Two-state quantum system - Revision history

en>Eli4ph: word correction

2015-01-07T07:43:15Z

word correction

← Older revision		Revision as of 09:43, 7 January 2015
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87.151.237.228: /* Eigenvalues of the Hamiltonian, Basis vectors and Time evolution */

2014-02-23T20:21:38Z

Eigenvalues of the Hamiltonian, Basis vectors and Time evolution

← Older revision		Revision as of 22:21, 23 February 2014
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	In [[mathematics]], the '''Iverson bracket''', named after [[Kenneth E. Iverson]]<!-- mentioned here so it's clear that it's an upper-case 'i' not an 'l'. -->, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,		I love harmony<br><br>Here is my blog - [http://westcoast.co.kr/xe/?document_srl=485227 Phone Sex]
	:<~~math~~>~~[P] = \begin{cases} 1 & \text{if } P \text{ is true;} \\ 0 & \text{otherwise.} \end{cases}~~<~~/math~~>
	~~where {{math\|''P''}}~~ is a [[statement (logic)\|statement]] that can be true or false. This notation was introduced by [[Kenneth E. Iverson]] in his programming language [[APL (programming language)\|APL]],<ref>[[Ronald Graham]], [[Donald Knuth]], and [[Oren Patashnik]]. ''[[Concrete Mathematics]]'', Section 2.2: Sums and Recurrences.</ref> while the specific restriction to square brackets was advocated by [[Donald Knuth]] to avoid ambiguity in parenthesized logical expressions.<ref>Knuth 1992.</ref>

	~~==Uses==~~
	The Iverson bracket converts a [[Boolean data type\|Boolean value]] to an integer value through the natural map <math>\textbf{false}\mapsto 0; \textbf{true}\mapsto1</math>, which allows counting to be represented as summation. For instance, the [[Euler phi function]] that counts the number of positive integers up to ''n'' which are [[coprime]] to ''n'' can be expressed by
	~~: <math> \phi(n)=\sum_{i=1}^{n}[\gcd(i,n)=1],\qquad\text{for }n\in\mathbb N^+.</math>~~
	More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example
	~~: <math>\sum_{1\le i \le 10} i^2 = \sum_{i} i^2[1 \le i \le 10].</math>~~
	In the first sum, the index <math>i</math> is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where ''i'' is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions.
	<!-~~- add an example of such a manipulation -->~~

	~~Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula~~
	~~:<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)</math>~~

	which is valid for {{math\|''n'' > 1}} but which is off by {{sfrac\|1\|2}} for {{math\|''n'' {{=}} 1}}. To get an identity valid for all positive integers ''n'' (i.e., all values for which <math>\phi(n)</math> is defined), a correction term involving the Iverson bracket may be added:

	~~:<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n(\varphi(n)+~~[~~n=1])</math>~~

	~~==Special cases==~~
	~~The [[Kronecker delta]] notation is a specific case of Iverson notation when the condition is equality. That is,~~
	: ~~<math>\delta_{ij} = [i=j].\,<~~/~~math>~~

	~~The [[indicator function]], another specific case, has set membership as its condition:~~
	~~: <math>\mathbf{I}_A(x) = [x\in A].</math>~~

	~~The [[sign function]] and [[Heaviside step function]] are also easily expressed in this notation:~~
	~~: <math> \sgn(x) = [x > 0] - [x < 0] \,<~~/~~math>~~
	~~: <math> H(x) = [x > 0]~~.~~</math>~~

	~~The [[floor and ceiling functions]] can be expressed:~~
	~~: <math> \lfloor x \rfloor = \sum_{n=-\infty}^{\infty}n[n \le x < n+1],</math>~~
	~~: <math> \lceil x \rceil = \sum_{n=-\infty}^{\infty}n[n-1 < x \le n]~~.</~~math>~~

	~~And the [[Trichotomy (mathematics)\|trichotomy]] of the reals can be expressed:~~
	~~:<math>[a < b] + [a = b] + [a > b] = 1. \, <~~/~~math>~~

	~~==See also==~~
	*[[Macaulay brackets]]

	~~== Notes ==~~
	~~{{reflist}}~~

	~~== References =~~=
	* Donald Knuth, "Two Notes on Notation", ''[[American Mathematical Monthly]]'', Volume 99, Number 5, May 1992, pp. 403–422. ([http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz {{TeX}}], {{arxiv\|math/9205211}})
	* Kenneth E. Iverson, ''A Programming Language'', New York: Wiley, p. 11, 1962.

	~~[[Category:Mathematical notation]~~]

en>WereSpielChequers: know known

2013-12-15T21:26:06Z

know known

← Older revision		Revision as of 23:26, 15 December 2013
Line 1:		Line 1:
	~~Yesterday I woke up and noticed~~ - I'~~ve been single for~~ a ~~little while today~~ and ~~following much bullying from buddies I now find myself registered for on line dating~~. ~~They promised me~~ that ~~there are a lot of pleasant~~, ~~ordinary~~ and [~~http~~:/~~/Www~~.~~Adobe~~.~~com~~/~~cfusion/search/index.cfm?term~~=~~&enjoyable+individuals&loc~~=~~en_us&siteSection~~=~~home enjoyable individuals~~] to ~~fulfill~~, [~~http://lukebryantickets.lazintechnologies.com meet and greet tickets luke bryan~~] ~~therefore~~ the ~~pitch is gone~~ by ~~here!~~<br>~~I make an effort to maintain~~ as ~~physically healthy as possible being at~~ the ~~fitness center several times a week. I love my athletics and strive~~ to ~~perform or see because many a possible. Being winter I shall frequently at Hawthorn fits~~. ~~Notice~~: ~~If you would contemplated buying~~ [~~http:~~//~~www~~.~~banburycrossonline.com cheap tickets luke bryan] a hobby I really don~~'~~t mind~~, ~~I have seen~~ the ~~carnage~~ of ~~fumbling suits at stocktake sales~~.<br>~~My family and friends are amazing and spending some time~~ with ~~them at tavern gigabytes or dinners~~ is ~~obviously imperative~~. ~~I haven~~'~~t ever been into night clubs as I locate that one~~ may ~~do not~~ [~~http:~~/~~/lukebryantickets.flicense.com information on luke bryan~~] ~~have~~ a ~~significant conversation against~~ the ~~noise~~. I ~~also got 2 undoubtedly cheeky~~ and ~~very cunning canines who~~ are ~~constantly ready to meet new folks~~.<br><br>~~My web site~~ :: [http://~~lukebryantickets~~.~~iczmpbangladesh~~.~~org luke bryan tour dates for 2014~~]		In [[mathematics]], the '''Iverson bracket''', named after [[Kenneth E. Iverson]]<!-- mentioned here so it's clear that it's an upper-case 'i' not an 'l'. -->, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,
			:<math>[P] = \begin{cases} 1 & \text{if } P \text{ is true;} \\ 0 & \text{otherwise.} \end{cases}</math>
			where {{math\|''P''}} is a [[statement (logic)\|statement]] that can be true or false. This notation was introduced by [[Kenneth E. Iverson]] in his programming language [[APL (programming language)\|APL]],<ref>[[Ronald Graham]], [[Donald Knuth]], and [[Oren Patashnik]]. ''[[Concrete Mathematics]]'', Section 2.2: Sums and Recurrences.</ref> while the specific restriction to square brackets was advocated by [[Donald Knuth]] to avoid ambiguity in parenthesized logical expressions.<ref>Knuth 1992.</ref>

			==Uses==
			The Iverson bracket converts a [[Boolean data type\|Boolean value]] to an integer value through the natural map <math>\textbf{false}\mapsto 0; \textbf{true}\mapsto1</math>, which allows counting to be represented as summation. For instance, the [[Euler phi function]] that counts the number of positive integers up to ''n'' which are [[coprime]] to ''n'' can be expressed by
			: <math> \phi(n)=\sum_{i=1}^{n}[\gcd(i,n)=1],\qquad\text{for }n\in\mathbb N^+.</math>
			More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example
			: <math>\sum_{1\le i \le 10} i^2 = \sum_{i} i^2[1 \le i \le 10].</math>
			In the first sum, the index <math>i</math> is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where ''i'' is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions.
			<!-- add an example of such a manipulation -->

			Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula
			:<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)</math>

			which is valid for {{math\|''n'' > 1}} but which is off by {{sfrac\|1\|2}} for {{math\|''n'' {{=}} 1}}. To get an identity valid for all positive integers ''n'' (i.e., all values for which <math>\phi(n)</math> is defined), a correction term involving the Iverson bracket may be added:

			:<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n(\varphi(n)+[n=1])</math>

			==Special cases==
			The [[Kronecker delta]] notation is a specific case of Iverson notation when the condition is equality. That is,
			: <math>\delta_{ij} = [i=j].\,</math>

			The [[indicator function]], another specific case, has set membership as its condition:
			: <math>\mathbf{I}_A(x) = [x\in A].</math>

			The [[sign function]] and [[Heaviside step function]] are also easily expressed in this notation:
			: <math> \sgn(x) = [x > 0] - [x < 0] \,</math>
			: <math> H(x) = [x > 0].</math>

			The [[floor and ceiling functions]] can be expressed:
			: <math> \lfloor x \rfloor = \sum_{n=-\infty}^{\infty}n[n \le x < n+1],</math>
			: <math> \lceil x \rceil = \sum_{n=-\infty}^{\infty}n[n-1 < x \le n].</math>

			And the [[Trichotomy (mathematics)\|trichotomy]] of the reals can be expressed:
			:<math>[a < b] + [a = b] + [a > b] = 1. \, </math>

			==See also==
			*[[Macaulay brackets]]

			== Notes ==
			{{reflist}}

			== References ==
			* Donald Knuth, "Two Notes on Notation", ''[[American Mathematical Monthly]]'', Volume 99, Number 5, May 1992, pp. 403–422. ([http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz {{TeX}}], {{arxiv\|math/9205211}})
			* Kenneth E. Iverson, ''A Programming Language'', New York: Wiley, p. 11, 1962.

			[[Category:Mathematical notation]]

en>Zueignung: /* Dynamics for a time-independent Hamiltonian */ dab

2012-08-25T17:29:54Z

Dynamics for a time-independent Hamiltonian: dab

New page

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