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 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). However, they were not recognized as special cases of a general notion.
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).
 
Further progress was made in the 14th century. Two "schools of natural philosophy", at Oxford ([[William Heytesbury]], [[Richard Swineshead]]) and Paris ([[Nicole Oresme]]), trying to investigate natural phenomena mathematically, came to the idea that laws of nature should be formulated as functional relations between physical quantities. 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. Linear functions were well understood, but nonlinear functions remained intractable, except for few isolated marginal cases.
 
The name "function" was assigned to the new concept later, in 1698, by [[Johann Bernoulli]] and [[Gottfried Wilhelm Leibniz|Gottfried Leibniz]], and published by Bernoulli in 1718.
 
===Power series===
 
The sum of the [[geometric series]]
:<math> 1+x+x^2+x^3+\dots = \frac1{1-x} </math>
was calculated by [[Archimedes]], but only for ''x''=1/4, since only this value was needed, and of course not written in this form, since algebraic notation appeared only in the 16th century. New wonderful formulas with [[Series (mathematics)|infinite sums]] were discovered (and repeatedly rediscovered) in the 14–17 centuries: for [[arctangent]],
:<math> \arctan x = x - \frac{x^3}3 + \frac{x^5}5 - \dots </math>
([[Madhava of Sangamagramma]], around 1400; [[James Gregory]], 1671); for [[logarithm]],
:<math> \log (1+x) = x - \frac{x^2}2 + \frac{x^3}3 - \dots </math>
([[Nicholas Mercator]], 1668); and many others ([[Isaac Barrow]], [[Isaac Newton]], Gottfried Leibniz, ...) Nonlinear functions, desperately needed for the study of motion ([[Johannes Kepler]], [[Galileo Galilei]]) and geometry ([[Pierre de Fermat|Pierre Fermat]], [[René Descartes]]), became tractable via such infinite sums now called [[power series]].
<blockquote>Newton understood by analysis the investigation of equations by means of infinite series. In other words, Newton's basic discovery was that everything had to be expanded in infinite series.(Arnold, page 35)</blockquote>
<blockquote>These studies [on power series] stand in the same relation to algebra as the studies of decimal fractions to ordinary arithmetic. (Newton)</blockquote>
Power series became a ''de facto'' standard of function, since on one hand, all functions needed in applications were successfully developed into power series, and on the other hand, only functions developed into power series were tractable in the theory. It was not unusual, to claim a theorem for an arbitrary function, and then, in the proof, to consider its development into a power series.
 
===Trigonometric 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; [[Sine|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, [[Isaac Barrow]], [[Isaac Newton]] and others. Moreover, all these formulas appeared to be special cases of a much more [[Taylor series|general formula]] found by Brook 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 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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