Quantum mechanics I

  1. From polarisation to quantum mechanics: states, observables, Born's law 
  2. Projection operators, generalised Born's rule, position basis, wavefunction
  3. Dealing with eigenspaces; noncommuting variables and another postulate
  4. Position, momentum bases and operators, Fourier transform, uncertainty
  5. Systems and sub-systems, tensor product space, quantum entanglement
  6. Mixed states I: density matrix, partial trace, the most general Born rule
  7. Mixed states II: decoherence; important measures of purity and entropy
  8. Time evolution, Schrodinger and Heisenberg pictures, Noether's theorem
  9. ħ→0: Emergence of classical phenomena, waves in quantum theory
  10. Important operators: angular momentum, spin and other related
  11. Examples
  12. Open quantum systems
  13. Interesting no-go theorems
  14. Symmetries and superselection
  15. Comments on the axiomatisation of quantum mechanics and theory
  16. Obligatory comment on quantum foundations
Plan for article 16:
  • (copy over the rant from article 1) -- Wigner's friend. Exclusivity of states is a way to reproduce the same results of the "hidden reality" of classical mechanics without having one.
  • "Shut up and calculate" because logical positivism 
  • What exactly is superposition? "Or" not "and" and it talks about our knowledge of the state (our knowledge is probabilistic)
  • What exactly makes a theory quantum (commutators/bounds on uncertainty)?
  • Superpositions vs mixed states -- they are completely different and unrelated. It doesn't matter the language we use to describe them may be sometimes similar (e.g. "the photon has a 50% probability of being vertically polarised..."), they are mathematically different, and physically different, which means they can be tested empirically (by using a diagonal filter). This is a limitation of language, not of physics.
  • Quantum entanglement -- "or" and logical positivism clarify things -- look at stuff from each/any observer's perspective.
Plan for article 15:
  • axiomatisation vs motivation -- e.g. Born's law, eigenvalues as observables, etc.
  • state axioms, separated into fundamental axioms and one that has to do with our theory
  • quantum theory vs a specific quantum theory
  • de Broglie stuff -- do we derive momentum from de Broglie or from Noether's theorem? In terms of mathematical axiomatisation and in terms of physical motivation (give the original motivation)
  • Gleason's theorem
  • Rigged Hilbert spaces
Other stuff: path integrals, variational QM, raising and lowering operators, creation and annihilation operators, ground states, Fock space, entanglement monogomy, Bloch sphere, quantum computing [1], Wigner-Weyl transform, Wigner quasiprobability distribution, quantisation... https://motls.blogspot.com/2011/06/density-matrix-and-its-classical.html

Mixed states II
https://en.wikipedia.org/wiki/Purity_(quantum_mechanics)
Decoherence, https://motls.blogspot.com/2014/10/redundant-synonyms-of-decoherence.html
https://motls.blogspot.com/2014/10/redundant-synonyms-of-decoherence.html
https://physics.stackexchange.com/questions/317126/off-diagonal-elements-of-density-matrix-measurement-of-coherence
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Theorems
no-cloning,
no-communication,
https://en.wikipedia.org/wiki/Quantum_entanglement#Classification_of_entanglement,
https://en.wikipedia.org/wiki/Quantum_entanglement#Entropy
Bell's theorem, enwp.org/Bell_statemotls.blogspot.com/2019/06/bells-inequality-is-straightforward.htmlhttps://motls.blogspot.com/2017/09/why-vanishing-commutators-imply-theres.html
Does Bob create hidden variables for Alice? What does Bell's experiment say here?
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Classical limits
https://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html
Classical approximation -- Ehrneferst, hbar approach zero, decoherence, just looking at probabilities, quantisation
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Examples
box, well, tunneling, H atom, double-slit
double-slit weirdness
https://qchu.wordpress.com/2011/07/16/the-heisenberg-picture-of-quantum-mechanics/ -- worked out examples
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Evolution and unitarity
The energy-time uncertainty principle
https://motls.blogspot.com/2016/03/measurement-isnt-violation-of-unitarity.html
quora.com/What-is-a-tensor-product-in-quantum-mechanics/answer/Charuhas-Shiveshwarkar
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Open quantum systems
https://en.wikipedia.org/wiki/Open_quantum_system
https://en.wikipedia.org/wiki/Lindbladian
https://en.wikipedia.org/wiki/Belavkin_equation
https://motls.blogspot.com/2017/02/lindblad-equation-cant-solve-any.html

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