User:Boris Tsirelson/Sandbox1: Difference between revisions

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===Birth and infancy of the idea===
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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.


Some tables compiled by ancient Babylonians may be treated now as
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).
tables of some functions. Also, some arguments of ancient Greeks may
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),
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.
 
===Power series===
 
Further progress appears 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''}}?
Is it "composed of ''x''"? "In any manner whatever" is now interpreted much more widely than it was possible in 17th century.
 
<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>
 
Surely, {{nowrap|sin ''x''}} is not a polynomial of ''x''. However, it is the sum of a power series:
:<math> \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \dots </math>
which was found already by James Gregory in 1667. Many other functions were developed into power series by him, Barrow, [[Isaac Newton]] and others. Moreover, all these formulas appeared to be special cases of a much more general formula found by Taylor in 1715.
 
== ... ==
 
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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