Bra-ket notation - Revision history

FredericBernard: Change category to Quantum Mechanics

2017-04-06T12:20:03Z

Change category to Quantum Mechanics

DavidB4-bot: Spelling/Grammar Check, typos fixed: often times → oftentimes, Enlish → English

2016-07-14T14:30:17Z

Spelling/Grammar Check, typos fixed: often times → oftentimes, Enlish → English

DavidB4-bot: clean up & uniformity

2016-07-13T03:35:47Z

clean up & uniformity

AugustO at 13:48, April 2, 2013

2013-04-02T13:48:40Z

AugustO: /* Orthogonality of Bras and Kets */ ...

2013-04-02T13:46:35Z

Orthogonality of Bras and Kets: ...

AugustO: sp

2013-04-02T13:44:19Z

HelpJazz: /* Othogonality of Bras and Kets */ spelling

2008-01-08T20:36:47Z

Othogonality of Bras and Kets: spelling

Jaques at 22:36, April 4, 2007

2007-04-04T22:36:01Z

Jaques at 22:33, April 4, 2007

2007-04-04T22:33:42Z

Linus M.: Some cleanup

2007-03-09T04:37:41Z

Some cleanup

@@ Line 60: / Line 60: @@
 ==References==
 <references/>
-[[Category:Physics]]

@@ Line 1: / Line 1: @@
-'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by the Enlish physicist Paul Dirac and is named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.
+'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by the English physicist Paul Dirac and is named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.
 In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>.  In bra-ket notation, the symbol <math>\left|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question.  This vector is called a "ket".  However, by [[Riesz Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:
@@ Line 18: / Line 18: @@
 == Orthogonality of Bras and Kets ==
-Because the bracket represents an inner product, certain concepts from [[linear algebra]] will continue to play a role.  One is that of [[orthogonality]].  By using the [[Gram-Schmidt Process]], and set of linearly independent kets can be orthogonalized, and we may often times assume that such a procedure has been carried out.  In addition, because of the probabilistic interpretation of [[wave mechanics]], we may actually take the kets to be normalized.  In the Dirac Notation, the normalization condition reads:
+Because the bracket represents an inner product, certain concepts from [[linear algebra]] will continue to play a role.  One is that of [[orthogonality]].  By using the [[Gram-Schmidt Process]], and set of linearly independent kets can be orthogonalized, and we may oftentimes assume that such a procedure has been carried out.  In addition, because of the probabilistic interpretation of [[wave mechanics]], we may actually take the kets to be normalized.  In the Dirac Notation, the normalization condition reads:
 <math>\left\langle\psi|\psi\right\rangle = 1</math>

@@ Line 13: / Line 13: @@
 <math>\left\langle\varphi|\psi\right\rangle = \left\langle\psi,\varphi\right\rangle</math>
-The notation is deceptively simple.  The elegant nature of the Dirac Bra-ket notation allows physicists to treat linear functionals represented as bras in a very intuitive fashion (they preserve nearly all of the familiar algebraic properties of numbers except commutativity).  Computations of inner products are naturally suggested, and the problem is not bogged down in excessive notation -- it is essentially distilled down to its algebraic content alone.
+The notation is deceptively simple.  The elegant nature of the Dirac Bra-ket notation allows physicists to treat linear functionals represented as bras in a very intuitive fashion (they preserve nearly all of the familiar algebraic properties of numbers except commutativity).  Computations of inner products are naturally suggested, and the problem is not bogged down in excessive notation—it is essentially distilled down to its algebraic content alone.
@@ Line 60: / Line 60: @@
 ==References==
 <references/>
-[[category:physics]]
+[[Category:Physics]]

← Older revision		Revision as of 13:48, April 2, 2013
Line 1:		Line 1:
−	'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by ~~a man named~~ Paul Dirac and ~~originally~~ named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.	+	'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by the Enlish physicist Paul Dirac and is named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.

	In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>. In bra-ket notation, the symbol <math>\left\|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question. This vector is called a "ket". However, by [[Riesz Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:		In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>. In bra-ket notation, the symbol <math>\left\|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question. This vector is called a "ket". However, by [[Riesz Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:

@@ Line 46: / Line 46: @@
 <math>Q=\sum_{n}\lambda_n\left|n\right\rangle\left\langle n\right|</math>
-The kets may also represent a [[continuous]] set of states.  In such circumstances (which would include, for example, a [[free particle]]), Dirac also found it necessary to develop what is known as the [[Dirac Delta Function]] as an analogue to the [[Kronecker Delta Function]].  For a continuous set of complete kets indexed by the continuous variables <math>x^\prime</math> and <math>x^{\prime\prime}</math>:
+The kets may also represent a [[continuous]] set of states.  In such circumstances (which would include, for example, a [[free particle]]), Dirac also found it necessary to develop what is known as the [[Dirac delta]] function as an analogue to the [[Kronecker Delta|Kronecker delta]] function.  For a continuous set of complete kets indexed by the continuous variables <math>x^\prime</math> and <math>x^{\prime\prime}</math>:
 <math>\left\langle x^\prime|x^{\prime\prime}\right\rangle = \delta\left(x^\prime-x^{\prime\prime}\right)</math>

@@ Line 1: / Line 1: @@
 '''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by a man named Paul Dirac and originally named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.
-In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>.  In bra-ket notation, the symbol <math>\left|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question.  This vector is called a "ket".  However, by [[Reisz' Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:
+In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle , \rangle</math>.  In bra-ket notation, the symbol <math>\left|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question.  This vector is called a "ket".  However, by [[Riesz Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:
 <math>f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}</math>

← Older revision		Revision as of 20:36, January 8, 2008
Line 16:		Line 16:


−	== ~~Othogonality~~ of Bras and Kets ==	+	== Orthogonality of Bras and Kets ==

	Because the bracket represents an inner product, certain concepts from [[linear algebra]] will continue to play a role. One is that of [[orthogonality]]. By using the [[Gram-Schmidt Process]], and set of linearly independent kets can be orthogonalized, and we may often times assume that such a procedure has been carried out. In addition, because of the probabilistic interpretation of [[wave mechanics]], we may actually take the kets to be normalized. In the Dirac Notation, the normalization condition reads:		Because the bracket represents an inner product, certain concepts from [[linear algebra]] will continue to play a role. One is that of [[orthogonality]]. By using the [[Gram-Schmidt Process]], and set of linearly independent kets can be orthogonalized, and we may often times assume that such a procedure has been carried out. In addition, because of the probabilistic interpretation of [[wave mechanics]], we may actually take the kets to be normalized. In the Dirac Notation, the normalization condition reads:

← Older revision		Revision as of 22:33, April 4, 2007
Line 60:		Line 60:
	==References==		==References==
	<references/>		<references/>
		+	[[category:physics]]

@@ Line 1: / Line 1: @@
-'''Bra-ket Notation''' is essentially the language of quantum mechanics.  It was invented by a man named Paul Dirac and originally named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref>  Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.
+'''Bra-ket notation''', also known as '''Dirac notation''', is essentially the language of quantum mechanics. It was invented by a man named Paul Dirac and originally named after him.<ref>http://www.quantiki.org/wiki/index.php/Bra-ket_notation</ref> Although observable quantities are associated with [[linear operators]], and states are represented by vectors, the required computations can be greatly simplified through the use of the Dirac Bracket Notation.
-In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle \cdot,\cdot \rangle</math>.  In the Dirac Notation, we use the symbol <math>\left|\psi\right\rangle</math> to represent an element of the Hilbert Space in question.  This vector is called a "ket".  However, by [[Reisz' Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:
+In non-relativistic quantum mechanics, states are said to reside in a [[Hilbert Space]] <math>\mathcal{H}</math> which, by definition, has an [[inner product]], typically denoted by <math>\langle \cdot,\cdot \rangle</math>.  In bra-ket notation, the symbol <math>\left|\psi\right\rangle</math> is used to represent an element of the Hilbert Space in question.  This vector is called a "ket".  However, by [[Reisz' Representation Theorem]], each element <math>\psi</math> of the Hilbert space also uniquely defines a [[linear functional]] which resides in the [[dual space]] in terms of the inner product, as follows:
 <math>f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}</math>
-In the Dirac Notation, the functional defined by <math>\psi</math> is instead represented by:
+In the Dirac notation, the functional defined by <math>\psi</math> is instead represented by:
 <math>\left\langle\psi\right| \dot= f_\psi\left(x\right)</math>
@@ Line 13: / Line 13: @@
 <math>\left\langle\varphi|\psi\right\rangle = \left\langle\psi,\varphi\right\rangle</math>
-The notation is deceptively simple.  The elegant nature of the Dirac Bracket Notation allows physicists to treat linear functionals represented as bras in a very intuitive fashion (they preserve nearly all of the familiar algebraic properties of numbers except commutativity).  Computations of inner products are naturally suggested, and the problem is not bogged down in excessive notation -- it is essentially distilled down to its algebraic content alone.
+The notation is deceptively simple.  The elegant nature of the Dirac Bra-ket notation allows physicists to treat linear functionals represented as bras in a very intuitive fashion (they preserve nearly all of the familiar algebraic properties of numbers except commutativity).  Computations of inner products are naturally suggested, and the problem is not bogged down in excessive notation -- it is essentially distilled down to its algebraic content alone.
@@ Line 57: / Line 57: @@
 <math>\psi\left(x\right)=\left\langle x|\psi\right\rangle</math>
 ==References==
 <references/>