# Bra–ket notation

**Bra–ket notation**, also called **Dirac notation**, is a notation for linear algebra over complex vector spaces, introduced by [Paul Dirac](https://www.edgepedia.org/paul-dirac) to give a concise way of performing and describing the calculations of quantum mechanics. A state vector is written as a **ket** such as \( |\psi\rangle \), its dual as a **bra** such as \( \langle\psi| \), and their inner product as a **braket** \( \langle\varphi|\psi\rangle \), a wordplay on "bracket" that gives the notation its name.<sup>[1](https://www.theochem.ru.nl/~pwormer/Knowino/knowino.org/w/index393f.html?stable=1&title=Bra-ket_notation)</sup><sup> • </sup><sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)</sup>

| Key fact | Detail |
|---|---|
| Origin | Formulated by Paul Dirac, published in *The Principles of Quantum Mechanics* (Oxford University Press, 1930), for the linear algebra of matrix mechanics<sup>[1](https://www.theochem.ru.nl/~pwormer/Knowino/knowino.org/w/index393f.html?stable=1&title=Bra-ket_notation)</sup> |
| Ket | A vector \( |\psi\rangle \) in a complex Hilbert space \( \mathcal{H} \), the space of quantum states<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)</sup> |
| Bra | The dual vector \( \langle\psi| = (|\psi\rangle)^{\dagger} \), an element of the dual space \( \mathcal{H}^{*} \)<sup>[3](https://qm1.quantumtinkerer.tudelft.nl/9_formalism/)</sup> |
| Inner product | \( \langle\varphi|\psi\rangle \), conjugate-symmetric: \( \langle v|w\rangle = \langle w|v\rangle^{*} \)<sup>[4](https://www.cs.tufts.edu/~ablumer/QuantumLogic/DiracNotation.html)</sup> |
| Wavefunction | The position representation of a state: \( \psi(x) = \langle x|\psi\rangle \)<sup>[5](https://paradigms.oregonstate.edu/act/2665)</sup> |
| Born rule | Probability of measurement outcome \( a_n \) is \( p(a_n) = |\langle n|\psi\rangle|^{2} \)<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)</sup> |
| Completeness relation | \( \int |x\rangle\langle x|\, dx = \hat{I} \) for a continuous basis; \( \sum_n |\phi_n\rangle\langle\phi_n| = \hat{I} \) for a discrete one<sup>[6](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/33%3A_Appendices/33.02%3A_Bra-Ket_Notation)</sup> |

## Kets, bras, and the inner product

In quantum mechanics the state of a system is a vector in a complex [Hilbert space](https://www.edgepedia.org/hilbert-space) \( \mathcal{H} \), a vector space over the complex numbers with an inner product. Dirac notation writes such a vector as a ket \( |\psi\rangle \). The corresponding bra \( \langle\psi| \) is the [Hermitian adjoint](https://www.edgepedia.org/hermitian-adjoint) \( (|\psi\rangle)^{\dagger} \), an element of the dual space \( \mathcal{H}^{*} \) that maps kets to complex numbers. In a finite-dimensional basis a ket is a column vector of coefficients and its bra is the conjugate-transposed row vector.<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)</sup><sup> • </sup><sup>[3](https://qm1.quantumtinkerer.tudelft.nl/9_formalism/)</sup><sup> • </sup><sup>[7](https://books.physics.oregonstate.edu/LinAlg/mnote.html)</sup>

Placing a bra against a ket produces the inner product \( \langle\varphi|\psi\rangle \), a complex number that is conjugate-symmetric, \( \langle v|w\rangle = \langle w|v\rangle^{*} \). The norm of a state is the square root of \( \langle\psi|\psi\rangle \), which for a 3-component vector \( (a, b, c) \) equals \( \sqrt{|a|^{2} + |b|^{2} + |c|^{2}} \).<sup>[4](https://www.cs.tufts.edu/~ablumer/QuantumLogic/DiracNotation.html)</sup><sup> • </sup><sup>[7](https://books.physics.oregonstate.edu/LinAlg/mnote.html)</sup>

**States are rays, not kets.** Multiplying a ket by a nonzero complex number changes only its normalization and overall phase, which have no physical significance, so a pure physical state corresponds to a one-dimensional subspace (a ray) rather than to any particular ket in it.<sup>[8](https://bohr.physics.berkeley.edu/classes/221/notes/hilbert.pdf)</sup>

## Operators and observables

Physical properties such as position, momentum, energy, and angular momentum are represented by linear Hermitian operators acting on kets from the left, \( \hat{A}|\psi\rangle \). A Hermitian operator equals its own adjoint and has an orthonormal basis of eigenvectors with real eigenvalues, which is why the possible measured values of an observable are real.<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)</sup><sup> • </sup><sup>[4](https://www.cs.tufts.edu/~ablumer/QuantumLogic/DiracNotation.html)</sup>

The expression \( \langle\varphi|\hat{A}|\psi\rangle \) is called a matrix element: the operator acts on the ket, and the result is then projected onto the bra. The diagonal element with \( \varphi = \psi \) is the expectation value,

\[ \langle \hat{A} \rangle = \langle\psi|\hat{A}|\psi\rangle, \]

which in the position representation becomes the familiar integral \( \int \psi^{*}(\vec{r})\, \hat{A}\, \psi(\vec{r})\, d\vec{r} \).<sup>[6](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/33%3A_Appendices/33.02%3A_Bra-Ket_Notation)</sup><sup> • </sup><sup>[9](https://web.gps.caltech.edu/~gab/ch21b/docs/Ch21b_Lecture01_2018.pdf)</sup>

## Completeness relations and changing representation

The practical power of the notation comes from the completeness relation, the statement that a complete orthonormal basis sums to the identity operator. For a discrete basis \( \{|\phi_n\rangle\} \) this is \( \sum_n |\phi_n\rangle\langle\phi_n| = \hat{I} \); for the continuous position basis it is \( \int |x\rangle\langle x|\, dx = \hat{I} \). Inserting such a relation into any braket lets a calculation be moved from one basis to another; applying it in this way is called a projection onto a basis.<sup>[6](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/33%3A_Appendices/33.02%3A_Bra-Ket_Notation)</sup><sup> • </sup><sup>[10](https://paradigms-new.physics.oregonstate.edu/courses/ph425/Winter-2026/5487.html)</sup>

**The wavefunction is a representation.** Inserting the position completeness relation into a ket gives

\[ |\psi\rangle = \int_{\text{all space}} |x\rangle \langle x|\psi\rangle\, dx = \int_{\text{all space}} \psi(x)\, |x\rangle\, dx, \]

so the wavefunction is the collection of coefficients \( \psi(x) = \langle x|\psi\rangle \) of the state in the position basis. The bra corresponds to the [complex conjugate](https://www.edgepedia.org/complex-conjugate), \( \langle\psi| \doteq \psi^{*}(x) \). The same state can equally be written in the momentum basis, where \( \phi(p) = \langle p|\psi\rangle \), or as a column of expansion coefficients in an energy basis; the ket itself is independent of the choice.<sup>[5](https://paradigms.oregonstate.edu/act/2665)</sup><sup> • </sup><sup>[3](https://qm1.quantumtinkerer.tudelft.nl/9_formalism/)</sup><sup> • </sup><sup>[8](https://bohr.physics.berkeley.edu/classes/221/notes/hilbert.pdf)</sup>

Because the inner product is a scalar, its value does not depend on the representation used to compute it. In the position representation it is the integral \( \langle a|b\rangle = \int \psi_a^{*}(x)\, \psi_b(x)\, dx \); in a discrete eigenbasis it is the row-times-column product of the coefficient vectors.<sup>[3](https://qm1.quantumtinkerer.tudelft.nl/9_formalism/)</sup>

## Probabilities and normalization

The [Born rule](https://www.edgepedia.org/born-rule) connects the notation to measurement: if an observable has eigenvalue \( a_n \) with eigenvector \( |n\rangle \), the probability of obtaining \( a_n \) when measuring a system in state \( |\psi\rangle \) is

\[ p(a_n) = |\langle n|\psi\rangle|^{2}. \]

For a continuous observable such as position, \( |\psi(x)|^{2} \) is a probability density, and the probability of finding the particle between \( a \) and \( b \) is \( \int_a^b |\psi(x)|^{2}\, dx \). Normalization of the state reads \( \langle\psi|\psi\rangle = 1 \), which in wavefunction notation is \( \int |\psi(x)|^{2}\, dx = 1 \).<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)</sup><sup> • </sup><sup>[5](https://paradigms.oregonstate.edu/act/2665)</sup>

The probability amplitude for an energy measurement illustrates the translation between forms. In bra-ket notation it is simply \( c_n = \langle E_n|\psi\rangle \); inserting the position completeness relation turns it into the overlap integral \( c_n = \int E_n^{*}(x)\, \psi(x)\, dx \), with the modulus squared taken outside the integral.<sup>[5](https://paradigms.oregonstate.edu/act/2665)</sup>

## Why the notation is used

The Schrödinger wavefunction \( \psi(x) \), the momentum wavefunction \( \phi(p) \), and the sequence of energy coefficients \( (c_1, c_2, \ldots) \) are all equivalent descriptions of the same state, related by invertible, norm-preserving transformations. Since none of these representations is more fundamental than the others, a notation that does not presuppose any one of them is preferable; the ket \( |\psi\rangle \) names the abstract state while any concrete wavefunction is obtained by projecting onto a chosen basis. This representation-independent viewpoint, and the transformation theory connected with changes of representation, was worked out in the early days of quantum theory by Dirac, Born, Jordan, von Neumann, and others.<sup>[8](https://bohr.physics.berkeley.edu/classes/221/notes/hilbert.pdf)</sup>

The notation also compresses bookkeeping. Writing out integrals repeatedly is cumbersome, and the bra-ket form \( \langle\psi|\hat{A}|\psi\rangle \) replaces each such integral with a compact expression whose structure, state, operator, and projection, is visible at a glance. The same notation extends naturally to quantum computing, where kets label the basis states of qubit registers.<sup>[9](https://web.gps.caltech.edu/~gab/ch21b/docs/Ch21b_Lecture01_2018.pdf)</sup><sup> • </sup><sup>[11](https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes2.pdf)</sup>

## Teaching and rigor

Dirac notation is a standard part of the undergraduate quantum curriculum. MIT's graduate-level course 8.05 devotes a full lecture, taught by Barton Zwiebach, professor of physics at MIT, to building the notation from inner products through projection operators, the adjoint of a linear operator, and Hermitian and unitary operators.<sup>[12](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/lecture-9-diracs-bra-and-ket-notation/)</sup> Mathematically, the notation is informal in places: position kets \( |x\rangle \) are generalized eigenvectors that are not normalizable and do not belong to the Hilbert space proper, and a fully rigorous treatment places the bra-ket formalism in a rigged Hilbert space rather than the Hilbert space alone.<sup>[13](https://arxiv.org/abs/quant-ph/0502053)</sup>

## References

1. [Bra-ket notation, Knowino](https://www.theochem.ru.nl/~pwormer/Knowino/knowino.org/w/index393f.html?stable=1&title=Bra-ket_notation)
2. [Lecture notes, Chapter 2, 22.02 Introduction to Applied Nuclear Physics, MIT OpenCourseWare](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)
3. [The Formalism of Quantum Mechanics, TN2304 Kwantummechanica 1, TU Delft](https://qm1.quantumtinkerer.tudelft.nl/9_formalism/)
4. [Hilbert spaces and Dirac bra-ket notation, Tufts University](https://www.cs.tufts.edu/~ablumer/QuantumLogic/DiracNotation.html)
5. [Compare & Contrast Kets & Wavefunctions, Paradigms in Physics, Oregon State University](https://paradigms.oregonstate.edu/act/2665)
6. [33.2: Bra-Ket Notation, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/33%3A_Appendices/33.02%3A_Bra-Ket_Notation)
7. [Bra-Ket Notation, Paradigms in Physics linear algebra notes, Oregon State University](https://books.physics.oregonstate.edu/LinAlg/mnote.html)
8. [Notes 1: Mathematical Formalism, Physics 221, UC Berkeley](https://bohr.physics.berkeley.edu/classes/221/notes/hilbert.pdf)
9. [Ch21b Lecture 1, Caltech](https://web.gps.caltech.edu/~gab/ch21b/docs/Ch21b_Lecture01_2018.pdf)
10. [Quantum Fundamentals: Matrix Elements and Completeness Relations, Oregon State University](https://paradigms-new.physics.oregonstate.edu/courses/ph425/Winter-2026/5487.html)
11. [Dirac Notation and Basic Linear Algebra for Quantum Computing, CSE 599d, University of Washington](https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes2.pdf)
12. [Lecture 9: Dirac's Bra and Ket Notation, 8.05 Quantum Physics II, MIT OpenCourseWare](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/lecture-9-diracs-bra-and-ket-notation/)
13. [The role of the rigged Hilbert space in Quantum Mechanics, arXiv](https://arxiv.org/abs/quant-ph/0502053)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › State vectors and Hilbert-space states › Dirac (bra–ket) notation*

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