User:Boris Tsirelson/Sandbox1: Difference between revisions

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In [[Newtonian mechanics]], coordinates of moving bodies are functions of time. For example, the classical equation for a falling body; its height ''h'' at a time ''t'' is
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:<math> h = f(t) = h_0 - 0.5 g t^2 </math>
The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.
(here ''h''<sub>0</sub> is the initial height, and ''g'' is the [[acceleration due to gravity]]). Infinitely many corresponding values of ''t'' and ''h'' are embraced by a single function ''f''.


{{Image|Moving wave.gif|right||<small>Vibrating string: a function changes in time</small>}}
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
The instantaneous shape of a vibrating string is described by a function (the displacement ''y'' as a function of the coordinate ''x''), and this function changes in time:
:<math> y = f_t (x). </math>
Infinitely many functions ''f''<sub>''t''</sub> are embraced by a single function ''f'' of two variables,
:<math> y = f(x,t). </math>
 
After some speculations by Galileo (the ?? century) and mathematical interpretation by Taylor (1715), the mathematical theory of vibrating string was started by d'Alembert (communicated in 1746, published in 1749). His approach is equivalent to a [[partial differential equation]] written out by Euler in 1755,
:<math>
\frac{\partial^2 f(x,t)}{\partial x^2} =  \frac{\partial^2 f(x,t)}{\partial t^2},
</math>
now well-known as the one-dimensional [[Wave equation (classical physics)|wave equation]].

Latest revision as of 02:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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