamagazine.org/how-our-reality-may-be-a-sum-of-all-possible-realities-20230206/
7. ↑ Richard Feynman PhD thesis (1942) http://files.untiredwithloving.org/thesis.pdf
8. ↑ ^{8.0 8.1} David Bohm, Quantum Theory Prentice Hall (1951)
9. ↑ Emanuele Pesaresi, Uncertainty Principle Derivation from Fourier Analysis https://www.linkedin.com/pulse/uncertainty-principle-derivation-from-fourier-emanuele-pesaresi
10. ↑ Wolfgang Pauli,
11. ↑ ^{11.0 11.1 11.2} David Lindley, Uncertainty Doubleday ISBN 9780385515061
12. ↑ ^{12.0 12.1 12.2 12.3} John von Neumann, 1932, Mathematische Grundlagen der Quantenmechanik, Berlin: Springer Verlag; English translation by R.T. Beyer, 1955, Mathematical Foundations of Quantum Mechanics, Princeton: Princeton University Press ISBN 9780691178561
13. ↑ Jenann Ismael , Quantum Mechanics Plato (2020-09-20) https://plato.stanford.edu/ENTRIES/qm/
14. ↑ Guillen Barroso, “All models are wrong, but some are useful”. George E. P. Box (2019-07-01) https://www.lacan.upc.edu/admoreWeb/2018/05/all-models-are-wrong-but-some-are-useful-george-e-p-box/
15. ↑ S. N. Bose, Warmegleichgewicht im Strahlungsfeld bei Anwesenheit von Materie, Zeitschrift fur Physik, vol. 27, pp. 384-393, 1924. English translation in O. Theimer and B. Ram, The beginning of quantum statistics, American Journal of Physics, vol. 44, pp. 1056-1057, 1976.
16. ↑ Paula Mallol Travesset, Origin and foundations of Bose-Einstein statistical mechanics University of Barcelona https://diposit.ub.edu/dspace/bitstream/2445/125078/1/Mallol%20Travesset%20Paula.pdf
17. ↑ Werner Heisenberg, Quantum=Thoretical Re-interpretaton of Kinematic and Mechanical Relations (1925) in English Sources of Quantum Mechanics Dover ISBN 0486618811
18. ↑ Pradeep Kumar, Heisenberg’s Invention of Matrices https://www.ias.ac.in/article/fulltext/reso/022/04/0399-0405
19. ↑ Erwin Schrödinger,Quantization as an eigenvalue problem (1926-01-27 for the first paper.) http://ofp.cosmo-ufes.org/uploads/1/3/7/0/13701821/quantisation_as_an_eigenvalue_problem.pdf
20. ↑ Louis de Broglie, The wave nature of the electron Nobel Lecture, (1929-12-12) https://www.nobelprize.org/uploads/2016/04/broglie-lecture.pdf
21. ↑ Erwin Schrödinger, Quantization as a problem of Proper Values (Part 4) in Hawking, The Dreams that Stuff is made of
22. ↑ University of Washington The 'breath' between atoms—a new building block for quantum technology (2023-06-01) https://phys.org/news/2023-06-atomsa-block-quantum-technology.html
23. ↑ Louis De Broglie, The Revolution in Physics (1953 in English) https://books.google.com/books/about/The_Revolution_in_Physics.html?id=l5kfAQAAMAAJ ASIN‎ B0007HDM8I
24. ↑ What exactly is the difference between entanglement and correlations? https://physics.stackexchange.com/questions/561382/what-exactly-is-the-difference-between-entanglement-and-correlations
25. ↑ What is the difference between coherence and entanglement? Physics https://physics.stackexchange.com/questions/728449/what-is-the-difference-between-coherence-and-entanglement
26. ↑ Entanglement and coherence https://physics.stackexchange.com/questions/204100/entanglement-and-coherence