Bra-ket notation 1

Bra-ket notationIn quantum mechanics, Bra-ket notation is a standard notation for describing quantum states, composed of anglebrackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is socalled because the inner product (or dot product) of two states is denoted by a ⟨bra|c|ket⟩;

,consisting of a left part, , called the bra (  /brɑː/), and a right part, , called the ket (  /kɛt/). Thenotation was introduced in 1939 by Paul Dirac[1] and is also known as Dirac notation, though the notation hasprecursors in Grassmann's use of the notation for his inner products nearly 100 years previously.[2]

Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantummechanics—including a large portion of modern physics — is usually explained with the help of bra-ket notation.The expression is typically interpreted as the probability amplitude for the state ψ to collapse into the stateϕ.

Vector spaces

Background: Vector spacesIn physics, basis vectors allow any vector to be represented geometrically using angles and lengths, in differentdirections, i.e. in terms of the spatial orientations. It is simpler to see the notational equivalences between ordinarynotation and bra-ket notation, so for now; consider a vector A as an element of 3-d Euclidean space using the field ofreal numbers, symbolically stated as .The vector A can be written using any set of basis vectors and corresponding coordinate system. Informally basisvectors are like "building blocks of a vector", they are added together to make a vector, and the coordinates are thenumber of basis vectors in each direction. Two useful representations of a vector are simply a linear combination ofbasis vectors, and column matrices. Using the familiar cartesian basis, a vector A is written;

Illustration of cartesian vectors, bases, coordinates and components. The coordinates ofthe vector are equal to the projections of the vector (yellow) onto the x-component basis

vector (green) - using the dot product (a special case of an inner product, see below).

Bra-ket notation 2

respectively, where ex, ey, ez denotes the cartesian basis vectors (all are orthogonal unit vectors) and Ax, Ay, Az are thecorresponding coordinates, in the x, y, z directions. In a more general notation, for any basis in 3d space we write;

Generalizing further, consider a vector A in an N dimensional vector space over the field of complex numbers ,symbolically stated as . The vector A is still conventionally represented by a linear combination of basisvectors or a column matrix:

though the coordinates and vectors are now all complex-valued.Even more generally, A can be a vector in a complex Hilbert space. Some Hilbert spaces, like , have finitedimension, while others have infinite dimension. In an infinite-dimensional space, the column-vector representationof A would be a list of infinitely many complex numbers.

Bra-ket notation 3

Ket notation for vectors

Rather than boldtype, over/under-arrows, underscores etc. conventionally used elsewhere; , Dirac'snotation for a vector uses vertical bars and angular brackets; . When this notation is used, these vectors arecalled ket, read as "ket-A".[3] This applies to all vectors, the resultant vector and the basis. The previous vectors arenow written

Illustration of cartesian vectors, bases, coordinates and components. The coordinates ofthe vector are equal to the projections of the vector (yellow) onto the x-component basis

vector (green) - using the inner product (see below).

or in a more easily generalizednotation,

The last one may be written in short as

Notice how any symbols, letters,numbers, or even words — whateverserves as a convenient label — can beused as the label inside a ket. In otherwords, the symbol "   " has aspecific and universal mathematicalmeaning, but just the "A" by itself doesnot. Nevertheless, for convenience,there is usually some logical schemebehind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics with alist of their quantum numbers.

Inner products and brasAn inner product is a generalization of the dot product. The inner product of two vectors is a complex number.Bra-ket notation uses a specific notation for inner products:

For example, in three-dimensional complex Euclidean space,

where denotes the complex conjugate of . A special case is the inner product of a vector with itself, which issquare of its norm (magnitude):

Bra-ket notation splits this inner product (also called a "bracket") into two pieces, the "bra" and the "ket":

where is called a bra, read as "bra-A", and is a ket as above.The purpose of "splitting" the inner product into a bra and a ket is that both the bra and the ket aremeaningful on their own, and can be used in other contexts besides within an inner product. There are two mainways to think about the meanings of separate bras and kets:

Bra-ket notation 4

Bras and kets as row and column vectors

For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrixmultiplication of a row vector with a column vector:

Based on this, the bras and kets can be defined as:

and then it is understood that a bra next to a ket implies matrix multiplication.The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice-versa:

because if one starts with the bra

then performs a complex conjugation, and then a matrix transpose, one ends up with the ket

Bras as linear operators on kets

A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to saythat bras are linear functionals on kets, i.e. operators that input a ket and output a complex number. The bra operatorsare defined to be consistent with the inner product.In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and correspondingbras and kets are related by the Riesz representation theorem.

Non-normalizable states and non-Hilbert spacesBra-ket notation can be used even if the vector space is not a Hilbert space.In quantum mechanics, it is common practice to write down kets which have infinite norm, i.e. non-normalisablewavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves.These do not, technically, belong to the Hilbert space itself. However, the definition of "Hilbert space" can bebroadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). Thebra-ket notation continues to work in an analogous way in this broader context.For a rigorous treatment of the Dirac inner product of non-normalizable states see the definition given by D. Carfì in[4] and.[4] For a rigorous definition of basis with a continuous set of indices and consequently for a rigorousdefinition of position and momentum basis see.[4] For a rigorous statement of the expansion of an S-diagonalizableoperator - observable - in its eigenbasis or in another basis see.[4]

Bra-ket notation 5

Banach spaces are a different generalization of Hilbert spaces. In a Banach space B, the vectors may be notated bykets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate thevectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have themeaning of an inner product, because the Riesz representation theorem does not apply.

Usage in quantum mechanicsThe mathematical structure of quantum mechanics is based in large part on linear algebra:• Wave functions and other quantum states can be represented as vectors in a complex Hilbert space. (The exact

structure of this Hilbert space depends on the situation.) In bra-ket notation, for example, an electron might be in"the state ". (Technically, the quantum states are rays of vectors in the Hilbert space, as correspondsto the same state for any nonzero complex number c.)

• Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in thestate is in a quantum superposition of the states and .

• Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.• Dynamics is also described by linear operators on the Hilbert space. For example, in the Schrödinger picture,

there is a linear operator U with the property that if an electron is in state right now, then in one minute itwill be in the state , the same U for every possible .

• Wave function normalization is scaling a wave function so that its norm is 1.Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, andoften does involve, bra-ket notation. A few examples follow:

Position-space wave function

The Hilbert space of a spin-0 point particle is spanned by a "position basis" , where the label r extends overthe set of all points in space. Since there are infinitely many vectors in the basis, this is an infinite-dimensionalHilbert space.Starting from any ket in this Hilbert space, we can define a complex scalar function of r, known as awavefunction:

On the left side, is a function mapping any point in space to a complex number; on the right side, is a ket.It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by

For instance, the momentum operator p has the following form:

One occasionally encounters an expression like

though this is something of a (fairly common) abuse of notation. The differential operator must be understood to bean abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression isprojected into the position basis:

Bra-ket notation 6

Overlap of states

In quantum mechanics the expression is typically interpreted as the probability amplitude for the state tocollapse into the state . Mathematically, this means the coefficient for the projection of onto .

Changing basis for a spin-1/2 particleA stationary spin-½ particle has a two-dimensional Hilbert space. One orthonormal basis is:

where is the state with a definite value of the spin operator Sz equal to +1/2 and is the state with adefinite value of the spin operator Sz equal to -1/2.Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantumsuperposition) of these two states:

where are complex numbers.A different basis for the same Hilbert space is:

defined in terms of Sx rather than Sz.Again, any state of the particle can be expressed as a linear combination of these two:

In vector form, you might write

depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.

There is a mathematical relationship between ; see change of basis.

Misleading usesThere are few conventions and notation abuses generally accepted on the physical community which might confusethe non initiated.It is common among physicists to use the same symbol for labels and constants on the same equation. It supposedlybecomes easier to identify that the constant is related to the labeled object, and is claimed that the divergent nature ofeach will eliminate any ambiguity and no further differentiation is required. A common example is the usualdefinition of a quantum coherent state as satisfying where is both the name of the state andits eigenvalue(a complex number) with respect to the lowering operator.Something similar occurs in component notation of vectors. While (uppercase) is traditionally associated withwavefunctions, (lowercase) may be used to denote a label, a wave function or complex constant in the samecontext, usually differentiated only by a subscript.The main abuses are including operations inside the vector labels. This is usually done for a fast notation of scalingvectors. E.g. if the vector is scaled by , it might be denoted by , which makes no sense since

is a label, not a function or a number, so you can't make operations on it.This is specially common when denoting vectors as tensor products, where part of the labels are moved outside thedesigned slot. E.g. . Here part of the labeling that should state that all three vectorsare different was moved outside the kets, as subscripts 1 and 2. And a further abuse occurs, since is meant to referto the norm of the first vector - which is a label is denoting a value.

Bra-ket notation 7

Linear operators

Linear operators acting on ketsA linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to havecertain properties.) In other words, if A is a linear operator and is a ket, then is another ket.

In an N-dimensional Hilbert space, can be written as an N×1 column vector, and then A is an N×N matrix withcomplex entries. The ket can be computed by normal matrix multiplication.Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities arerepresented by self-adjoint operators, such as energy or momentum, whereas transformative processes arerepresented by unitary linear operators such as rotation or the progression of time.

Linear operators acting on brasOperators can also be viewed as acting on bras from the right hand side. Specifically, if A is a linear operator and

is a bra, then is another bra defined by the rule

.

(in other words, a function composition). This expression is commonly written as (cf. energy inner product)

In an N-dimensional Hilbert space, can be written as a 1×N row vector, and A (as in the previous section) is anN×N matrix. Then the bra can be computed by normal matrix multiplication.If the same state vector appears on both bra and ket side,

then this expression gives the expectation value, or mean or average value, of the observable represented by operatorA for the physical system in the state .

Outer products

A convenient way to define linear operators on H is given by the outer product: if is a bra and is a ket, theouter product

denotes the rank-one operator that maps the ket to the ket (where is a scalar multiplyingthe vector ).For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:

The outer product is an N×N matrix, as expected for a linear operator.

One of the uses of the outer product is to construct projection operators. Given a ket of norm 1, the orthogonalprojection onto the subspace spanned by is

Bra-ket notation 8

Hermitian conjugate operator

Just as kets and bras can be transformed into each other (making into ) the element from the dual spacecorresponding with is where A† denotes the Hermitian conjugate (or adjoint) of the operator A. Inother words,

if and only if .If A is expressed as an N×N matrix, then A† is its conjugate transpose.Self-adjoint operators, where A=A†, play an important role in quantum mechanics; for example, an observable isalways described by a self-adjoint operator. If A is a self-adjoint operator, then is always a real number(not complex). This implies that expectation values of observables are real.

PropertiesBra-ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of theproperties that allow this manipulation are listed herein. In what follows, c1 and c2 denote arbitrary complexnumbers, c* denotes the complex conjugate of c, A and B denote arbitrary linear operators, and these properties areto hold for any choice of bras and kets.

Linearity•• Since bras are linear functionals,

• By the definition of addition and scalar multiplication of linear functionals in the dual space,[5]

AssociativityGiven any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators(but not addition), written in bra-ket notation, the parenthetical groupings do not matter (i.e., the associative propertyholds). For example:

and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguouslybecause of the equalities on the left. Note that the associative property does not hold for expressions that includenon-linear operators, such as the antilinear time reversal operator in physics.

Hermitian conjugationBra-ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †)of expressions. The formal rules are:•• The Hermitian conjugate of a bra is the corresponding ket, and vice-versa.•• The Hermitian conjugate of a complex number is its complex conjugate.• The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is

itself—i.e.,

.

Bra-ket notation 9

•• Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators,written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components,and taking the Hermitian conjugate of each.

These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are asfollows:•• Kets:

•• Inner products:

•• Matrix elements:

•• Outer products:

Composite bras and ketsTwo Hilbert spaces V and W may form a third space by a tensor product. In quantum mechanics, this isused for describing composite systems. If a system is composed of two subsystems described in V and Wrespectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception tothis is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.)

If is a ket in V and is a ket in W, the direct product of the two kets is a ket in . This is writtenvariously as

or or or

The unit operatorConsider a complete orthonormal system (basis), , for a Hilbert space H, with respect to the normfrom an inner product . From basic functional analysis we know that any ket can be written as

with the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars nowfollows that

must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affectingits value, for example

where in the last identity Einstein summation convention has been used.

In quantum mechanics it often occurs that little or no information about the inner product of two arbitrary(state) kets is present, while it is possible to say something about the expansion coefficients and

of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful toinsert the unit operator into the bracket one time or more (for more information see Resolution of the identity).

Bra-ket notation 10

Notation used by mathematiciansThe object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner productspace).

Let be a Hilbert space and is a vector in . What physicists would denote as is the vector itself.That is

.Let be the dual space of . This is the space of linear functionals on . The isomorphism is defined by where for all we have

,where and are just different notations for expressing an inner product between twoelements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises whenidentifying and with and respectively. This is because of literal symbolic substitutions. Let

and let . This gives

One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since weare dealing with linear operators and composition acts like a ring multiplication.Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the secondone, and they don't use the *-symbol, but an overline (which the physicists reserve to averages) to denoteconjugate-complex numbers, i.e. for scalar products mathematicians usually write

whereas physicists would write for the same quantity

References and notes[1] PAM Dirac (1939). "A new notation for quantum mechanics" (http:/ / journals. cambridge. org/ action/ displayAbstract?fromPage=online&

aid=2031476). Mathematical Proceedings of the Cambridge Philosophical Society 35 (3): pp. 416-418. doi:10.1017/S0305004100021162. .[2] H. Grassmann (1862). Extension Theory. History of Mathematics Sources. American Mathematical Society, London Mathematical Society,

2000 translation by Lloyd C. Kannenberg.[3][3] Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9[4] Carfì, David (April 2003). "Dirac-orthogonality in the space of tempered distributions" (http:/ / portal. acm. org/ citation. cfm?id=774918).

Journal of Computational and Applied Mathematics 153 (1-2): 99–107. Bibcode 2003JCoAM.153...99C.doi:10.1016/S0377-0427(02)00634-9. .

[5] Lecture notes by Robert Littlejohn (http:/ / bohr. physics. berkeley. edu/ classes/ 221/ 1112/ notes/ hilbert. pdf), eqns 12 and 13

Bra-ket notation 11

Further reading• Feynman, Leighton and Sands (1965). The Feynman Lectures on Physics Vol. III. Addison-Wesley.

ISBN 0-201-02115-3.

External links• Richard Fitzpatrick, "Quantum Mechanics: A graduate level course" (http:/ / farside. ph. utexas. edu/ teaching/

qm/ lectures/ lectures. html), The University of Texas at Austin.• 1. Ket space (http:/ / farside. ph. utexas. edu/ teaching/ qm/ lectures/ node7. html)• 2. Bra space (http:/ / farside. ph. utexas. edu/ teaching/ qm/ lectures/ node8. html)• 3. Operators (http:/ / farside. ph. utexas. edu/ teaching/ qm/ lectures/ node9. html)• 4. The outer product (http:/ / farside. ph. utexas. edu/ teaching/ qm/ lectures/ node10. html)• 5. Eigenvalues and eigenvectors (http:/ / farside. ph. utexas. edu/ teaching/ qm/ lectures/ node11. html)

• Robert Littlejohn, Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra-ketnotation. (http:/ / bohr. physics. berkeley. edu/ classes/ 221/ 0708/ notes/ hilbert. pdf)

Article Sources and Contributors 12

Article Sources and ContributorsBra-ket notation  Source: http://en.wikipedia.org/w/index.php?oldid=525032852  Contributors: A5, AManWithNoPlan, AlexChurchill, Ancheta Wis, Anders Kaseorg, Andris, Arjen Dijksman,ArnoldReinhold, Ashchap, AugPi, Awakeforever, AxelBoldt, Belovedfreak, BenFrantzDale, Bevo, Brazmyth, Brews ohare, CYD, Cap'n Refsmmat, Carl Turner, CesarB, Charles Matthews,Chetvorno, Chris Howard, Clickingban, CompuChip, Conversion script, Custos0, DGDanforth, DMacks, DannyWilde, Dauto, Davidcarfi, Dcoetzee, Decumanus, Deflective, Dextercioby,Discospinster, Dolda2000, DragonflySixtyseven, Eddy 1000, Edsanville, Eriatarka, F=q(E+v^B), Fastfission, Ffangs, Fibonacci, FilipeS, Fornita, Freiddie, Fresheneesz, Gauge, Gblandst, GeneWard Smith, Gianluigi, Giftlite, Gulmammad, H2g2bob, Harold f, Headbomb, Hydrogen Iodide, JabberWok, Japo, Jcchurch, Jheald, Jim1138, Jmhodges, John of Reading, JoonasD6, Jordgette,Julian Mendez, Justin545, KHamsun, KnightRider, Kwamikagami, Larsobrien, Laurascudder, Laussy, Leafyplant, Lomask, Looxix, Lupin, Lysdexia, M0M3NTUM, Magioladitis, Mako098765,MarkSweep, Maschen, MathKnight, Mathchem271828, Mct mht, Meier99, Michael Hardy, Mikegretton, Montyy0, NNemec, Nehahaha, Netheril96, Nikai, Ninly, Oleg Alexandrov, Pashtron,Passw0rd, Patrick, Philipp Wetzlar, PhotoBox, Phys, Pj.de.bruin, PowerUserPCDude, Ravedave, Rich Farmbrough, Rjwilmsi, Robert littlejohn, RockMagnetist, Rossami, SCZenz, Salgueiro,Samhg, Sbyrnes321, Seb, SebastianHelm, Shahram 77, Skyezx, Smjg, Stepa, Stephen B Streater, Stromal, Swishka, Teply, The Anome, Theresa knott, Tobias Bergemann, Torchiest, Treekids,Tregoweth, Trewornan, TylerWilliamRoss, Ubiquity, UncleDouggie, Vald, Vanished user 47736712, WISo, Wayne Slam, Weialawaga, Widr, Widsith, Wikiborg, WillowW, Wnt, Xyzzyva,Y!qtr9f, 153 anonymous edits

Image Sources, Licenses and ContributorsFile:Loudspeaker.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Loudspeaker.svg  License: Public Domain  Contributors: Bayo, Gmaxwell, Gnosygnu, Husky, Iamunknown,Mirithing, Myself488, Nethac DIU, Omegatron, Rocket000, Shanmugamp7, The Evil IP address, Wouterhagens, 23 anonymous editsFile:Cartesian vector components bases projection.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Cartesian_vector_components_bases_projection.svg  License: Public Domain Contributors: User:MaschenFile:Ket vector components bases projection.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ket_vector_components_bases_projection.svg  License: Public Domain  Contributors:User:Maschen

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/