In energy conservation situations where the transient probability is additive, irrational numbers such as 2^1/2 have no place.
However, especially in QM as I understood it, the probability amplitude is multiplicative rather than additive.
The question is why in particular is 2^1/2 so important in QM?
An irrational such as you give has no known direct relevance to QM. Its math not physics. However, enough confusion have I heard recently as to ask how you reach this idea?
Many people may think that an irrational such as 2^1/2 is mathematical, not physical, and has no direct connection to quantum mechanics (QM).
On the other hand, we guess that's a great question even though no one really knows the exact answer.
We offer the following:
For the interpretation of probabilities in QM to make sense, the wave function Ψ must satisfy certain conditions.
An extremely important and yet rarely mentioned condition is,
Ψ squared = Ψ* squared=Ψ.Ψ* must always be positive and real.
This is the required answer.
Matrix transition chains B (solving the heat diffusion/conduction equation as a function of time) suggests finding an adequate alternative complex transition matrix (satisfying this condition) to solve the Schrödinger equation as a function of time.
what is quite striking is that 2^1/2 should appear explicitly and be expressed numerically as 1.142... in order to construct the required complex transition matrix.
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