Understanding Superposition Physically and Mathematically

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SUMMARY

This discussion focuses on the concept of superposition in both classical and quantum mechanics, emphasizing the transition from classical logic to quantum logic. It highlights the representation of probability distributions for classical states using vectors on an n-dimensional sphere, as discussed in Richard C. Henry's article. The conversation introduces a Markov chain example to illustrate the complexity of transitioning to quantum mechanics, specifically through the derivation of a matrix B that satisfies B^2 = A. The application of Gleason's Theorem is also mentioned as a crucial step in developing quantum mechanics from classical principles.

PREREQUISITES
  • Understanding of classical logic and set theory
  • Familiarity with linear algebra, particularly matrix operations
  • Knowledge of quantum mechanics fundamentals
  • Awareness of probability distributions and their representations
NEXT STEPS
  • Study the application of Gleason's Theorem in quantum mechanics
  • Explore the mathematical foundations of Markov chains
  • Learn about the representation of quantum states using vectors
  • Investigate the implications of classical versus quantum logic
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Students and researchers in physics, particularly those interested in quantum mechanics, mathematicians focusing on linear algebra, and anyone seeking to understand the foundational concepts of superposition in both classical and quantum contexts.

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Classical logic is concretely expressed using the algebra of sets.
Reference https://www.physicsforums.com/insights/understanding-superposition/

As I remember the article Quantum mechanics made transparent. by Richard C. Henry, it makes the argument that probability distributions for classical states can be conveniently represented by vectors that lie on the surface of an n-dimensional sphere (or equivalence classes of rays that pass through such points).

Perhaps there is a way to present the transistion between classical logic and quantum mechanics in a more gradual manner instead of the jump from the logic of sets to the methods of QM. The properties of probability distributions on classical states could be an intermediate step.
 
After thinking about this for a number of years now I finally decided on the following as a reasonable motivation for QM. Consider a simple Markov chain for turning a coin over each second. Its matrix, A, is dead simple, 0's on the main diagonal and 1's otherwise. Now we ask a simple question - what happens if we want to generalise this to what's going on at 1/2 second. We need to find the matrix B such that B^2 = A. Thats not a hard exercise in linear algebra, but low and behold, it's complex. Apply it to the starting state of the Markov chain and what do you get - a complex state. How are we to make sense of this? Well we define this thing called a POV and apply the modern easier version of Gleason's Theorem. From that you basically get the two axioms in Ballentine and QM can be developed from that. But - and this is a key point - you need to show how to apply the formalism to problems just like anything in applied math. The following is a good start along those lines:
https://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill
 

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