Difference between revisions of "Quantum Mechanics"

Revision as of 17:18, 29 May 2023

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.

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 Schrodinger developed another model based on the flow of a quantum from one (potentially observable) event to another known as the wave equation. It was later shown 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.^[1]

Mathematical Basis for Theories

We’re spoiled by how straightforward modern theories are. When you study quantum field theories, for instance, you have the particle concept. 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. And Causal Dynamical Triangulations is part of this trend of going back to basics.^[2]

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 and 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 world, Hamilton's focus 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

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 naturally. The certainty as to position and velocity that worked in Newton's model are no longer possible with the Hamiltonian model.

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.^[3]

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.^[4] 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.^[5] 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 Plank to employ quanta to light radiation.^[6] 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.^[7] It is not known if a version of Quantum Mechanics can be formulated without Fourier transforms.

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. After 32 years, 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 success of Quantum Mechanics was demonstrated when Pauli was able^[8] 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."^[9](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 Schoedinger'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^[6] 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."^[10]

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 an can be related to the intensities of the corresponding harmonics. In the high quantum number limit, the hydrogen atom do 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 Schrodinger 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?

Law of excluded middle can be violated by some quantum operations. So in that sense we can say the Aristotle's rules of logic (aka Boolean logic) was not up the the challenge of describing how Quantum Mechanics worked.

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.^[9] (page113)

Solutions

At its core, Quantum mechanics is simply a mathematical model for predicting the behaviors of microscopic particles — or, at least, of the measuring instruments we use to explore those behaviors — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had.^[11] But still, it is just a model and so false, even though it is extremely useful.^[12]

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

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 solutions 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^[9]

Consequences

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.^[13]

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.

References

1. ↑ Max Born. The Statistical Interpretation of Quantum Mechanics, (1954-12-11) https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf
2. ↑ 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/
3. ↑ https://www.britannica.com/science/quantum
4. ↑ 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/
5. ↑ Richard Feynman PhD thesis
6. ↑ ^{6.0 6.1} David Bohm, Quantum Theory Prentice Hall (1951)
7. ↑ Emanuele Pesaresi, Uncertainty Principle Derivation from Fourier Analysis https://www.linkedin.com/pulse/uncertainty-principle-derivation-from-fourier-emanuele-pesaresi
8. ↑ Wolfgang Pauli,
9. ↑ ^{9.0 9.1 9.2} David Lindley, Uncertainty Doubleday ISBN 9780385515061
10. ↑ 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
11. ↑ Jenann Ismael , Quantum Mechanics Plato (2020-09-20) https://plato.stanford.edu/ENTRIES/qm/
12. ↑ 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/
13. ↑ https://physics.stackexchange.com/questions/561382/what-exactly-is-the-difference-between-entanglement-and-correlations