Understanding Superposition Physically and Mathematically

[jambaugh]

What is superposition
Superposition is a fundamental concept in physics and mathematics, particularly in the field of wave theory and quantum mechanics. It describes how multiple wave-like phenomena can combine or interact with each other.
In physics, superposition can be summarized as follows:

Wave Superposition: When two or more waves overlap in space and time, the resulting wave is the algebraic sum of the individual waves’ amplitudes at each point in space and time. This principle applies to various types of waves, including electromagnetic waves (such as light), sound waves, and water waves.
Quantum Mechanics: In quantum mechanics, superposition refers to the ability of quantum systems (such as particles) to exist in multiple states simultaneously. According to the principle of superposition, a quantum system can exist in a linear combination of its possible states until observed or measured. This concept is often exemplified by the famous...

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[Stephen Tashi]

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.

 

[bhobba]

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