User:Boris Tsirelson/Sandbox1: Difference between revisions

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Some tables compiled by ancient Babylonians may be treated now as
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tables of some functions. Also, some arguments of ancient Greeks may
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.
be treated now as integration of some functions. Thus, in ancient
times some functions were used implicitly, without being recognized as
special cases of a general notion.


Further progress was made in the 11th century by Al-Biruni (Persia),
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).
and in 14th century by the "schools of natural philosophy" at Oxford
(William Heytesbury, Richard Swineshead) and Paris (Nicole
Oresme). The concept of function was born, including a curve as a
graph of a function of one variable and a surface - for two
variables. However, the new concept was not yet widely exploited
either in mathematics or in its applications.
 
Further progress appears only in the 17th century
from the study of motion (Johannes Kepler, Galileo Galilei) and
geometry (P. Fermat, R. Descartes).
 
A formulation by Descartes (La Geometrie, 1637) appeals to graphic
representation of a functional dependence and does not involve
formulas (algebraic expressions):
 
<blockquote>If then we should take successively an infinite number of different
values for the line ''y'', we should obtain an infinite number of
values for the line ''x'', and therefore an infinity of different
points, such as ''C'', by means of which the required curve could be
drawn.</blockquote>
 
The term ''function'' is adopted by Leibniz and Jean Bernoulli between
1694 and 1698, and disseminated by Bernoulli in 1718:
 
<blockquote>One calls here a function of a variable a quantity composed in any
manner whatever of this variable and of constants.</blockquote>
 
This time a formula is required, which restricts the class of
functions. However, what is a formula? Surely, {{nowrap|''y'' &#061; 2
''x''<sup>2</sup> - 3}} is allowed; what about {{nowrap|''y'' &#061; sin ''x''}}?
 
<blockquote>... little by little, and often by very subtle detours, various
transcendental operations, the logarithm, the exponential, the
trigonometric functions, quadratures, the solution of differential
equations, passing to the limit, the summing of series, acquired the
right of being quoted. (Bourbaki, p. 193)</blockquote>
 
But on the first stage the notion of an algebraic expression is quite
restrictive. More general, possibly ill-behaving functions have to
wait for the 19th century.

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).

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