### Quantum mechanics

Introduction to quantum mechanics
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
Quantum information/computing/probability/logic

Philosophy of quantum mechanics
1. Comments on the axiomatisation of quantum mechanics and theory
2. Quantum mechanics as a replacement for probability theory
3. Obligatory comment on quantum foundations
Plan for quantum foundations article:
• (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 axiomatization article:
• 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