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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 act. More generally the notation uses the angle brackets (the ⟨ and ⟩ symbols) and a vertical bar (the | symbol), for a ket /kɛt/ like to denote a vector in an abstract (usually complex) vector space and a bra, /brɑː/ like to denote a linear functional on , i.e. a co-vector, an element of the dual vector space . The natural pairing of a linear functional with a vector is then written as . On Hilbert spaces, the scalar product (with anti linear first argument) gives an (anti-linear) identification of a vector ket with a linear functional bra . Using this notation, the scalar product . For the vector space , kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using matrix multiplication. If has the standard hermitian inner product , under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the hermitian conjugate .

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 on a two dimensional space of spinors, has eigenvalues with eigen spinors . In bra-ket notation one typically denotes this as , and . 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 introduced in 1939 by Paul Dirac[1][2] and is also known as the Dirac notation. However the bra-ket notation has a precursor in Hermann Grassmann's use of the notation for his inner products nearly 100 years earlier.[3][4]

IntroductionEdit

Bra–ket notation is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread. Many phenomena that are explained using quantum mechanics are explained using bra–ket notation.


Vector spacesEdit

Vectors vs ketsEdit

In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" is much more specific: "vector" refers almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of space time. Such vectors are typically denoted with over arrows ( ), boldface ( ) or indices ( ).

In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. Since the term "vector" is already used for something else (see previous paragraph), and physicists tend to prefer conventional notation to stating what space something is an element of, it is common and useful to denote an element   of an abstract complex vector spaces as a ket   using vertical bars and angular brackets and refer to them as "kets" rather than as vectors and pronounced "ket- " or "ket-A" for |A. Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the   making clear that the label indicates a vector in vector space. In other words, the symbol "|A" has a specific and universal mathematical meaning, while just the "A" by itself does not. For example, |1⟩ + |2⟩ might or might not be equal to |3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers.

Ket notation was invented by Dirac [5]

Bra-Ket notationEdit

Since kets are just vectors in a vector space they can be manipulated using the usual rules of linear algebra, for example:

 

Note how the last line above involves infinitely many different kets, one for each real number x.

If the ket is an element of a vector space, a bra   is an element of its dual space, i.e a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts

A bra   and a ket   (i.e. a functional and a vector), can be combined to an operator   of rank one with outer product

 

Inner product and bra-ket identification on Hilbert spaceEdit

The bra-ket notation is particularly useful in Hilbert spaces which have an inner product[6] that allows Hermitian conjugation and identifying a vector with a linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem). The inner product on Hilbert space   (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti linear) identification between the space of kets and that of bras in the bra ket notation: for a vector ket   define a functional (i.e. bra)   by

 


Bras and kets as row and column vectorsEdit

In the simple case where we consider the vector space  , a ket can be identified with a column vector, and a bra as a row vector. If moreover we use the standard hermitian innerproduct on  , the bra corresponding to a ket, in particular a bra m| and a ket |m with the same label are conjugate transpose. Moreover, conventions are set up in such a way tha writing bras, kets, and linear operators next to each other simply imply matrix multiplication.[7]. In particular the outer product   of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix).


For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication 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

 

Writing elements of a finite dimensional (or mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a basis. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "|m" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "|-" and "|+".


Non-normalizable states and non-Hilbert spacesEdit

Bra–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-normalizable wavefunctions. 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 be broadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). The bra–ket notation continues to work in an analogous way in this broader context.

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

Usage in quantum mechanicsEdit

The 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" |ψ. (Technically, the quantum states are rays of vectors in the Hilbert space, as c|ψ corresponds to 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 the state |1⟩ + i |2⟩ is in a quantum superposition of the states |1⟩ and |2⟩.
  • Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.
  • Dynamics are also described by linear operators on the Hilbert space. For example, in the Schrödinger picture, there is a linear time evolution operator U with the property that if an electron is in state |ψ right now, at a later time it will be in the state U|ψ, the same U for every possible |ψ.
  • 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, and often does involve, bra–ket notation. A few examples follow:

Spinless position–space wave functionEdit

Discrete components Ak of a complex vector |A = ∑k Ak |ek, which belongs to a countably infinite-dimensional Hilbert space; there are countably infinitely many k values and basis vectors |ek.
Continuous components ψ(x) of a complex vector |ψ = ∫ dx ψ(x)|x, which belongs to an uncountably infinite-dimensional Hilbert space; there are infinitely many x values and basis vectors |x.
Components of complex vectors plotted against index number; discrete k and continuous x. Two particular components out of infinitely many are highlighted.

The Hilbert space of a spin-0 point particle is spanned by a "position basis" { |r }, where the label r extends over the set of all points in position space. This label is the eigenvalue of the position operator acting on such a basis state,  . Since there are an uncountably infinite number of vector components in the basis, this is an uncountably infinite-dimensional Hilbert space. The dimensions of the Hilbert space (usually infinite) and position space (usually 1, 2 or 3) are not to be conflated.

Starting from any ket |Ψ⟩ in this Hilbert space, one may define a complex scalar function of r, known as a wavefunction,

 

On the left-hand side, Ψ(r) is a function mapping any point in space to a complex number; on the right-hand side, |Ψ⟩ = ∫ d3r Ψ(r) |r is a ket consisting of a superposition of kets with relative coefficients specified by that function.

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   has the following form,

 

One occasionally encounters an expression such as

 

though this is something of an abuse of notation. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis,   even though, in the momentum basis, this operator amounts to a mere multiplication operator (by p). That is, to say,

 

or

 

Overlap of statesEdit

In quantum mechanics the expression φ|ψ is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ. Mathematically, this means the coefficient for the projection of ψ onto φ. It is also described as the projection of state ψ onto state φ.

Changing basis for a spin-1/2 particleEdit

A stationary spin-1/2 particle has a two-dimensional Hilbert space. One orthonormal basis is:

 

where |↑z is the state with a definite value of the spin operator Sz equal to +1/2 and |↓z is the state with a definite 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., quantum superposition) of these two states:

 

where aψ and bψ 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 aψ, bψ, cψ and dψ; see change of basis.

Misleading usesEdit

There are a few conventions and abuses of notation that are generally accepted by the physics community, but which might confuse the non-initiated.

It is common to use the same symbol for labels and constants in the same equation. For example, α̂ |α = α |α, where the symbol α is used simultaneously as the name of the operator α̂, its eigenvector |α and the associated eigenvalue α.

Something similar occurs in component notation of vectors. While Ψ (uppercase) is traditionally associated with wavefunctions, ψ (lowercase) may be used to denote a label, a wave function or complex constant in the same context, usually differentiated only by a subscript.

The main abuses are including operations inside the vector labels. This is done for a fast notation of scaling vectors. E.g. if the vector |α is scaled by 2, it might be denoted by |α/2, which makes no sense since α is a label, not a function or a number, so you can't perform operations on it.

This is especially common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. |α = |α/21|α/22. Here part of the labeling that should state that all three vectors are different was moved outside the kets, as subscripts 1 and 2. And a further abuse occurs, since α is meant to refer to the norm of the first vector—which is a label denoting a value.

Linear operatorsEdit

Linear operators acting on ketsEdit

A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have certain properties.) In other words, if A is a linear operator and |ψ is a ket, then A|ψ is another ket.

In an N-dimensional Hilbert space, |ψ can be written as an N × 1 column vector, and then A is an N × N matrix with complex entries. The ket A|ψ can be computed by normal matrix multiplication.

Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.

Linear operators acting on brasEdit

Operators can also be viewed as acting on bras from the right hand side. Specifically, if A is a linear operator and φ| is a bra, then φ|A 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, φ| can be written as a 1 × N row vector, and A (as in the previous section) is an N × N matrix. Then the bra φ|A 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 operator A for the physical system in the state |ψ.

Outer productsEdit

A convenient way to define linear operators on a Hilbert space H is given by the outer product: if ϕ| is a bra and |ψ is a ket, the outer product

 

denotes the rank-one operator with the rule

 .

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 |ψ of norm 1, the orthogonal projection onto the subspace spanned by |ψ is

 

Hermitian conjugate operatorEdit

Just as kets and bras can be transformed into each other (making |ψ into ψ|), the element from the dual space corresponding to A|ψ is ψ|A, where A denotes the Hermitian conjugate (or adjoint) of the operator A. In other words,

 

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 is always described by a self-adjoint operator. If A is a self-adjoint operator, then ψ|A|ψ is always a real number (not complex). This implies that expectation values of observables are real.

PropertiesEdit

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

LinearityEdit

  • Since bras are linear functionals,
 
  • By the definition of addition and scalar multiplication of linear functionals in the dual space,[8]
 

AssociativityEdit

Given 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 property holds). For example:

 

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

Hermitian conjugationEdit

Bra–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.,
 
  • 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 as follows:

  • Kets:
 
  • Inner products:
 
Note that φ|ψ is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e.
 
  • Matrix elements:
 
  • Outer products:
 

Composite bras and ketsEdit

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

If |ψ is a ket in V and |φ is a ket in W, the direct product of the two kets is a ket in VW. This is written in various notations:

 

See quantum entanglement and the EPR paradox for applications of this product.

The unit operatorEdit

Consider a complete orthonormal system (basis),

 

for a Hilbert space H, with respect to the norm from an inner product ⟨·,·⟩.

From basic functional analysis, it is known that any ket |ψ can also be written as

 

with ⟨·|·⟩ the inner product on the Hilbert space.

From the commutativity of kets with (complex) scalars, it follows that

 

must be the identity operator, which sends each vector to itself.

This, then, can be inserted in any expression without affecting its value; for example

 

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

In quantum mechanics, it often occurs that little or no information about the inner product ψ|φ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients ψ|ei = ei|ψ* and ei|φ of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more.

For more information, see Resolution of the identity,

  •     1 = ∫ dx |xx| = ∫ dp |pp|, where |p = ∫ dx eixp/ħ|x/2πħ.

Since x′|x = δ(xx′), plane waves follow, x|p = eixp/ħ/2πħ.[9]

Typically, when all matrix elements of an operator such as

 

are available, this resolution serves to reconstitute the full operator,

 

Notation used by mathematiciansEdit

The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space).

Let H be a Hilbert space and hH a vector in H. What physicists would denote by |h is the vector itself. That is,

 .

Let H* be the dual space of H. This is the space of linear functionals on H. The isomorphism Φ : HH* is defined by Φ(h) = φh, where for every gH we define

 ,

where IP(·,·), (·,·), ⟨·,·⟩ and ⟨·|·⟩ are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises when identifying φh and g with h| and |g respectively. This is because of literal symbolic substitutions. Let φh = H = h| and let g = G = |g. This gives

 

One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we are 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 second one, and they usually use not an asterisk but an overline (which the physicists reserve for averages and the Dirac spinor adjoint) to denote complex conjugate numbers; i.e., for scalar products mathematicians usually write

 

whereas physicists would write for the same quantity

 

See alsoEdit

NotesEdit

  1. ^ Dirac 1939
  2. ^ Shankar 1994, Chapter 1
  3. ^ Grassmann 1862
  4. ^ Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.
  5. ^ McMahon, D. (2006). Quantum Mechanics Demystified. McGraw-Hill. ISBN 0-07-145546-9.
  6. ^ Lecture 2 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind on inner product, 2006-10-02.
  7. ^ Gidney, Craig (2017). Bra–Ket Notation Trivializes Matrix Multiplication
  8. ^ Lecture notes by Robert Littlejohn, eqns 12 and 13
  9. ^ In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate   in the momentum representation, i.e.,  . Consequently, the corresponding wavefunction is a constant,  , and  .

ReferencesEdit

External linksEdit