Bra-ket notation
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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. It is also the notation of choice in quantum computing.
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[edit] Bras and kets
[edit] Most common use: Quantum mechanics
In quantum mechanics, the state of a physical system is identified with a vector in a complex Hilbert space, <math>H</math>. Each vector is called a "ket" and written as <math>|\psi\rangle</math>, which would be read as "psi ket". The ket can be viewed as a column vector and (given a basis for the Hilbert space) written out in components,
- <math>|\psi\rangle = (c_0, c_1, c_2, ...)^T, </math>
when the considered Hilbert space is finite-dimensional. In infinite-dimensional spaces there are infinitely many components (possibly even uncountably many) and the ket may be written in function notation, for example
- <math>|\phi\rangle = |\phi(x)\rangle = e^{- x^2}.</math>
Every ket <math>|\psi\rangle</math> has a dual bra, written as <math>\langle\psi|</math>. For example, the bra corresponding to the ket <math>|\psi\rangle</math> above would be the row vector
- <math>\langle\psi| = (c_0^*, c_1^*, c_2^*, ...). </math>
This is a continuous linear function from <math>H</math> to the complex numbers <math>\mathbb{C}</math>, defined by:
- <math>\langle\psi| : H \to \mathbb{C}: \langle \psi | \left( |\rho\rangle \right) = \operatorname{IP}\left( |\psi\rangle \;,\; |\rho\rangle \right)</math> for all kets <math>|\rho\rangle</math>
where <math>\operatorname{IP}( \cdot , \cdot )</math> denotes the inner product defined on the Hilbert space. Here an advantage of the bra-ket notation becomes clear: when we drop the brackets (as is common with linear functionals) and melt the bars together we get <math>\langle\psi|\rho\rangle</math>, 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.
In quantum mechanics the expression <math>\langle\phi|\psi\rangle</math> is typically interpreted as the probability amplitude for the state <math>\psi\!</math> to collapse into the state <math>\phi.\!</math>
[edit] More general uses
The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa. 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. However, this is not always the case; 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. This does not hinder quantum mechanics because all physically realistic wave functions are square integrable.
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 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.
[edit] Properties
Because each ket is a vector in a complex Hilbert space and each bra-ket is an inner product, it follows directly that 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>
[edit] Linear operators
If A : H → H 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, observable physical quantities are represented by hermitian operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.
Operators can also be viewed as acting on bras from the right hand side. Composing the bra <math>\langle\phi|</math> with the operator A 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>
Just as kets and bras can be transformed into each other (making <math>|\psi\rangle</math> into <math>\langle\psi|</math>) the element from the dual space corresponding with <math>A|\psi\rangle</math> is <math>\langle \psi | A^\dagger</math> where A† denotes the Hermitian conjugate of the operator A.
It is usually taken as a postulate or axiom of quantum mechanics, that any operator corresponding to an observable quantity (shortly called observable) is Hermitian, that is, it satisfies A† = A. Then the identity
- <math> \langle \psi | A | \psi \rangle^\star = \langle \psi |A^\dagger |\psi \rangle = \langle \psi | A | \psi \rangle </math>
holds (for the first equality, use the scalar product's conjugate symmetry and the conversion rule from the preceding paragraph). This implies that expectation values of observables are real.
[edit] 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> or <math>|\psi ,\phi\rangle.</math>
[edit] 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). 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}) \ \stackrel{\mathrm{def}}{=}\ \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}) \ \stackrel{\mathrm{def}}{=}\ \lang \mathbf{x}|A|\psi\rang.</math>
For instance, the momentum operator p has the following form:
- <math>\mathbf{p} \psi(\mathbf{x}) \ \stackrel{\mathrm{def}}{=}\ \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.
[edit] The unit operator
Consider a complete orthonormal system (basis), <math>\{ e_i \ | \ i \in \mathbb{N} \}</math>, for a Hilbert space H, with respect to the norm from an inner product <math>\langle\cdot,\cdot\rangle</math>. From basic functional analysis we know that any ket <math>|\psi\rangle</math> can be written as
- <math>|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,</math>
with <math>\langle\cdot,\cdot\rangle</math> the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars now follows that
- <math>\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \hat{1}</math>
must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affecting it's value, for example
- <math> \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 w | e_i \rangle \langle e_i | e_j \rangle \langle e_j | v \rangle </math>
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 <math>\langle\psi|\phi\rangle</math> of two arbitrary (state) kets is present, while it is possible to say something about the expansion coefficients <math>\langle\psi|e_i\rangle = \langle e_i|\psi\rangle^*</math> and <math>\langle e_i|\phi\rangle</math> of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful to insert the unit operator into the bracket one or multiple times.
[edit] Notation used by mathematicians
The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product space).
Let <math> \mathcal{H} </math> be a Hilbert space and <math> h\in\mathcal{H} </math>. What physicists would denote as <math> |h\rangle </math> is the vector itself. That is
- <math> (|h\rangle)\in \mathcal{H} </math>.
Let <math> \mathcal{H}^* </math> be the dual space of <math> \mathcal{H} </math>. This is the space of linear operators on <math>\mathcal{H}</math>. The isomorphism <math> \Phi:\mathcal{H}\to\mathcal{H}^* </math> is defined by <math> \Phi(h) = \phi_h </math> where for all <math> g\in\mathcal{H} </math> we have
- <math> \phi_h(g) = \mbox{IP}(h,g) = (h,g) = \langle h,g \rangle = \langle h|g \rangle </math>,
Where
- <math> \mbox{IP}(\cdot,\cdot), (\cdot,\cdot),\langle \cdot,\cdot \rangle, \langle \cdot | \cdot \rangle </math>
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 <math> \phi_h </math> and <math> g </math> with <math> \langle h | </math> and <math>|g \rangle </math> respectively. This is because of literal symbolic substitutions. Let <math> \phi_h = H = \langle h| </math> and <math> g=G=|g\rangle </math>. This gives
- <math> \phi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|(
|g\rangle) </math>
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.
[edit] Further reading
- Feynman, Leighton and Sands (1965). The Feynman Lectures on Physics Vol. III. Addison-Wesley. ISBN 0-201-02115-3.
[edit] References
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