User:Boris Tsirelson/Sandbox1: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Boris Tsirelson
No edit summary
No edit summary
 
(15 intermediate revisions by one other user not shown)
Line 1: Line 1:
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
{{AccountNotLive}}
:<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 and mathematical interpretation by [[Brook Taylor]] (1715), the mathematical theory of vibrating string was started by [[Jean le Rond d'Alembert|Jean d'Alembert]] (communicated in 1746, published in 1749). His approach is equivalent to a [[partial differential equation]] written out by [[Leonhard Euler]] in 1755,
:<math>
\frac{\partial^2}{\partial x^2} f(x,t) = \frac{\partial^2}{\partial t^2} f(x,t),
</math>
now well-known as the one-dimensional [[Wave equation (classical physics)|wave equation]].
 
The initial shape of the string is given by the function ''f''<sub>0</sub>. It was a controversial question in the 18th century, whether ''f''<sub>0</sub> must develop into a power series, or not necessarily.
 
D'Alembert held the opinion that the ''de-facto'' standard mentioned above still applies; ''f''<sub>0</sub> must be represented by a single equation. (He changed his opinion in 1780.)
 
The old standard was repudiated by Euler in 1744. He introduced "mixed" functions, given by different equations on two or more intervals. Moreover, he admitted functions that do not comply with any analytical law, whose graphs are traced by a free stroke of the hand.

Latest revision as of 03:25, 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).

Retrieved from "https://citizendium.org/wiki/index.php?title=User:Boris_Tsirelson/Sandbox1&oldid=904361"