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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.^[1]: 154

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

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.
  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.
  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"

Spin (physics) notes

Sebens

Ref^[2] 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.^[3]

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.^[4] 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. ^[5]: 38 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.^[6] 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".^[7]

Decoherence theory

Decoherence theory grew out of Zeh's extensions^[8] to Hugh Everett III's "Relative states" interpretation of quantum mechanics.^[9]

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

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.

Unused material follows

A two dimensional pattern of ridges alternating with flat areas on

• Optically shaped matter waves[12]

• 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.^[13]

This effect has been used to demonstrate atomic

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.^[15]

Buchanan light-weight review of matter wave diffraction.^[16]