In summary, classical logic is represented by the algebra of sets, while quantum mechanics uses vectors and linearity to express probability distributions. The transition between the two can be understood through the properties of probability distributions on classical states. One approach to understanding QM is through the concept of a POV, which can be derived from Gleason's Theorem. However, in order to apply the formalism to problems, further development and understanding is needed. This can be achieved by studying and applying the principles outlined in Scott Aaronson's lecture on quantum mechanics.
jambaugh
Gold Member
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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bhobba, Mentz114, fresh_42 and 1 other person
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

## 1. What is superposition and why is it important in science?

Superposition is the principle that states when two or more waves or particles interact, the resulting wave or particle is the sum of the individual waves or particles. This is important in science because it helps us understand and predict the behavior of complex systems, such as quantum mechanics and electromagnetic fields.

## 2. How does superposition work mathematically?

Mathematically, superposition is represented by the principle of linearity, which states that the sum of two solutions to a linear equation is also a solution. In other words, the mathematical representation of superposition is the addition of individual solutions to a problem to obtain the overall solution.

## 3. Can you give an example of superposition in everyday life?

One example of superposition in everyday life is the interference pattern produced by two overlapping waves, such as when you throw two pebbles into a pond and the ripples from each pebble interact. The resulting pattern is the superposition of the individual wave patterns.

## 4. How does superposition relate to the concept of quantum states?

In quantum mechanics, superposition is used to describe the state of a quantum system. This means that a system can exist in multiple states at the same time, and the overall state is a superposition of these individual states. This is a fundamental concept in understanding the behavior of particles on a quantum level.

## 5. Are there any limitations to the principle of superposition?

While superposition is a useful tool in understanding the behavior of complex systems, it does have limitations. It only applies to linear systems, and in some cases, it may not accurately predict the behavior of non-linear systems. Additionally, the principle of superposition does not take into account the effects of external factors such as friction or air resistance.

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