This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 10: The emergence of quantum mechanics
Table of contents
10.1: An introduction to quantum theory10.2: Function space
10.3: Complex numbers, periodic functions and time
10.4: The double slit experiment
10.5: How does it work?
10.6: Superposition
10.7: Which slit does the particle go through?
10.8: Feynman's jewel
10.9: Wigner's mystery
10.1 An introduction to quantum theory
We are attempting to construct a universe from a primordial quantum of action. In broadest terms we might imagine that the mechanism for this construction has two phases, familiar to us from Darwinian evolution: variation and selection. Quantum mechanics is a mathematical theory which provides us with both these elements of evolution.
The theory involves two distinct and apparently incompatible processes. The first, described by the Schrödinger equation, is believed to be a continuous error free deterministic process which describes the evolution of undisturbed quantum systems through time. This is described on this page and leads to the discussion of error free communication on page 11: Quantization: the mathematical theory of communication. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics, Schrödinger equation - Wikipedia
The second describes the interruption of this process by observation which leads, after some preparation on page 12: The quantum creation of Minkowski space and page 13: Is Hilbert space independent of Minkowski space?, to a discussion of the so called measurement problem on page 14: Measurement: the interface between Hilbert and Minkowski spaces
10.2: Function space
A Hilbert space is a function space, which means every point in the space represents a function, f. By establishing an origin in this space we may imagine every point in it as a vector from the origin (that is zero) to the point representing the function.
We may divide functions into real and complex, depending on whether the domain of the function are real or complex numbers. Complex numbers are often written z = x + iy or z = w + iv where x, y, w and v are real numbers and i is the imaginary unit i = √-1 (i2 = -1). Two outstanding features of complex numbers are that they do not have a natural order, and that they are naturally periodic. Real number - Wikipedia, Complex number - Wikipedia
10.3: Complex numbers, periodic functions and time
Points and vectors in the Hilbert space of quantum mechanics are represented by complex numbers. Complex numbers may be represented as vectors on the complex plane with one real dimension x and one complex dimension iy. Complex plane - Wikipedia
The complex plane is a simple application of the coordinate system developed by Descartes to create geometric representations of algebraic functions. A simple real linear function like f = ax + b appears on the Cartesian plane as a family of straight line whose steepness or gradient depends on a. If b = 0 this line runs through the origin. As b varies, the point at which the line intercepts the y axis moves up and down the axis. Linear function (calculus) - Wikipedia
The next most interesting family of function are those which involve x2, known as the conic sections since we can visualize them by slicing through a three dimensional cone in various ways. The simplest of these is the circle, where we take slices off the cone parallel to its base. Conic section - Wikipedia
The circle provides us with another and more informative polar graphical representation of complex numbers using a circle on the complex plane. A circle is the locus of line which lies at a fixed distance r, from a point, the centre of the circle. Quantum mechanics (aka wave mechanics) describes periodic motion in some domain, which maybe a mathematical abstraction like Hilbert space or real spacetime. The cyclic nature of complex numbers is perfect for describing such motion. Since we understand Hilbert space here to be a mathematical ideal which exists outside space and time, we must imagine the variables in quantum mechanics to be pure numbers without any specific relationship to physical space and time. Page 13: Is Hilbert space independent of Minkowski space?, explains that Hilbert and quantum mechanics precede the existence of space-time. "Distances" in quantum mechanics are measured by angle or phase which are conveniently represented by the polar form of complex numbers. Polar coordinate system - Wikipedia
Complex numbers have no natural order, unlike the real numbers 1, 2, 3, . . .. We can however assign a length to a complex vector z called its absolute value, symbol |z|, which measures the distance of the point z from the origin of coordinates . The equation of a circle with radius r is x2 + y2 = r2, which shows that |z| = √ x2 + y2.
We designate a specific point z on this circle by the angle θ between the positive x axis and the radius ending at z. This angle is called the argument or phase of z. The complex number z = x + iy can therefore be represented by the equation z = |z| θ. The phase θ, measured in radians, may take any real value. In pure quantum mechanics |z| is normalized to 1 so that only phase of the vector is significant.
The abstract world of quantum mechanics is kinematic and formal rather than dynamic and concrete. We understand kinematic processes, like the motions we see in a projected movie, as processes without causality. The motions we see in movies are artificially constructed and have no need to follow the usual rules of physical causality. Here we understand motions in the Hilbert space of the Universe in a similar way. It is driven by its source, the initial singularity, which plays the roles historically attributed to divinities and angels. It is not physically constrained like events in Minkowski space. It acts as a source of variation from which the actual physical motion in Minkowski space are selected by the laws or symmetries of physics as we shall see on page 14 referenced above. This situation enables the existence of a form of evolution by variation and selection in the physical domain. Warner Brothers Loony Tunes
The quantum energy equation, which we write E = ℏ ω where ω is the angular frequency in radians per second dθ/dt, measures the rate at which the vector representing a quantum state is rotating in the complex plane. We equate one quantum of action to one full turn θ = 2π. Although we use the terms energy, action and angular frequency these terms simply point to numbers in ideal Hilbert space some of which will come to represent observable quantitites in real Minkowski space.
Since the action of the quantum of action is to act, we imagine that there is always formal kinematic energy present in the quantum world whose local value is the rate at which state vectors are rotating. This is represented by another form of the quantum energy equation, ∂ψ/∂t = Hψ where t is a number and H is the energy matrix representing the energy of each element in a superposition of quantum states. Quantum superposition - Wikipedia, Hamiltonian (quantum mechanics) - Wikipedia
We are inclined to think of frequency as the rate of repetition of some event. Another way to look at it is its inverse, the period between events. From this point of view, high energy physics may be seen as the physics of very short periods or fast processes. One of the beauties of Hilbert space is that it is ideally suited for representing periodic functions like music. Much of the content of music is more easily felt and understood as intervals rather than as frequencies.
10.4: The double slit experiment
A common simple illustration of the difference between the behaviour of microscopic quantum particles and macroscopic classical particles is provided by the double slit thought experiment. If we spray real bullets at random at a barrier with two holes, we see that some bullets go through one or other hole and strike a target behind the holes in line with the holes.
Quantum particles, on the other hand, appear to go through both holes and interact with themselves to generate an interference pattern on a screen behind the holes. Double-slit experiment - Wikipedia
In his lectures on the double slit experiment Richard Feynman summarizes the quantum experiment in three propositions:
1. The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number φ which is called the probability amplitude:
P = probability,
φ = probability amplitude,
P = |φ|2.2. When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference:
φ = φ1 + φ2,
P = |φ1 + φ2|2.3. If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost:
P = P1 + P2
From this we might conclude that interference is something which occurs in the invisible complex Hilbert domain rather than the observable real classical domain. The fact that the particles appears to go through both holes indicates that this process occurs in a domain other Minkowski spacetime. Feynman, Lectures on Physics vol. III chapter 1: Quantum Behaviour
10.5: How Does it Work?
Feynman then goes on to ask:
“How does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced.
10.6: Superposition
This statement seems to be a bit pessimistic. The explanation, devised in the first decades of the twentieth century, is quantum mechanics. The key idea, identified by Paul Dirac in his treatise on quantum mechanics, is superposition, which simply means the addition of vectors representing physical states in Hilbert space. Paul Dirac (1930,1983): The Principles of Quantum Mechanics (4th ed)We meet a simple classical version of superposition in the bathroom. A bath or shower is supplied with two sources of water, one painfully hot, the other painfully cold. We adjust the taps until the temperature if just right by adding cold water and hot water.
We understand the amplitudes added by quantum superposition are waves represented by the complex numbers described in section 10.3 above. The quantum world is in perpetual motion, the frequency of the waves depending on their energy. Although in the two slit experiment quantum particles interfere with themselves, a large number of them must have the same energy, that is the same frequency, to create a clear pattern on the screen. The random frequencies in white light do not produce the effect.
We can get an idea of the addition of moving waves by throwing stones into smooth pond. When two stones fall simultaneously close by we see that when the expanding circles of waves spreading from each impact meet they add and subtract from one another to form a complex patterns which seem to pass through one another unaffected. Jessica Haines: Two stones wave patterns
The phenomenon is also clear in sound. We can usually distinguish voices of different instruments or people sounding together. No musical instrument generates a perfectly pure note. They all produce overtones related to the fundamental frequency of each note, and it is these overtones that make the difference between C on a piano and C on a trumpet. Listening to an orhestra we can often pick out the contributions from different instruments even though they are playing the same note. Catherine & Johnathan Karoly: Heitor Villa-Lobos: The Jet Whistle
The quantum probability amplitudes referred to here are invisible and undetectable, but can be calculated and represented mathematically to give results that match our experimental observations (see page 15 Quantum amplitudes and logical processes are invisible). We assume that the invisible amplitudes of quantum theory behave mathematically just like the visible and audible interference of real physical waves of water and sound. In the early days of wave mechanics, physicists often found themselves studying sound waves to gain insight into quantum waves.
Each slit in the two slit experiment emits a quantum wave. Because the slits are separated, the paths taken by the waves to any point on the screen, except the exact centre, have different lengths. When the waves add at a given point sometimes they are in phase and the added amplitudes have a corresponding large absolute value giving a high probability of observing a particle (Feynman's proposition 1 above). When they are out of phase and cancel one another, the probability is low and it is unlikely that a particle will appear. The observed pattern of particle impacts on the screen is the result.
The kinematic behaviour of vectors in Hilbert space has an effect on the dynamic behaviour of events in Minkowski space. We might see a similar effect at the movies. Although the moving pictures and sounds may be purely kinematic and artificially created, they can nevertheless have the real emotional effects on the audience that the movie makers seek. The movie is a signal carrying information which relies on the intelligence of the viewers to be realized as physical emotion.
10.7: Which slit does the particle go through?
It seems intuitively obvious that a real particle would go through one slit or the other. If we block one slit, or devise a way to decide which slit the particle goes through, however, the interference pattern is lost. (Feynman's proposition 3). How can this be?
The answer proposed here, to be explained on page 12: The quantum creation of Minkowski space and page 13: Is Hilbert space independent of Minkowski space?, is that the formalism of quantum mechanics operates at a level that lies beneath the spacetime familiar to us in everyday life.
From an intuitive point of view, we can say that Hilbert space is the realm of the imagination of the Universe, just as our own minds are the realm of our imagination. Imagination has a significant input into what we do just as quantum theory has a significant input into what the Universe does. In the case of quantum mechanics, this input is not deterministic. The heart of the measurement problem discussed on page 14: Measurement: the interface between Hilbert and Minkowski is the uncertainty arising in the choice and timing of the outcomes that arise from a particular experiment. The theory is capable of predicting the exact nature of each of these random outcomes. What is cannot predict is the exact moment at which a quantum event will occur. Nor is our imagination deterministic. It provides us with many options from which we ultimately select what we actually do. This changes as our experience changes.
Here we touch on the heart of Einstein's difficulty with quantum mechanics. His theories of relativity are based on the notion that the behaviour of the physical world is independent of observers. This idea is implemented mathematically by general covariance which maintains the independence of phenomena as the frames of reference used by observers change. This principle holds in the classical world, but in the quantum world interactions are more like conversations where two sources of communication interact with one another, sharing messages that change the state of both. In effect, to observe is to be observed, and the Universe is in effect conscious because it observes itself. When we meet, we are both changed, General covariance - Wikipedia
This idea becomes more plausible as we progress. It lies at the heart of cognitive cosmology, the idea that we can produce a comprehensive theory of everything by interpreting the Universe as a mind.
10.8: Feynman's jewel
Quantum mechanics would never have been possible without the discovery of complex numbers which are perfect for repreenting the periodicity of the musical source of the Universe. In section 3 we equated the absolute value of a complex number with the radius of a circle in the complex plane. This equation is consequence of the Pythagorean theorem, the square on the hypotenuse of a right angled triangle in equal to the sum of the squares on the other two sides. Pythagorean theorem - Wikipedia
Very early in a course on trigonometry, we learn that in a right angled triangle, the sine of each of the acute angles is the ratio of the length of the side opposite the angle to the hypotenuse, and the cosine of that angle is the ratio of the side adjacent to the angle to the hypotenuse. This leads us to the conclusion that for any angle θ sin2 + cos2 = 1.
In the field of complex numbers, this insight can be taken a step further.
Feynman writes:
In our study of oscillating systems we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics. From the physicist’s point of view we could bring forth this formula in two minutes or so, and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes on this amazing jewel, we shall surround the jewel by its proper setting in the grand design of that branch of mathematics which is called elementary algebra.
His first conclusion is that, given the invention of complex numbers, every algebraic equation can be solved. This is the fundamental theorem of algebra. On page 14: The measurement problem we will see that this theorem makes the solution of quantum mechanical problems possible. Fundamental theorem of algebra - Wikipedia
He then goes on to conclude:
we summarize this with the most remarkable formula in mathematics:ei θ = cos(θ) + i sin(θ)
This result is known as Euler's formula. Here, we might say, is the formal mathematical backbone of quantum mechanics. Feynman Lectures on Physics, Vol 1, chapter 22: Algebra, Euler's formula - Wikipedia
10.9: Wigner's mystery
What does this remarkable collusion between mathematics and physics mean? After a critical discussion, Wigner writes:
Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. Eugene Wigner (1960): The Unreasonable Effectiveness of Mathematics in the Natural Sciences
I feel that there is a theological reason for this gift which was expressed by Aquinas in his explanation of the limits to God's omnipotence:
. . . God is called omnipotent because He can do all things that are possible absolutely; . . . For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey. Aquinas, Summa I, 25, 3
The formalist approach to mathematics proposed by Hilbert, which justifies the existence of Cantor's Paradise, puts a similar bound on the "omnipotence" of mathematics: every mathematical statement is acceptable as long as it does not involve a contradiction. God and mathematics are playing the same game, and this may be why, in a cognitive universe, mathematics, physics and theology have a lot in common. Formalism (mathematics) - Wikipedia, Cantor's paradise - Wikipedia
Feynman's first rule tells us that the real probabilities that we measure are determined by the square of the absolute value of the complex probability amplitudes computed by quantum theory. We calculate this square by multiplying a complex number by its complex conjugate. In a sense we might say that a complex number represents something that is half real and its complex conjugate represents the other half, and the multiplication providing us with the whole reality. This idea will gain substance as we go along. Ultimately we will conclude that Hilbert space is the scene in which the Universe observes itself, that is where it is conscious of itself, talking to itself. This is analogous to the way we consciously talk to ourselves to decide what to do. Hilbert space is the imagination of the Universe, the source of the variation which makes its evolution possible. Von Neumann, discussing quantum measurement, concludes that measurement increases the entropy of the Universe, it is the source of creation. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics, Chapter V
(revised on Tuesday 6 A ugust 2024)
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Notes and references
Further reading
Books
Dirac (1930,1983), P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)
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Links
Aquinas, Summa I, 25, 3, Is God omnipotent?, '. . . God is called omnipotent because He can do all things that are possible absolutely; which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.' back |
Cantor's paradise - Wikipedia, Cantor's paradise - Wikipedia, the fre encyclopedia, 'Cantor's paradise is an expression used by David Hilbert . . . in describing set theory and infinite cardinal numbers developed by Georg Cantor. The context of Hilbert's comment was his opposition to what he saw as L. E. J. Brouwer's reductive attempts to circumscribe what kind of mathematics is acceptable; see Brouwer–Hilbert controversy.' back |
Catherine & Johnathan Karoly, Heitor Villa-Lobos: The Jet Whistle, ' Villa-Lobos never forgot his “musical education” – the Rio street bands, the trips to the Amazon, and the music of the movie halls and theaters of his teen-age years. He fused these diverse influences into a powerfully nationalist musical voice. Villa-Lobos composed Assobio a Jato (The Jet Whistle) in New York in 1950. The composer named his work to describe the technique he calls on the flutist to use during its last movement. To produce the effect, the player blows directly and forcefully into the flute with his or her mouth almost covering the mouthpiece. Combined with a glissando, the resulting whistle sounds like a jet taking off.' back |
Complete theory - Wikipedia, Complete theory - Wikipedia - Wikipedia, the free encyclopedia, 'I In mathematical logic, a theory is complete if, for every closed formula in the theory's language, that formula or its negation is demonstrable. Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. ' back |
Complex number - Wikipedia, Complex number - Wikipedia, the free encyclopedia, 'A complex number is a number that can be expressed in the form a + bi, where a. and b are real numbers and |
Complex plane - Wikipedia, Complex plane - Wikipedia, the free encuclopedia, ' In mathematics, the complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.' back |
Computability theory - Wikipedia, Computability theory - Wikipedia, the free encyclopedia, 'Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory. The basic questions addressed by recursion theory are "What does it mean for a function from the natural numbers to themselves to be computable?" and "How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched.' back |
Conic section - Wikipedia, Conic section - Wikipedia, the free encyclopedia, ' In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. back |
Double-slit experiment - Wikipedia, Double-slit experiment - Wikipedia, the free encyclopedia, 'In the double-slit experiment, light is shone at a solid thin plate that has two slits cut into it. A photographic plate is set up to record what comes through those slits. One or the other slit may be open, or both may be open. . . . The most baffling part of this experiment comes when only one photon at a time is fired at the barrier with both slits open. The pattern of interference remains the same as can be seen if many photons are emitted one at a time and recorded on the same sheet of photographic film. The clear implication is that something with a wavelike nature passes simultaneously through both slits and interferes with itself — even though there is only one photon present. (The experiment works with electrons, atoms, and even some molecules too.)' back |
Eugene Wigner (1960), The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back |
Euler's formula - Wikipedia, Euler's formula - Wikipedia, the free encyclopedia, 'Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics". (Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley, p. 22-10. ISBN 0-201-02010-6.) back |
Feynman, Leighton & Sands FLP III_1, The Feynman Lectures on Physics volume III: Chapter 1: Quantum Behaviour, ' “Quantum mechanics” is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.' back |
Feynman, Leighton & Sands FLP III:08, Chapter 8: The Hamiltonian Matrix, 'One problem then in describing nature is to find a suitable representation for the base states. But that’s only the beginning. We still want to be able to say what “happens.” If we know the “condition” of the world at one moment, we would like to know the condition at a later moment. So we also have to find the laws that determine how things change with time. We now address ourselves to this second part of the framework of quantum mechanics—how states change with time.' back |
Feynman, Leighton & Sands: FLP I:22, The Feynman Lectures on Physics: I:22 Algebra , ' In our study of oscillating systems we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics. From the physicist’s point of view we could bring forth this formula in two minutes or so, and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes on this amazing jewel, we shall surround the jewel by its proper setting in the grand design of that branch of mathematics which is called elementary algebra.' back |
Formalism (mathematics) - Wikipedia, Formalism (mathematics) - Wikipedia, the free encyclopedia, ' In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules. For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.' back |
Fundamental theorem of algebra - Wikipedia, Fundamental theorem of algebra - Wikipedia, the free encyclopedia, 'The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.' back |
General covariance - Wikipedia, General covariance - Wikipedia, the free encyclopedia, In theoretical physics, general covariance (also known as diffeomorphism covariance or general invariance) is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.' back |
Hamiltonian (quantum mechanics) - Wikipedia, Hamiltonian (quantum mechanics) - Wikipedia, the free encyclopedia, 'I In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system (this addition is the total energy of the system in most of the cases under analysis). It is usually denoted by H . . .. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who created a revolutionary reformulation of Newtonian mechanics, now called Hamiltonian mechanics, which is also important in quantum physics. ' back |
Jessica Haines, Two Stones Wave Patterns, Wave patterns from two stones thrown into calm water. back |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Linear function (calculus) - Wikipedia, Linear function (calculus) - Wikipedia, the free encyclopedia, ' In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane.The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input. Linear functions are related to linear equations.' back |
Pythagorean theorem - Wikipedia, Pythagorean theorem - Wikipedia, the free encyclopedia, ' In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. . . .. The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem.' back |
Quantum harmonic oscillator - Wikipedia, Quantum harmonic oscillator - Wikipedia, the free encyclopedia, 'The quantum harmonic oscillator is the quantum-mechanical analogue of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.' back |
Quantum state - Wikipedia, Quantum state - Wikipedia, the free encyclopedia, 'In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.' back |
Quantum superposition - Wikipedia, Quantum superposition - Wikipedia, the free encyclopedia, 'Quantum superposition is the application of the superposition principle to quantum mechanics. The superposition principle is the addition of the amplitudes of waves from interference. In quantum mechanics it is the sum of wavefunction amplitudes, or state vectors. It occurs when an object simultaneously "possesses" two or more possible values for an observable quantity (e.g. the position or energy of a particle)' back |
Real number - Wikipedia, Real number - Wikipedia, the free encyclopedia, 'In mathematics, a real number is a value that represents a quantity along a continuous line. . . . The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field (R ; + ; · ; <), up to an isomorphism,' back |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, ' In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . . In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Warner Brothers Loony Tunes, Roadrunner VS Coyote Compilation, ' No amount of ACME products can prepare Wile E Coyote for Road Runner's tricks!' back |