User:Xenwolf/Bra–ket notation - Revision history

Xenwolf: /* References */ +1

2020-06-02T13:20:11Z

References: +1

← Older revision		Revision as of 13:20, 2 June 2020
Line 30:		Line 30:
	* [https://de.wikipedia.org/wiki/Dirac-Notation Dirac-Notation – Wikipedia]		* [https://de.wikipedia.org/wiki/Dirac-Notation Dirac-Notation – Wikipedia]
	* [https://www.youtube.com/watch?v=2vvjrBbcTZU Dualraum - intuitiv erklärt! \| Math Intuition - YouTube] (German)		* [https://www.youtube.com/watch?v=2vvjrBbcTZU Dualraum - intuitiv erklärt! \| Math Intuition - YouTube] (German)
			* [https://www.youtube.com/watch?v=KK_fHodz-lQ LINEARE ABBILDUNG / Homomorphismus einfach erklärt! \| Math Intuition - YouTube] (German)
	* [https://www.youtube.com/watch?v=TjAFH6hWg1I Bilinearform einfach erklärt :) \| Math Intuition - YouTube] (German)		* [https://www.youtube.com/watch?v=TjAFH6hWg1I Bilinearform einfach erklärt :) \| Math Intuition - YouTube] (German)

Xenwolf: /* References */ +2 YouTube links

2020-06-02T08:59:36Z

References: +2 YouTube links

← Older revision		Revision as of 08:59, 2 June 2020
Line 29:		Line 29:
	* [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation Bra–ket notation - Wikipedia]		* [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation Bra–ket notation - Wikipedia]
	* [https://de.wikipedia.org/wiki/Dirac-Notation Dirac-Notation – Wikipedia]		* [https://de.wikipedia.org/wiki/Dirac-Notation Dirac-Notation – Wikipedia]
			* [https://www.youtube.com/watch?v=2vvjrBbcTZU Dualraum - intuitiv erklärt! \| Math Intuition - YouTube] (German)
			* [https://www.youtube.com/watch?v=TjAFH6hWg1I Bilinearform einfach erklärt :) \| Math Intuition - YouTube] (German)

Xenwolf: added German version and ref section

2020-05-30T09:37:36Z

added German version and ref section

@@ Line 6: / Line 6: @@
 Bra–ket notation was effectively established in 1939 by Paul Dirac (Dirac 1939)(Shankar 1994) and is thus also known as the '''Dirac notation'''. (Still, the bra-ket notation has a precursor in Hermann Grassmann's use of the notation <math>[\phi{\mid}\psi]</math> for his inner products nearly 100 years earlier. (Grassmann 1862),([[#Video]]))
 == Video ==
 [https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561 Lecture 2 | Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.
@@ Line 15: / Line 24: @@
 <div>
 </div>

Xenwolf: removed links except youtube link (for references see Wikipedia article), embedded youtube video

2020-05-30T08:56:22Z

removed links except youtube link (for references see Wikipedia article), embedded youtube video

← Older revision		Revision as of 08:56, 30 May 2020
Line 1:		Line 1:
	In [[quantum mechanics]], '''bra–ket notation''' is a common notation for [[quantum ~~state]]s~~, i.e. vectors in a complex [[Hilbert space]] on which an algebra of observables acts. More generally the notation uses the [[angle ~~bracket]]s~~ (the ⟨ and ⟩ symbols) and a [[vertical bar]] (the \| symbol), for a '''ket''' ~~{{IPAc-en\|k\|ɛ\|t}}~~ (for example, <math>\|v \rangle</math> ) to denote a [[vector ~~space\|vector]]~~ in an abstract (usually complex) [[vector space]] <math>V</math> and a '''bra''', ~~{{IPAc-en\|b\|r\|ɑː}}~~ (for example, <math>\langle f\|</math> ) to denote a [[linear functional]] on <math>V</math>, i.e. a co-vector, an element of the [[dual vector space]] <math> V^\vee</math>. The natural pairing of a linear functional <math>f = \langle f\|</math> with a vector <math>v = \|v\rangle</math> is then written as <math>\langle f\| v\rangle</math>. On Hilbert spaces, the [[scalar product]] <math>(\ , \ )</math> (with anti linear first argument) gives an (anti-linear) identification of a vector ket <math>\phi = \|\phi\rangle</math> with a linear functional bra <math>(\phi, \ ) = \langle\phi\|</math>. Using this notation, the scalar product <math> (\phi, \psi) = \langle\phi\|\psi\rangle</math>.		In quantum mechanics, '''bra–ket notation''' is a common notation for quantum states, i.e. vectors in a complex Hilbert space on which an algebra of observables acts. More generally the notation uses the angle brackets (the ⟨ and ⟩ symbols) and a vertical bar (the \| symbol), for a '''ket''' (/kɛt/) (for example, <math>\|v \rangle</math> ) to denote a vector in an abstract (usually complex) vector space <math>V</math> and a '''bra''', (/brɑː/) (for example, <math>\langle f\|</math> ) to denote a linear functional on <math>V</math>, i.e. a co-vector, an element of the dual vector space <math> V^\vee</math>. The natural pairing of a linear functional <math>f = \langle f\|</math> with a vector <math>v = \|v\rangle</math> is then written as <math>\langle f\| v\rangle</math>. On Hilbert spaces, the scalar product <math>(\ , \ )</math> (with anti linear first argument) gives an (anti-linear) identification of a vector ket <math>\phi = \|\phi\rangle</math> with a linear functional bra <math>(\phi, \ ) = \langle\phi\|</math>. Using this notation, the scalar product <math> (\phi, \psi) = \langle\phi\|\psi\rangle</math>.
	For the vector space <math>\mathbb{C}^n</math>, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using [[matrix multiplication]]. If <math>\mathbb{C}^n</math> has the standard hermitian inner product <math>(v, w) = v^\dagger w</math>, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the hermitian conjugate <math> \dagger</math>.		For the vector space <math>\mathbb{C}^n</math>, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using matrix multiplication. If <math>\mathbb{C}^n</math> has the standard hermitian inner product <math>(v, w) = v^\dagger w</math>, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the hermitian conjugate <math> \dagger</math>.

	It is common to suppress the vector or functional from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator <math>\sigma_z</math> on a two dimensional space <math>\Delta</math> of [[spinors]], has eigenvalues <math>\pm</math>½ with eigenspinors <math>\psi_+,\psi_- \in \Delta</math>. In bra-ket notation one typically denotes this as <math>\psi_+ = \|+\rangle</math>, and		It is common to suppress the vector or functional from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator <math>\sigma_z</math> on a two dimensional space <math>\Delta</math> of spinors, has eigenvalues <math>\pm</math>½ with eigenspinors <math>\psi_+,\psi_- \in \Delta</math>. In bra-ket notation one typically denotes this as <math>\psi_+ = \|+\rangle</math>, and
	<math>\psi_- = \|-\rangle</math>. Just as above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular when also identified with row and column vectors, kets and bras with the same label are identified with [[Hermitian conjugate]] column and row vectors.		<math>\psi_- = \|-\rangle</math>. Just as above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors.

	Bra–ket notation was effectively established in 1939 by [[Paul Dirac~~]]<ref name="Dirac">{{harvnb\|~~Dirac\|1939~~}}</ref><ref>{{harvnb\|~~Shankar\|1994~~\|loc=Chapter 1}}</ref>~~ and is thus also known as the '''Dirac notation'''. (Still, the bra-ket notation has a precursor in [[Hermann Grassmann]]'s use of the notation <math>[\phi{\mid}\psi]</math> for his inner products nearly 100 years earlier.~~<ref name="Grassmann">{{harvnb\|~~Grassmann\|1862~~}}</ref><ref>~~[https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561 Lecture 2 \| Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.</~~ref~~>)		Bra–ket notation was effectively established in 1939 by Paul Dirac (Dirac 1939)(Shankar 1994) and is thus also known as the '''Dirac notation'''. (Still, the bra-ket notation has a precursor in Hermann Grassmann's use of the notation <math>[\phi{\mid}\psi]</math> for his inner products nearly 100 years earlier. (Grassmann 1862),([[#Video]]))

			== Video ==
			[https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561 Lecture 2 \| Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.

			<youtube alignment="center" container="frame" description="
			Lecture 2 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 2, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.
			">https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561</youtube>
			<div>
			</div>

Xenwolf: raw copy from https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation

2020-05-30T08:17:09Z

raw copy from https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation

New page

In [[quantum mechanics]], '''bra–ket notation''' is a common notation for [[quantum state]]s, i.e. vectors in a complex [[Hilbert space]] on which an algebra of observables acts. More generally the notation uses the [[angle bracket]]s (the ⟨ and ⟩ symbols) and a [[vertical bar]] (the | symbol), for a '''ket''' {{IPAc-en|k|ɛ|t}} (for example, <math>|v \rangle</math> ) to denote a [[vector space|vector]] in an abstract (usually complex) [[vector space]] <math>V</math> and a '''bra''', {{IPAc-en|b|r|ɑː}} (for example, <math>\langle f|</math> ) to denote a [[linear functional]] on <math>V</math>, i.e. a co-vector, an element of the [[dual vector space]] <math> V^\vee</math>. The natural pairing of a linear functional <math>f = \langle f|</math> with a vector <math>v = |v\rangle</math> is then written as <math>\langle f| v\rangle</math>. On Hilbert spaces, the [[scalar product]] <math>(\ , \ )</math> (with anti linear first argument) gives an (anti-linear) identification of a vector ket <math>\phi = |\phi\rangle</math> with a linear functional bra <math>(\phi, \ ) = \langle\phi|</math>. Using this notation, the scalar product <math> (\phi, \psi) = \langle\phi|\psi\rangle</math>.
For the vector space <math>\mathbb{C}^n</math>, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using [[matrix multiplication]]. If <math>\mathbb{C}^n</math> has the standard hermitian inner product <math>(v, w) = v^\dagger w</math>, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the hermitian conjugate <math> \dagger</math>.

It is common to suppress the vector or functional from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator <math>\sigma_z</math> on a two dimensional space <math>\Delta</math> of [[spinors]], has eigenvalues <math>\pm</math>½ with eigenspinors <math>\psi_+,\psi_- \in \Delta</math>. In bra-ket notation one typically denotes this as <math>\psi_+ = |+\rangle</math>, and
<math>\psi_- = |-\rangle</math>. Just as above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular when also identified with row and column vectors, kets and bras with the same label are identified with [[Hermitian conjugate]] column and row vectors.

Bra–ket notation was effectively established in 1939 by [[Paul Dirac]]<ref name="Dirac">{{harvnb|Dirac|1939}}</ref><ref>{{harvnb|Shankar|1994|loc=Chapter 1}}</ref> and is thus also known as the '''Dirac notation'''. (Still, the bra-ket notation has a precursor in [[Hermann Grassmann]]'s use of the notation <math>[\phi{\mid}\psi]</math> for his inner products nearly 100 years earlier.<ref name="Grassmann">{{harvnb|Grassmann|1862}}</ref><ref>[https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561 Lecture 2 | Quantum Entanglements, Part 1 (Stanford)], Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.</ref>)