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1

It looks like Pythagoras, but it is only remotely related. The important concept, as presented in SteveB's answer, is that the variables are considered to be independent, i.e. one does not affect the other. In mathematics, independent parameters are said to be orthogonal , and can thus be assigned to separate axes in Cartesian N-space. It just so happens ...


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The general formula for error propagation is: $$\Delta f(x_1,x_2,\ldots)=\sqrt{(\frac{\partial f}{\partial x_1}\Delta x_1)^2 + (\frac{\partial f}{\partial x_2}\Delta x_2)^2 + \cdots}$$ Where does this come from? We assume that the errors are relatively small (ignore $\Delta x_i \Delta x_j$ terms etc.), and that the errors are independent (in the ...


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The formula $$\frac{\Delta{A}}{A} \approx \frac{\Delta{X}}{X} + \frac{\Delta{Y}}{Y} $$ is an approximation because you are ignoring $\Delta X$$\Delta Y$ A better approximation would be $$\Delta A=\frac{\partial A}{\partial X}\Delta X+\frac{\partial A}{\partial Y}\Delta Y$$ Since errors always add we take the absolute magnitude of $\frac{\partial ...


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The square root is there as a better estimator of the error than just adding the errors together. If you add the errors together you are finding the maximum possible error which will happen when both quantities are a maximum(or minimum) together. This is an unlikely event compared with all the other domination of errors. The square root formula you you ...


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An observable in quantum mechanics is a measurable quantity in an experiment or observation. A postulate for building the mathematical theory of quantum mechanics is that 2.With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that ...


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Random variables satisfy the Kolmogorov axioms for probability; quantum observables do not. In particular, any four-tuple of binary random variables (with any joint distribution) satisfies Bell's Inequality, while there are four-tuples of quantum observables that don't.


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An observable in quantum mechanics is an operator (say $\widehat{\mathcal{O}}$) on the Hilbert space (Say $\mathcal{H}$) of physical states, such that eigenkets in (say $\widehat{\mathcal{O}}$) in $\mathcal{H}$ span $\mathcal{H}$. The eigenvalues of $\widehat{\mathcal{O}}$ are then the observable values of some classical variable $\mathcal{O}$, even though ...


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What you say is quite reasonable. At the risk of being slightly more pedantic, I would say that physical observables are only those random variables that are Hermitean. Any operator (Hermitean or not) is a random variable -- in quantum mechanics these might be various properties of a particular state like spin, energy, etc. In quantum field theory, the ...


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About the physical implications, what this exercise shows you, is that the radiactive nucleus as a 50% chance to decay during its half-life period, but if it has not decayed, it remains unchanged: it has same number of protons and neutrons, same instability, and so during the next period it still have a 50% probability to decay, and so on.


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It means that after every half-life of time there is a 50% probability that any given nucleus will decay. So after one half life, there is a 50% probability that a particular nucleus will have decayed. But after that time, if your particular nucleus has not decayed, then there is a further 50% probability that it will decay after another half life. Thus the ...



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