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where = is the reduced Planck constant.. The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time.
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
Similarly, uncertainty is propagated through calculations so that the calculated value has some degree of uncertainty depending upon the uncertainties of the measured values and the equation used in the calculation. [27] In physics, the Heisenberg uncertainty principle forms the basis of modern quantum mechanics. [17]
In practical experiments, these values will be estimated from observed data, i.e., measurements. These measurements are averaged to produce the estimated mean values to use in the equations, e.g., for evaluation of the partial derivatives. Thus, the variance of interest is the variance of the mean, not of the population, and so, for example,
In this equation, and denote the uncertainties in position and momentum, respectively. The term ℏ {\displaystyle \hbar } represents the reduced Planck constant, while β {\displaystyle \beta } is a parameter that embodies the minimal length scale predicted by the GUP.
A comprehensive list of equations used in nuclear and particle physics.
In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. Physics is derived of formulae only.
The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle—between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty relation corresponds to the symplectic form.