Note on Quantum Probability

Makoto Ohya

JAYDIP DATTA

Wave Function , Probability Density Function , Multiplicative Probability , Partition function .In this note we correlate a quantum normalized probabilistic approach with Algebric approach of Probability . The Probability Density [ Shi ]^2 may be equated as additive wave functions ie [Shi A] +[ Shi B ] . In real probability algebra we also know that P(A) +P(B )= P ( A*B ) . This is termed as multiplicative rule of Probability . Greater is P ( A*B ) = [ Shi ] ^2 greater will be the Probability Density F( x ) .

Kalyanapuram Rangachari Parthasarathy

Francesco R. Ruggeri

May 1, 2026 Correction: In the example of = dW/dx / W for W= Sum over p>=0 a(p)exp(ipx), Real is finite when W(x)=0, but Im( ) tends to infinite if one uses l'Hopital's rule. In the note, only the real part of is considered, but the Im part leads to 0/0. In previous notes, we argued that the Lorentz invariant -Et+px, allows for x,t and x+hbar/p, t+hbar/p solutions. We suggested this might give rise to the quantum mechanical probability exp(-iEt+ipx). In other notes, we sought a probability which conserves energy and momentum and respects Lorentz invariance, again leading to exp(-iEt+ipx). In a couple of notes, we argued that the uncertainty in x arises from a rest mass at x=0 jumping to x’/t’ = v as seen in the moving frame. This uncertainty in x forces one to introduce probabilities. Here we combine the latter idea with the notion that special relativity yields formulas for the interaction variables E and p. We argue that one may replace the notion of a force reaction as well as the idea of frames moving at constant speed relative to each other by demanding that an interaction be represented by an AND (product) of probabilities and this must necessarily conserve energy and momentum. This probability is in the location and time of interactions. This leads to a skewed dynamic probability exp(-iEt+ipx) because exp(ipx) and ex(-ipx) are not the same. Furthermore, as we have noted before, one uses exp(ipx) as an interaction probability which leads to complex average kinetic energy (nonrelativistic case) and momentum if one only uses p>=0. The Schrodinger equation (which is based on conservation of energy) bypasses this problem by combining p>=0 and p<0. Nevertheless, we note that there can exist W(x) = Sum over p>0 a(p)exp(ipx) such that if W(x) = 0 dW/dx and d/dx dW/dx are also zero at these points even though the average kinetic energy and momentum are complex. We finally show that a complex average can be used in a physical calculation, namely one dimensional reflection-refraction.

PierGianLuca Porta Mana, A. Maansson, Gunnar G. Bjork

This is the first part of a three-note study which starts from an analysis of "probabilities of probabilities" to arrive at old and new state-assignment methods in classical and quantum mechanics. In this note, probability-like parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of 'circumstance'. The idea is basically Laplace's and Jaynes', and rests on a theorem from probability theory which shows that a set of propositions can be uniquely parametrised by probability distributions. This parametrisation is invariant with respect to changes in the probabilities of the propositions themselves.

Igor Evgenievich Pris