Bra-ket notation - Citizendia

Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons It can also be used to denote abstract vectors and linear functionals in pure mathematics. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and It is so called because the inner product (or dot product) of two states is denoted by a bracket, $\langle\phi|\psi\rangle$ , consisting of a left part, $\langle\phi|$ , called the bra, and a right part, $|\psi\rangle$ , called the ket. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R The notation was invented by Paul Dirac,^[1] and is also known as Dirac notation.

1 Bras and kets
- 1.1 Most common use: Quantum mechanics
- 1.2 More general uses
2 Linear operators
3 Properties
4 Composite bras and kets
5 Representations in terms of bras and kets
6 The unit operator
7 Notation used by mathematicians
8 References and notes
9 Further reading
10 External links

Bras and kets

Most common use: Quantum mechanics

In quantum mechanics, the state of a physical system is identified with a unit ray in a complex separable Hilbert space, $\mathcal{H}$ , or, equivalently, by a point in the projective Hilbert space of the system. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n This article assumes some familiarity with Analytic geometry and the concept of a limit. Each vector in the ray is called a "ket" and written as $|\psi\rangle$ , which would be read as "ket psi ". In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added For other uses see Psi. Psi (uppercase Ψ, lowercase ψ) is the 23rd letter of the Greek alphabet and has a (The ψ can be replaced by any symbols, letters, numbers, or even words—whatever serves as a convenient label for the ket. ) The ket can be viewed as a column vector and (given a basis for the Hilbert space) written out in components,

$|\psi\rangle = (c_0, c_1, c_2, ...)^T,$

when the considered Hilbert space is finite-dimensional. In infinite-dimensional spaces there are infinitely many components and the ket may be written in complex function notation, by prepending it with a bra (see below). For example,

$\langle x|\psi\rangle = \psi(x)\ = c e^{- ikx}.$

Every ket $|\psi\rangle$ has a dual bra, written as $\langle\psi|$ . This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional For example, the bra corresponding to the ket $|\psi\rangle$ above would be the row vector

$\langle\psi| = (c_0^*, c_1^*, c_2^*, ...).$

This is a continuous linear functional from $H$ to the complex numbers $\mathbb{C}$ , defined by:

$\langle\psi| : H \to \mathbb{C}: \langle \psi | \left( |\rho\rangle \right) = \operatorname{IP}\left( |\psi\rangle \;,\; |\rho\rangle \right)$ for all kets $|\rho\rangle$

where $\operatorname{IP}( \cdot , \cdot )$ denotes the inner product defined on the Hilbert space. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. This article assumes some familiarity with Analytic geometry and the concept of a limit. Here an advantage of the bra-ket notation becomes clear: when we drop the parentheses (as is common with linear functionals) and meld the bars together we get $\langle\psi|\rho\rangle$ , which is common notation for an inner product in a Hilbert space. This combination of a bra with a ket to form a complex number is called a bra-ket or bracket.

The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically conjugate isomorphic. There are several well-known theorems in Functional analysis known as the Riesz representation theorem. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals Thus, each bra corresponds to exactly one ket, and vice versa.

Note that this only applies to states that are actually vectors in the Hilbert space. Non-normalizable states, such as those whose wavefunctions are Dirac delta functions or infinite plane waves, do not technically belong to the Hilbert space. In Quantum mechanics, Wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose So if such a state is written as a ket, it will not have a corresponding bra according to the above definition. This problem can be dealt with in either of two ways. First, since all physical quantum states are normalizable, one can carefully avoid non-normalizable states. Alternatively, the underlying theory can be modified and generalized to accommodate such states, as in the Gelfand-Naimark-Segal construction or rigged Hilbert spaces. In Functional analysis, given a C*-algebra A, the Gelfand-Naimark-Segal construction establishes a correspondence between cyclic *-representations of In Mathematics, a rigged Hilbert space ( Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the In fact, physicists routinely use bra-ket notation for non-normalizable states, taking the second approach either implicitly or explicitly.

In quantum mechanics the expression $\langle\phi|\psi\rangle$ (mathematically: the coefficient for the projection of $\psi\!$ onto $\phi\!$ ) is typically interpreted as the probability amplitude for the state $\psi\!$ to collapse into the state $\phi.\!$

More general uses

Bra-ket notation can be used even if the vector space is not a Hilbert space. In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which Quantum systems apparently evolve according to the laws of This article assumes some familiarity with Analytic geometry and the concept of a limit. In any Banach space B, the vectors may be notated by kets and the continuous linear functionals by bras. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional 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.

Linear operators

If A : H → H is a linear operator, we can apply A to the ket $|\psi\rangle$ to obtain the ket $(A|\psi\rangle)$ . In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H →

Operators can also be viewed as acting on bras from the right hand side. Composing the bra $\langle\phi|$ with the operator A results in the bra $(\langle\phi|A)$ , defined as a linear functional on H by the rule

$\bigg(\langle\phi|A\bigg) \; |\psi\rangle = \langle\phi| \; \bigg(A|\psi\rangle\bigg)$ .

This expression is commonly written as

$\langle\phi|A|\psi\rangle.$

If the same state vector appears on both bra and ket side, 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 $|\psi\rangle$ , written as

$\langle\psi|A|\psi\rangle.$

A convenient way to define linear operators on H is given by the outer product: if $\langle\phi|$ is a bra and $|\psi\rangle$ is a ket, the outer product

$|\phi\rang \lang \psi|$

denotes the rank-one operator that maps the ket $|\rho\rangle$ to the ket $|\phi\rangle\langle\psi|\rho\rangle$ (where $\langle\psi|\rho\rangle$ is a scalar multiplying the vector $|\phi\rangle$ ). In Quantum mechanics, the expectation value is the predicted mean value of the result of an experiment In Linear algebra, the outer product typically refers to the tensor product of two vectors. One of the uses of the outer product is to construct projection operators. Given a ket $|\psi\rangle$ of norm 1, the orthogonal projection onto the subspace spanned by $|\psi\rangle$ is

$|\psi\rangle\langle\psi|.$

Just as kets and bras can be transformed into each other (making $|\psi\rangle$ into $\langle\psi|$ ) the element from the dual space corresponding with $A|\psi\rangle$ is $\langle \psi | A^\dagger$ where A^† denotes the Hermitian conjugate of the operator A. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator.

It is usually taken as a postulate or axiom of quantum mechanics, that any operator corresponding to an observable quantity (shortly called observable) is self-adjoint, that is, it satisfies A^† = A. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical Quantity is a kind of property which exists as magnitude or multitude Then the identity

holds (for the first equality, use the scalar product's conjugate symmetry and the conversion rule from the preceding paragraph). In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. This implies that expectation values of observables are real. In Quantum mechanics, the expectation value is the predicted mean value of the result of an experiment

Properties

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.

Linearity

Since bras are linear functionals,

By the definition of addition and scalar multiplication of linear functionals in the dual space,

Associativity

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. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals e. , the associative property holds). In Mathematics, associativity is a property that a Binary operation can have For example:

$\lang \psi| (A |\phi\rang) = (\lang \psi|A)|\phi\rang$

$(A|\psi\rang)\lang \phi| = A(|\psi\rang \lang \phi|)$

and so forth. The expressions can thus be written, unambiguously, with no parentheses whatsoever. Note that the associative property does not hold for expressions that include non-linear operators, such as the antilinear time reversal operator in physics. In Mathematics, a mapping f: V → W from a Complex vector space to another is said to be antilinear (or conjugate-linear T Symmetry is the symmetry of physical laws under a Time reversal transformation &mdash T t \mapsto -t

Hermitian conjugation

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. ,

$(x^\dagger)^\dagger=x$ .

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:

$\left(c_1|\psi_1\rangle + c_2|\psi_2\rangle\right)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2|.$

Inner products:

$\lang \phi | \psi \rang^* = \lang \psi|\phi\rang$

Matrix elements:

$\lang \phi| A | \psi \rang^* = \lang \psi | A^\dagger |\phi \rang$

$\lang \phi| A^\dagger B^\dagger | \psi \rang^* = \lang \psi | BA |\phi \rang$

Outer products:

Composite bras and kets

Two Hilbert spaces V and W may form a third space $V \otimes W$ by a tensor product. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector 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. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle In that case, the situation is a little more complicated. )

If $|\psi\rangle$ is a ket in V and $|\phi\rangle$ is a ket in W, the direct product of the two kets is a ket in $V \otimes W$ . This is written variously as

Representations in terms of bras and kets

In quantum mechanics, it is often convenient to work with the projections of state vectors onto a particular basis, rather than the vectors themselves. The reason is that the former are simply complex numbers, and can be formulated in terms of partial differential equations (see, for example, the derivation of the position-basis Schrödinger equation). Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system This process is very similar to the use of coordinate vectors in linear algebra. In Linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or equivalently as an Linear algebra is the branch of Mathematics concerned with

For instance, the Hilbert space of a zero-spin point particle is spanned by a position basis $\lbrace|\mathbf{x}\rangle\rbrace$ , where the label x extends over the set of position vectors. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Starting from any ket $|\psi\rangle$ in this Hilbert space, we can define a complex scalar function of x, known as a wavefunction:

$\psi(\mathbf{x}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{x}|\psi\rang.$

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by

$A \psi(\mathbf{x}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{x}|A|\psi\rang.$

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

$\mathbf{p} \psi(\mathbf{x}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{x} |\mathbf{p}|\psi\rang = - i \hbar \nabla \psi(x).$

One occasionally encounters an expression like

$- i \hbar \nabla |\psi\rang.$

This is something of an abuse of notation, though a fairly common one. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Mathematics, abuse of notation occurs when an author uses a Mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition 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 into the position basis:

$- i \hbar \nabla \lang\mathbf{x}|\psi\rang.$

For further details, see rigged Hilbert space. In Mathematics, a rigged Hilbert space ( Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the

The unit operator

Note: the Einstein summation convention of summing on repeated indices is used below. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational

Consider a complete orthonormal system (basis), $\{ e_i \ | \ i \in \mathbb{N} \}$ , for a Hilbert space H, with respect to the norm from an inner product $\langle\cdot,\cdot\rangle$ . In Linear algebra, two vectors in an Inner product space are orthonormal if they are orthogonal and both of unit length From basic functional analysis we know that any ket $|\psi\rangle$ can be written as

$|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,$

with $\langle\cdot,\cdot\rangle$ the inner product on the Hilbert space. For functional analysis as used in psychology see the Functional analysis (psychology article From the commutativity of kets with (complex) scalars now follows that

$\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \hat{1}$

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

$\langle v | w \rangle = \langle v | \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | w \rangle = \langle v | \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j | w \rangle = \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle$

where in the last identity Einstein summation convention has been used. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational

In quantum mechanics it often occurs that little or no information about the inner product $\langle\psi|\phi\rangle$ of two arbitrary (state) kets is present, while it is possible to say something about the expansion coefficients $\langle\psi|e_i\rangle = \langle e_i|\psi\rangle^*$ and $\langle e_i|\phi\rangle$ of those vectors with respect to a chosen (orthonormalized) basis. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In this case it is particularly useful to insert the unit operator into the bracket one time or more.

Notation used by mathematicians

The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product space). This article assumes some familiarity with Analytic geometry and the concept of a limit.

Let $\mathcal{H}$ be a Hilbert space and $h\in\mathcal{H}$ . What physicists would denote as $|h\rangle$ is the vector itself. That is

$(|h\rangle)\in \mathcal{H}$ .

Let $\mathcal{H}^*$ be the dual space of $\mathcal{H}$ . This is the space of linear functionals on $\mathcal{H}$ . The isomorphism $\Phi:\mathcal{H}\to\mathcal{H}^*$ is defined by $Φ(h) = φ h$ where for all $g\in\mathcal{H}$ we have

$\phi_h(g) = \mbox{IP}(h,g) = (h,g) = \langle h,g \rangle = \langle h|g \rangle$ ,

Where

$\mbox{IP}(\cdot,\cdot), (\cdot,\cdot),\langle \cdot,\cdot \rangle, \langle \cdot | \cdot \rangle$

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 $φ h$ and $g$ with $\langle h |$ and $|g \rangle$ respectively. This is because of literal symbolic substitutions. Let $\phi_h = H = \langle h|$ and $g=G=|g\rangle$ . This gives

$\phi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|(|g\rangle).$

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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

References and notes

^ PAM Dirac (1982). The principles of quantum mechanics, Fourth Edition, Oxford UK: Oxford University Press, pp. 18 ff. ISBN 0198520115.

External links

Richard Fitzpatrick, "Quantum Mechanics: A graduate level course", The University of Texas at Austin.
- 1. Ket space
- 2. Bra space
- 3. Operators
- 4. The outer product
- 5. Eigenvalues and eigenvectors
Robert Littlejohn, Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra-ket notation.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents