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Bra-ket notation

From Academic Kids

Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product of two states is denoted by a bracket, <math>\langle\phi|\psi\rangle<math>, consisting of a left part, <math>\langle\phi|<math>, called the bra, and a right part, <math>|\psi\rangle<math>, called the ket. The notation was invented by Paul Dirac, and is also known as Dirac notation.

Contents

Bras and kets

In quantum mechanics, the state of a physical system is identified with a vector in a complex Hilbert space, H. Each vector is called a "ket", and written as

<math>|\psi\rangle<math>

where ψ denotes the particular ket.

Every ket <math>|\psi\rangle<math> has a dual bra, written as

<math>\langle\psi|<math>

This is a continuous linear function from H to the complex numbers C, defined by:

<math>\langle\psi|\rho\rangle = \bigg( |\psi\rangle \;,\; |\rho\rangle \bigg)<math> for all kets <math>|\rho\rangle<math>

where ( , ) denotes the inner product defined on the Hilbert space. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically isomorphic. Thus, each bra corresponds to exactly one ket, and vice versa. This is not always the case however, on page 111 of Quantum Mechanics by Cohen-Tannoudji et. al. it is clarified that there is such a relationship between bras and kets so long as the defining functions used are square integrable. Consider a continuous basis and a Dirac delta function or a sine or cosine wave as a wave function. Such functions are not square integrable and therefore it arises that there are bras that exist with no corresponding ket. The reason this does not hinder quantum mechanics is because all wave functions are square integrable in reality.

Incidentally, bra-ket notation can be used even if the vector space is not a Hilbert space. In any Banach space B, the vectors may be notated by bras and the continuous linear functionals by kets. Over any vector space without topology, we may also notate the vectors by bras and the linear functionals by kets. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.

Applying the bra <math>\langle\phi|<math> to the ket <math>|\psi\rangle<math> results in a complex number, called a "bra-ket" or "bracket", which is written as

<math>\langle\phi|\psi\rangle<math>.

In quantum mechanics, this is the probability amplitude for the state ψ to collapse into the state φ.

Properties

Bras and kets can be manipulated in the following ways:

  • Given any bra <math>\langle\phi|<math>, kets <math>|\psi_1\rangle<math> and <math>|\psi_2\rangle<math>, and complex numbers c1 and c2, then, since bras are linear functionals,
<math>\langle\phi| \; \bigg( c_1|\psi_1\rangle + c_2|\psi_2\rangle \bigg) = c_1\langle\phi|\psi_1\rangle + c_2\langle\phi|\psi_2\rangle. <math>
  • Given any ket <math>|\psi\rangle<math>, bras <math>\langle\phi_1|<math> and <math>\langle\phi_2|<math>, and complex numbers c1 and c2, then, by the definition of addition and scalar multiplication of linear functionals,
<math>\bigg(c_1 \langle\phi_1| + c_2 \langle\phi_2|\bigg) \; |\psi\rangle = c_1 \langle\phi_1|\psi\rangle + c_2\langle\phi_2|\psi\rangle. <math>
  • Given any kets <math>|\psi_1\rangle<math> and <math>|\psi_2\rangle<math>, and complex numbers c1 and c2, from the properties of the inner product (with c* denoting the complex conjugate of c),
<math>

c_1|\psi_1\rangle + c_2|\psi_2\rangle<math> is dual to <math> c_1^* \langle\psi_1| + c_2^* \langle\psi_2|. <math>

  • Given any bra <math>\langle\phi|<math> and ket <math>|\psi\rangle<math>, an axiomatic property of the inner product gives
<math>\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*<math>.

Linear operators

If A : HH is a linear operator, we can apply A to the ket <math>|\psi\rangle<math> to obtain the ket <math>(A|\psi\rangle)<math>. Linear operators are ubiquitous in the theory of quantum mechanics. For example, hermitian operators are used to represent observable physical quantities, such as energy or momentum, whereas unitary linear operators represent transformative processes such as rotation or the progression of time.

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

<math>\bigg(\langle\phi|A\bigg) \; |\psi\rangle = \langle\phi| \; \bigg(A|\psi\rangle\bigg)<math>.

This expression is commonly written as

<math>\langle\phi|A|\psi\rangle.<math>

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

<math> |\phi\rang \lang \psi| <math>

denotes the rank one operator that maps the ket <math>|\rho\rangle<math> to the ket <math>|\phi\rangle\langle\psi|\rho\rangle<math> (where <math>\langle\psi|\rho\rangle<math> is a scalar multiplying the vector <math>|\phi\rangle<math>). One of the uses of the outer product is to construct projection operators. Given a ket <math>|\psi\rangle<math> of norm 1, the orthogonal projection onto the subspace spanned by <math>|\psi\rangle<math> is

<math>|\psi\rangle\langle\psi|<math>

Composite bras and kets

Two Hilbert spaces V and W may form a third space <math>V \otimes W<math> by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described by 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 <math>|\psi\rangle<math> is a ket in V and <math>|\phi\rangle<math> is a ket in W, the tensor product of the two kets is a ket in <math>V \otimes W<math>. This is written variously as

<math>|\psi\rangle|\phi\rangle<math> or <math>|\psi\rangle \otimes |\phi\rangle<math> or <math>|\psi \phi\rangle<math>.

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 Schrdinger equation). This process is very similar to the use of coordinate vectors in linear algebra.

For instance, the Hilbert space of a zero-spin point particle is spanned by a position basis <math>\lbrace|\mathbf{x}\rangle\rbrace<math>, where the label x extends over the set of position vectors. Starting from any ket <math>|\psi\rangle<math> in this Hilbert space, we can define a complex scalar function of x, known as a wavefunction:

<math>\psi(\mathbf{x}) \equiv \lang \mathbf{x}|\psi\rang<math>.

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

<math>A \psi(\mathbf{x}) \equiv \lang \mathbf{x}|A|\psi\rang<math>.

Although the operator A on the left hand side of this equation is, by convention, labelled in the same way as the operator on the right hand side, it should be borne in mind that the two are conceptually different entities: the first acts on wavefunctions, and the second acts on kets. For instance, the momentum operator p has the following form:

<math>\mathbf{p} \psi(\mathbf{x}) \equiv \lang \mathbf{x} |\mathbf{p}|\psi\rang = - i \hbar \nabla \psi(x) <math>.

One occasionally encounters an expression like

<math> - i \hbar \nabla |\psi\rang<math>.

This is something of an abuse of notation, though a fairly common one. 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:

<math> - i \hbar \nabla \lang\mathbf{x}|\psi\rang<math>.

For further details, see rigged Hilbert space.de:Bra-Ket fr:Notation bra-ket he:סימון דיראק it:Notazione bra-ket ko:브라-켓 표기법 sl:Diracov zapis

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