User:Johnjbarton/sandbox

Klein paradox

Notes on Cheng, T.; Su, Q.; Grobe, R. Two groups of resolution:

• Bjorken and Drell and others are purely QM on incoming electron, with pair-creation tacked on. pg 324
• Dombey and Calogeracos [63], by Hansen and Ravndal [48], describe pair-creation at barrier but no electron incoming. pg 325.
  • 63 says: "We hope that this discussion has demonstrated that the Klein step is pathological and therefore a misleading guide to the underlying physics."

Why potentials not forces

Feynman Lectures https://www.feynmanlectures.caltech.edu/I_14.html

• "The reason we bring this out is that the idea of force is not particularly suitable for quantum mechanics; there the idea of energy is most natural. We find that although forces and velocities “dissolve” and disappear when we consider the more advanced forces between nuclear matter and between molecules and so on, the energy concept remains. Therefore we find curves of potential energy in quantum mechanics books, but very rarely do we ever see a curve for the force between two molecules, because by that time people who are doing analyses are thinking in terms of energy rather than of force."

Goldstein p4. In a conservative system -V is the work done $$-V_{12}=W_{12}=\int \limits _{1}^{2}\mathbf {F} \cdot d\mathbf {s}$$ Goldstein p76. The inverse square law is the most important of all central force laws....

• Curtis, Lorenzo J. "A 21st century perspective as a primer to introductory physics." European journal of physics 32.5 (2011): 1259.
  • Introductory courses use concepts involving forces and accelerations that never enter in more advanced formulations."

Kibble uses energy to derive Kepler orbits.

• The Kepler Problem is a “soluble problem with a difference” in the sense that it stands in convincingly close approximation to some physical systems that matter —systems that lie at the heart both of celestial mechanics and of atomic physics. It was observational data that—by inspired analysis—led Kepler to the laws which now bear his name, but Kepler himself was never in position to formulate what we now call “the Kepler Problem,” for he inhabited a pre-dynamical world. It was Newton who was first in position to pose— and to solve—the dynamical “Kepler Problem,” and it was his success in this regard which most strongly recommended Newton’s Laws of Motion to the attention of the world. And it was a variant of this same physical model that in our own century inspired the work of, and lent credibility to, the successive accomplishments of Bohr, Heisenberg and Schr ̈odinger. The Kepler Problem has served as a primary stimulus to mathematical/physical invention for now more than 300 years, and still today retains many of its secrets.
  • Wheeler, N. "Kepler problem by descent from the Euler problem." (1995): 1-27.

Speiser, D. The Kepler Problem from Newton to Johann Bernoulli. Arch. Hist. Exact Sci. 50, 103–116 (1996). https://doi.org/10.1007/BF02327155

Plum pudding model

• Kragh, Helge. "Before Bohr: Theories of atomic structure 1850-1913." RePoSS: Research Publications on Science Studies 10 (2010). https://css.au.dk/fileadmin/reposs/reposs-010.pdf
  • "The vortex theory and J. J. Thomson’s electron theory were among the more successful of the pre-Bohr atomic theories."
  • "The English physician and chemist William Prout argued in 1815-1816 that the atomic weights indicated a common composition of the elements, namely that all the atoms were made up of hydrogen atoms."
  • Positive electrification had no mass in 1904-1906 model, only after the number of electrons decreased in 1910.
  • The model with lots of electrons had chemical periodicity but no the later model with one/one.
  • "According to Thomson, the alpha particle was of atomic dimensions and contained 10-12 electrons"
• Sinclair, Steve B. "JJ Thomson and the chemical atom: from ether vortex to atomic decay." Ambix 34.2 (1987): 89-116.
• Baily, C. Early atomic models – from mechanical to quantum (1904–1913). EPJ H 38, 1–38 (2013). https://doi.org/10.1140/epjh/e2012-30009-7
  • "The very existence of atomic radiation strongly suggested that atoms were not indivisible after all,"
  • Discusses Thomson's large number of electrons giving all of the mass in 1904 ppr: "This implied that the positive charge contributed nothing to the atomic mass. "
• Kragh, H. The First Subatomic Explanations of the Periodic System. Foundations of Chemistry 3, 129–143 (2001). https://doi.org/10.1023/A:1011448410646
  • "J. J. Thomson’s ‘plum pudding model’ of the atom is usually dated 1904, the year when he gave a detailed and mathematically elaborate version of it. However, the essence of the model can be found, if only qualitatively, in his famous paper of 1897 in which he announced the discovery of the ‘corpuscle'...
• Heilbron 1968
  • "The most persuasive and detailed exposition of the plenary atom is in Thomson's well-known paper of 1904. In this version a uniformly charged jelly filling the atomic volume retains the vast electronic hive which circulates freely throughout it."

Comparison to JJ Thomson's results

At the time of Rutherford's paper, JJ Thomson was the "undisputed world master in the design of atoms". Rutherford needed to compare his new approach to Thomson's. Thomson's model presented in 1910 relied on multiple or compound scattering from electrons ($\phi _{1}$) and a contribution ($\phi _{2}$) from the positive sphere surrounding them. Thomson modeled the electron collisions with hyperbolic orbits from his earlier paper in 1906, with an additional random walk component due to multiple collisions to be combined as $\phi ={\sqrt {\phi _{1}^{2}+\phi _{2}^{2}}}.$ The average deflection, quoted by Rutherford as: $$\phi _{1}={\frac {\pi }{4}}{\frac {NeE}{mu^{2}}}{\frac {1}{R}}$$ and $$\phi _{2}={\frac {16}{5}}{\frac {eE}{mu^{2}}}{\frac {1}{R}}{\sqrt {\frac {3N}{2}}}$$ The physical constants in these formula can be replaced by Rutherford's combination: $$b={\frac {2NeE}{mu^{2}}}$$ giving $$\phi _{1}={\frac {\pi }{8}}{\frac {b}{R}},\ \phi _{2}={\frac {8}{5}}{\sqrt {\frac {3}{2N}}}{\frac {b}{R}}$$ Rutherford computes his value for ($\phi _{1}$) value using single scattering from a compact charge and demonstrates that it is 3 times larger than Thomson's multiple scattering value: $$\phi _{1}={\frac {3\pi }{8}}{\frac {b}{R}}$$

Measurement problem notes

A thought experiment called Schrödinger's cat illustrates the measurement problem. A mechanism is arranged to kill a cat if a quantum event, such as the decay of a radioactive atom, occurs. The mechanism and the cat are enclosed in a chamber so the fate of the cat is unknown until the chamber is opened. Prior to observation, according to quantum mechanics, the atom is in a quantum superposition, a linear combination of decayed and intact states. Also according to quantum mechanics, the atom-mechanism-cat composite system is described by superpositions of compound states. Therefore, the cat would be described as in a superposition, a linear combination of two states an "intact atom-alive cat" and a "decayed atom-dead cat". However, when the chamber is opened the cat is either alive or it is dead: there is no superposition observed. After the measurement the cat is definitively alive or dead.

The cat scenario illustrates the measurement problem: how can an indefinite superposition yield a single definite outcome? It also illustrates other issues in quantum measurement. , including the Heisenberg cut (the boundary to between classical and quantum systems)? and what the role of the observer.

The story of the cat was originally invented by Edwin Schrodinger, in discussions with Albert Einstein, to explore the relationship between quantum mechanical wave functions and reality.BAGGOTT. Later it became way of illustrating other issues with quantum mechanics, including the measurement problem.PERES/Schoshaer.

The story describes an imaginary experiment: a cat is placed in a chamber with poison vial triggered by a Geiger counter sensitive to radioactivity. A radioactive substance is added and chamber is shut. The quantum model of the radioactive substance includes a superposition of decayed and intact atoms and, by the logic of quantum theory, the counter forms a composite system when it interacts with those atoms, so the composite would also be described by the states in an indeterminate superposition. The poison vial interacts with the counter and the vial with interacts with the cat, so the quantum description of the cat includes superposition of two states, one including a live cat and one with a dead cat. When chamber is opened however, only one definite state is expected.

Baggott pg 155. The cat scenario combines two rules of quantum mechanics: describing indefinite states by quantum superposition and the composition of two systems. Prior to measuring atomic disintegration the wave function of the atom of radioactive substance is indefinite, a combination of two states, one intact and one disintegrated. A Geiger counter tick announces that one atom has a definite state of disintegrated. But a Geiger counter is composed of atoms: we could consider the radioactive substance and the counter as a single quantum system. The quantum composition would include two states, one with counter registering a tick (atom disintegrated) and one without a tick. Schrodinger included the poison and the cat to create an even larger quantum system with two states. One state has a live cat and the other a dead cat: a 'quite ridiculous case'.

Wave function collapse notes

• Hance, J.R., Rarity, J. & Ladyman, J. Could wavefunctions simultaneously represent knowledge and reality?. Quantum Stud.: Math. Found. 9, 333–341 (2022). https://doi.org/10.1007/s40509-022-00271-3
  • However, we argue, nothing about the informal ideas of epistemic and ontic interpretations rules out wavefunctions representing both reality and knowledge.
  • 2022, 10 citations.
• Fuchs, Christopher A.; Peres, Asher (2000-03-01). "Quantum Theory Needs No 'Interpretation'". Physics Today. 53 (3): 70–71. doi:10.1063/1.883004. ISSN 0031-9228.
  • 439 citations.
  • Quantum theory has been accused of incompleteness because it cannot answer some questions that appear reasonable from the classical point of view.
  • Collapse is something that happens in our description of the system, not to the system itself.
• Frauchiger, Daniela, and Renato Renner. "Quantum theory cannot consistently describe the use of itself." Nature communications 9.1 (2018): 3711.
  • ... quantum theory cannot be extrapolated to complex systems, at least not in a straightforward manner.
  • a way test or categorize interpretations.
  • 456 cites
• Burt, M. G. "On The Peierls Interpretation of Quantum Mechanics." arXiv preprint arXiv:1805.11162 (2018).
  • The fundamental tenet of his work is that the wavefunction or density matrix represents the knowledge of an observer and that two observers of the same system may well have different knowledge and will use different density matrices to describe it
• Stamatescu, Ion-Olimpiu (2009). Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.). "Wave Function Collapse". Berlin, Heidelberg: Springer Berlin Heidelberg: 813–822. doi:10.1007/978-3-540-70626-7_230. ISBN 978-3-540-70622-9. {{cite journal}}: Cite journal requires |journal= (help)
  • Encyclopedic like article, not the clearest to be sure.
  • Sort formalism similar to our article.
  • Physical approaches cover three types, very nice.
• Griffiths 3rd
  • Pg 221: if you want to prepare a beam of atoms in a given spin configuration, you pass an unpolarized beam through a Stern–Gerlach magnet, and select the outgoing stream you are interested in (closing off the others
  • Footnote 49 p564 discusses collapse w/successive measurements.
• Hartle, James B. "The quantum mechanics of cosmology." arXiv preprint arXiv:1805.12246 (2018).
  • "The small numbers which estimate the failure of decoherence among familiar quasiclassical operators show how excellent the approximation of exact decoherence is in the ideal measurement model of the Copenhagen approximation to quantum mechanics."
  • Claims to describe a "post-Everett interpretation".
  • Says Everett incomplete
• Schlosshauer, Maximilian. "Decoherence, the measurement problem, and interpretations of quantum mechanics." Reviews of Modern physics 76.4 (2005): 1267.
  • von Neumann (1932) defined an ideal measurement scheme (also noted by Hartle). ** von Neumann not equivalent to Copenhagen which makes the apparatus classical.
• Ohanian, "Principles of QM".
  • P 352. Change of wavefunction not governed by Schrodinger eqn. Collapse is not produced by interaction between system and apparatus. Stern-Gerlach with long-wavelength laser detectors. Interaction leads to "superposition in which the spin-up and spin-down states are correlated with detector states". A second Stern-Gerlach in tandem with reversed field to restore original spin state (but not state of detectors). Thus collapse is not explained here. Ohanian describes a "popular interpretation" where the expectation values have cross terms that cancel.
• Alastair Rae, Quantum Physics illusion or reality. Cambridge.
  • Bell's theorem simplified.
• E. Merzbacher, Quantum Mechanics, 3rd ed. P406.
  • Multiple S-G experiments. If s_z is measured in the first expr., second experiment remeasures, finds only up in upper beam, down in lower beam, no further split. If the second S-G is rotated, result is now two equal intensity beams.
  • Simple SG is example of "ideal measurement" aka "measurement of the first kind".
  • SG device acts as a "spin filter"

Zero point energy; Casimir effect

• Jaynes, Edwin T. "Probability in quantum theory." Complexity, entropy, and the physics of information (1990): 381.
  • One sees the effect, like the van der Waals attraction, as arising from correlations in the state of electrons in the two plates, through the intermediary of their source fields (1). It do es not require ZP energy to reside throughout all space, any more than do es the van der Waals force.

Spin (physics) notes

Sebens

Ref Classical field theory before quanitization.

• Three obstacles for classical model
  • Superluminal velocity to get ang. mom.given classical radius
    • need to describe classical radius, where ang. mom value comes from
  • Superluminal velocity to get mag. mom.given classical radius
    • explain mag. mom.
  • Ratio wrong.
• Kronig et al. knew all of this.
• Free Dirac eqn as classical field, to be quantized.
• In Dirac field,
  • flow velocity automatically tops out at c
  • charge rotates at 2x mass, explaining gyromagnetic ratio.
• Reviews other spin models.

Giulini

Ref.

Compares spinning spheres to Pauli's two-level.

• First instance of quantum degree of freedom without the corresponding classical
• Comments that the historical rejection of classical electron models does not apply in modern times.
• Good on Pauli point of view.

Garraway Stenholm

Ref. Is spin intrinsic to electron? Can the free electron magnetic moment be measured.

• Mostly historical review.
  • Bohr claim that moment can't be measured for an electron in a (classical) trajectory.
• Uncertainty principle on momentum blurs trajectories needed to distinquish moment.

Juha Saats

Ref. Uses spin to argue about a form for scientific realism called "progress realism".

• Two page summary of the impact of spin model across fundamental to applications.
• Progress realism focuses on one area of knowledge without requiring models for everything at once.

Leader and Lorce

Ref. Review and analysis of QFT total angular momentum in interactions.

• Decomposition in to spin and orbit not unique; all equivalent.
  • Two classes, Belinfante and canonical.
  • Gauge produces different ones.
• Gluons, photons, electrons; QCD and QED, models of interactions.
• Good review ref for Belinfante.

Decoherence observation

Using double slits with extremely thin layer of Al metal, an electron double-slit experiment can be converted to a which-way experiment. Some electrons lose energy due to interaction with the thin layer: these 'inelastic' electrons clearly went through the corresponding slit. They show no interference. Electrons which do not lose energy do show interference. The interpretation is that "loss of coherence is related to the localization of the inelastic electrons within the slits".

Decoherence theory

Decoherence theory grew out of Zeh's extensions to Hugh Everett III's "Relative states" interpretation of quantum mechanics.

Messiah pg 155: double slit and complementarity. "Optical tests of complementarity" in depth analysis of double slit wrt complementarity.

History for Introduction to quantum mechanics

Main article: History of quantum mechanics

Quantum mechanics emerged from efforts to explain experimental results obtained in the last years of the 19th century. Maxwell's unification of electricity, magnetism, and even light in the 1880s lead to experiments on the interaction of light and matter. Some results defied the existing theories.

A black body radiator used in the GL Optic CARLO laboratory in Puszczykowo, Poland. The interior of the radiator is graphite in an atmosphere of Argon, heated to 3000K.

Unused material follows

A two dimensional pattern of ridges alternating with flat areas on silicon illuminated by a cold neon atomic beam acts as a reflection hologram.

• Optically shaped matter waves

• Thermal manipulation of matter wave wavelength. Advances in laser cooling have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal de Broglie wavelengths come into the micrometre range. Using Bragg diffraction of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold sodium atoms was explicitly measured and found to be consistent with the temperature measured by a different method.

This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometre resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

The effect has also been used to explain the spatial version of the quantum Zeno effect, in which an otherwise unstable object may be stabilised by rapidly repeated observations.

Buchanan light-weight review of matter wave diffraction.

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