User:Johnjbarton/sandbox

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."

Alpha particle scattering from Thomson's atom

Historically, Thomson and Crowther studied the scattering of beta particles from metal foils while Rutherford and Geiger used alpha particles. Alpha particles had an advantage unknown at the time: they had enough momentum to ignore interactions with atomic electrons but not enough energy to penetrated the nucleus. Prior to Rutherford's 1911 paper, there was no alpha particle scattering model.

Beiser has analyzed alpha scattering from a plum-pudding-like atomic model. The analysis is divided into three parts: scattering from electrons, scattering from the positive sphere, and the consequences of multiple scattering.

Scattering from atomic electrons

Alpha particle momentum change due to 90 degree collision with atomic electron, after Beiser, 1969.

Beiser notes that the electron is 7000 times lighter than the alpha particle. In any collision, the electron will be forcefully pushed aside. The largest deflection of the alpha particle would occur when the electron is pushed away at 90 degrees. For an upper limit on the sideways push, Beiser calculates the largest possible push for any angle: a direct head-on collision. That is, he uses the geometry of 90 degree scattering as shown in the diagram, but gets the size of $\Delta p$ from head-on collision as follows.

Using $M,V'$ for the alpha particle and $m,v'$ for the atomic electron, with unprimed velocities measured before collision, conservation of momentum means the electron momentum change equals the alpha particle momentum change: $$\Delta p=MV-MV'=mv'$$ Similarly, all of the energy lost by the alpha particle is gained by the electron: $$(1/2)MV^{2}-(1/2)MV'^{2}=(1/2)mv'^{2}$$ Separating the mass and velocity terms and using $V^{2}-V'^{2}=(V+V')(V-V')$ allows the energy formula to be divided by the momentum formula giving $$V+V'=v'$$ Next Beiser eliminates the alpha particle's final velocity $V'$ using the momentum conservation equation again: $$v'={\frac {2M}{M+m}}V$$ Since the mass of the alpha particle is much greater than the mass of the electron, $v'\approx 2V$ and the head-on momentum change is $\Delta p=mv'=2mV$.

From the geometry in the diagram $\sin \theta =\Delta p/p'$ but the angle is so small that we can use $\theta \approx \Delta p/p$, giving the angle for alpha scattering by the atomic electrons as $$\theta _{e-\alpha }={\frac {2mV}{MV}}={\frac {2}{7000}}\approx 3\times 10^{-4}{\text{radians}}=0.02^{\circ }.$$ Thus alpha particles scatter very little from the very light electrons.

Scattering from the positive sphere

Next Beiser considers the effect of scattering from the positive sphere. The alpha particle will experience maximum deflection if it just grazes the edge of the positive sphere because that is where the electric field is at its strongest. If the alpha particle were to pass through the sphere, not all of its positive charge would be pushing it out.

In the impulse model the average repelling force is applied for a time where that force is large. The alpha particle's lateral change in momentum Δp_y can be approximated using the Coulomb force over the distance equal to the radius of the atom, ${\frac {kq_{a}q_{g}}{r^{2}}}$, applied just for the time the alpha particle passes, which is equal to ⁠2r/v⁠:

$$\Delta p_{\text{y}}={\bar {F}}t\approx {\frac {kq_{a}q_{g}}{r^{2}}}\cdot {\frac {2r}{v}}$$

where

• q_g = positive charge of the gold atom = 79 e = 1.26×10⁻¹⁷ C
• q_a = charge of the alpha particle = 2 e = 3.20×10⁻¹⁹ C
• r = radius of the gold atom = 1.44×10⁻¹⁰ m
• v = speed of the alpha particle = 1.53×10⁷ m/s
• m = mass of the alpha particle = 6.64×10⁻²⁷ kg
• k = Coulomb constant = 8.987×10⁹ N·m²/C²

Geometrically, the tangent of the deflection angle θ is the ratio of the lateral and forward momentum but for small angles $\tan \theta \approx \theta$ giving: $$\theta _{+-\alpha }={\frac {\Delta p_{\text{y}}}{p_{\text{x}}}}\approx {\frac {kq_{a}q_{g}}{r^{2}}}\cdot {\frac {2r}{v}}\cdot {\frac {1}{mv}}=0.0003~{\text{radians}}~({\text{or}}~0.02^{\circ })$$ Alpha particle scatter very little from the relatively large, diffuse positive sphere.

Scattering by multiple collisions

Despite the extremely small scattering angles from either atomic electrons or a positive sphere, a gold foil like the one Rutherford and his colleagues used would be around 10,000 atoms thick. Beiser concludes his analysis of alpha particle scattering by considering the combination of many collisions.

Each collision may increase or decrease the total scattering angle. Only very rarely would a series of collisions all line up in the same direction. The result is similar to the standard statistical problem called a random walk. If the average deflection angle of the alpha particle in a single collision with an atom is ${\bar {\theta }}$, then the average deflection after n collisions is $${\bar {\theta }}_{n}={\bar {\theta }}{\sqrt {n}}$$ The probability that an alpha particle will be deflected by a total of more than 90° after n deflections is given by: $$e^{-(90/{\bar {\theta }}_{n})^{2}}$$ Where e is Napier's constant. If we assume an average deflection per collision of 0.01°, and therefore an average deflection of 1° after 10,000 collisions, then the probability of an alpha particle being deflected by more than 90° will be $$e^{-(90/1)^{2}}=e^{-8100}\approx 10^{-3518}$$ While in Thomson's "plum pudding" model it is possible that an alpha particle could be deflected by more than 90° after 10,000 collisions, the probability of such an event is so low as to be inconceivable. This extremely small number shows that Thomson's model of 1906 cannot explain the Geiger-Mardsen results of 1909.

Contemporary reactions

Rutherford's 1911 paper on alpha particle scattering contained largely the same points as described above and yet in the years immediate following its publication few scientists took note. The scattering model predictions were not considered definitive evidence against Thomson's plum pudding model. Thomson and Rutherford had pioneered scattering as a technique to probe atoms, its reliability and value were unproven. Before Rutherford's paper the alpha particle was considered an atom, not a compact mass. It was not clear why it should be a good probe. Rutherford's paper did not discuss the atomic electrons vital to practical problems like chemistry or atomic spectroscopy. Rutherford's nuclear model would only become widely accepted after the work of Niels Bohr.

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.

---