Difference between revisions of "Quantum Mechanics"

Revision as of 12:49, 23 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]

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.^[2] 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.^[3] It is now 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.

Observation

The success of Quantum Mechanics was demonstrated when Pauli was able^[4] 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."^[5](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.

Bell's Theorem

Describe 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."^[7]

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.

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 Observations 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.Cite error: Closing </ref> missing for <ref> tag But still, it is just a model and so false, even though it is extremely useful.^[8]

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.^[10]

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. ↑ David Bohm, Quantum Theory Prentice Hall (1951)
3. ↑ Emanuele Pesaresi, Uncertainty Principle Derivation from Fourier Analysis https://www.linkedin.com/pulse/uncertainty-principle-derivation-from-fourier-emanuele-pesaresi
4. ↑ Wolfgang Pauli,
5. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Lindley
6. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Bohm
7. ↑ 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
8. ↑ 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/
9. ↑ David Lindley, Uncertainty Doubleday ISBN 9780385515061
10. ↑ https://physics.stackexchange.com/questions/561382/what-exactly-is-the-difference-between-entanglement-and-correlations