Mechanism (philosophy)

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https://en.wikipedia.org/wiki/Mechanism_(philosophy)

Mechanism is the belief that natural wholes (principally living things) are similar to complicated machines or artifacts, composed of parts lacking any intrinsic relationship to each other.

The doctrine of mechanism in philosophy comes in two different flavors. They are both doctrines of metaphysics, but they are different in scope and ambitions: the first is a global doctrine about nature; the second is a local doctrine about humans and their minds, which is hotly contested. For clarity, we might distinguish these two doctrines as universal mechanism and anthropic mechanism.

There is no constant meaning in the history of philosophy for the word Mechanism. Originally, the term meant that cosmological theory which ascribes the motion and changes of the world to some external force. In this view material things are purely passive, while according to the opposite theory (i. e., Dynamism), they possess certain internal sources of energy which account for the activity of each and for its influence on the course of events; These meanings, however, soon underwent modification. The question as to whether motion is an inherent property of bodies, or has been communicated to them by some external agency, was very often ignored. With many cosmologists the essential feature of Mechanism is the attempt to reduce all the qualities and activities of bodies to quantitative realities, i. e. to mass and motion. But a further modification soon followed. Living bodies, as is well known, present at first sight certain characteristic properties which have no counterpart in lifeless matter. Mechanism aims to go beyond these appearances. It seeks to explain all "vital" phenomena as physical and chemical facts; whether or not these facts are in turn reducible to mass and motion becomes a secondary question, although Mechanists are generally inclined to favour such reduction. The theory opposed to this biological mechanism is no longer Dynamism, but Vitalism or Neo-vitalism, which maintains that vital activities cannot be explained, and never will be explained, by the laws which govern lifeless matter.

— "Mechanism" in Catholic Encyclopedia (1913)

Mechanical philosophy

The mechanical philosophy is a form of natural philosophy which compares the universe to a large-scale mechanism (i.e. a machine). The mechanical philosophy is associated with the scientific revolution of early modern Europe. One of the first expositions of universal mechanism is found in the opening passages of Leviathan by Thomas Hobbes, published in 1651.

Some intellectual historians and critical theorists argue that early mechanical philosophy was tied to disenchantment and the rejection of the idea of nature as living or animated by spirits or angels. Other scholars, however, have noted that early mechanical philosophers nevertheless believed in magic, Christianity and spiritualism.

Mechanism and determinism

Some ancient philosophies held that the universe is reducible to completely mechanical principles—that is, the motion and collision of matter. This view was closely linked with materialism and reductionism, especially that of the atomists and to a large extent, stoic physics. Later mechanists believed the achievements of the scientific revolution of the 17th century had shown that all phenomena could eventually be explained in terms of "mechanical laws": natural laws governing the motion and collision of matter that imply a determinism. If all phenomena can be explained entirely through the motion of matter under physical laws, as the gears of a clock determine that it must strike 2:00 an hour after striking 1:00, all phenomena must be completely determined, past, present or future.

Development of the mechanical philosophy

Further information: corpuscularianism

The natural philosophers concerned with developing the mechanical philosophy were largely a French group, together with some of their personal connections. They included Pierre Gassendi, Marin Mersenne and René Descartes. Also involved were the English thinkers Sir Kenelm Digby, Thomas Hobbes and Walter Charleton; and the Dutch natural philosopher Isaac Beeckman.

Robert Boyle used "mechanical philosophers" to refer both to those with a theory of "corpuscles" or atoms of matter, such as Gassendi and Descartes, and those who did without such a theory. One common factor was the clockwork universe view. His meaning would be problematic in the cases of Hobbes and Galileo Galilei; it would include Nicolas Lemery and Christiaan Huygens, as well as himself. Newton would be a transitional figure. Contemporary usage of "mechanical philosophy" dates back to 1952 and Marie Boas Hall.

In France the mechanical philosophy spread mostly through private academies and salons; in England in the Royal Society. In England it did not have a large initial impact in universities, which were somewhat more receptive in France, the Netherlands and Germany.

Hobbes and the mechanical philosophy

One of the first expositions of universal mechanism is found in the opening passages of Leviathan (1651) by Hobbes; the book's second chapter invokes the principle of inertia, foundational for the mechanical philosophy. Boyle did not mention him as one of the group; but at the time they were on opposite sides of a controversy. Richard Westfall deems him a mechanical philosopher.

Hobbes's major statement of his natural philosophy is in De Corpore (1655). In part II and III of this work he goes a long way towards identifying fundamental physics with geometry; and he freely mixes concepts from the two areas.

Descartes and the mechanical philosophy

Descartes was also a mechanist. A substance dualist, he argued that reality is composed of two radically different types of substance: extended matter, on the one hand, and immaterial mind, on the other. He identified matter with the spatial extension which is its only clear and distinct idea, and consequently denied the existence of vacuum. Descartes argued that one cannot explain the conscious mind in terms of the spatial dynamics of mechanistic bits of matter cannoning off each other. Nevertheless, his understanding of biology was mechanistic in nature:

"I should like you to consider that these functions (including passion, memory, and imagination) follow from the mere arrangement of the machine’s organs every bit as naturally as the movements of a clock or other automaton follow from the arrangement of its counter-weights and wheels." (Descartes, Treatise on Man, p.108)

His scientific work was based on the traditional mechanistic understanding which maintains that animals and humans are completely mechanistic automata. Descartes' dualism was motivated by the seeming impossibility that mechanical dynamics could yield mental experiences.

Beeckman and the mechanical philosophy

Isaac Beeckman's theory of mechanical philosophy described in his books Centuria and Journal is grounded in two components: matter and motion. To explain matter, Beeckman relied on atomism philosophy which explains that matter is composed of tiny inseparable particles that interact to create the objects seen in life. To explain motion, he supported the idea of inertia, a theory generated by Isaac Newton.

Newton's mechanical philosophy

Isaac Newton ushered in a weaker notion of mechanism that tolerated the action at a distance of gravity. Interpretations of Newton's scientific work in light of his occult research have suggested that he did not properly view the universe as mechanistic, but instead populated by mysterious forces and spirits and constantly sustained by God and angels. Later generations of philosophers who were influenced by Newton's example were nonetheless often mechanists. Among them were Julien Offray de La Mettrie and Denis Diderot.

The mechanist thesis

The French mechanist and determinist Pierre Simon de Laplace formulated some implications of the mechanist thesis, writing:

We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.

— Pierre Simon Laplace, A Philosophical Essay on Probabilities

Criticism

A study of mechanical philosophy shows that although this approach includes a wide range of useful observational and principled data, it has not adequately explained the world and its components, and there are weaknesses in its definitions. Among the criticisms made of this philosophy are:

• Experts in religious studies have criticized the philosophy that God's intervention in the management of the world seems unnecessary.
• Newton's mechanical philosophy, with all its positive effects on human life, ultimately leads to Deism.
• It is a stagnant worldview that cannot explain God's constant presence and favor in the world.
• At the height of this philosophy, God was viewed as a skilled designer, and for him the mental structure and human morality were conceived.
• The assumption that God tuned the world like a clock and left it to its own devices is in clear conflict with the God of the Bible, who is at all times directly and immediately involved in his creation.
• This philosophy abandons concepts such as essence, accident, matter, form, Ipso facto and potential that are used in ontology, and denies the involvement of transcendental affairs in the management of this world.
• This philosophy is incapable of explaining human spiritual experiences and the immaterial realms of the world.

Universal mechanism

The older doctrine, here called universal mechanism, is the ancient philosophies closely linked with materialism and reductionism, especially that of the atomists and to a large extent, stoic physics. They held that the universe is reducible to completely mechanical principles—that is, the motion and collision of matter. Later mechanists believed the achievements of the scientific revolution had shown that all phenomena could eventually be explained in terms of 'mechanical' laws, natural laws governing the motion and collision of matter that implied a thorough going determinism: if all phenomena could be explained entirely through the motion of matter under the laws of classical physics, then even more surely than the gears of a clock determine that it must strike 2:00 an hour after striking 1:00, all phenomena must be completely determined: whether past, present or future.

The French mechanist and determinist Pierre Simon de Laplace formulated the sweeping implications of this thesis by saying:

We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.

— Pierre Simon Laplace, A Philosophical Essay on Probabilities

One of the first and most famous expositions of universal mechanism is found in the opening passages of Leviathan by Thomas Hobbes (1651). What is less frequently appreciated is that René Descartes was a staunch mechanist, though today, in the philosophy of mind, he is remembered for introducing the mind–body problem in terms of dualism and physicalism.

Descartes was a substance dualist, and argued that reality was composed of two radically different types of substance: extended matter, on the one hand, and immaterial mind, on the other. Descartes argued that one cannot explain the conscious mind in terms of the spatial dynamics of mechanistic bits of matter cannoning off each other. Nevertheless, his understanding of biology was thoroughly mechanistic in nature:

I should like you to consider that these functions (including passion, memory, and imagination) follow from the mere arrangement of the machine’s organs every bit as naturally as the movements of a clock or other automaton follow from the arrangement of its counter-weights and wheels.

— René Descartes, Treatise on Man, p.108

His scientific work was based on the traditional mechanistic understanding that animals and humans are completely mechanistic automata. Descartes' dualism was motivated by the seeming impossibility that mechanical dynamics could yield mental experiences.

Isaac Newton ushered in a much weaker acceptation of mechanism that tolerated the antithetical, and as yet inexplicable, action at a distance of gravity. However, his work seemed to successfully predict the motion of both celestial and terrestrial bodies according to that principle, and the generation of philosophers who were inspired by Newton's example carried the mechanist banner nonetheless. Chief among them were French philosophers such as Julien Offray de La Mettrie and Denis Diderot (see also: French materialism).

Anthropic mechanism

The thesis in anthropic mechanism is not that everything can be completely explained in mechanical terms (although some anthropic mechanists may also believe that), but rather that everything about human beings can be completely explained in mechanical terms, as surely as can everything about clocks or the internal combustion engine.

One of the chief obstacles that all mechanistic theories have faced is providing a mechanistic explanation of the human mind; Descartes, for one, endorsed dualism in spite of endorsing a completely mechanistic conception of the material world because he argued that mechanism and the notion of a mind be logically incompatible. Hobbes, on the other hand, conceived of the mind and the will as purely mechanistic, completely explicable in terms of the effects of perception and the pursuit of desire, which in turn he held to be completely explicable in terms of the materialistic operations of the nervous system. Following Hobbes, other mechanists argued for a thoroughly mechanistic explanation of the mind, with one of the most influential and controversial expositions of the doctrine being offered by Julien Offray de La Mettrie in his Man a Machine (1748).

The main points of debate between anthropic mechanists and anti-mechanists are mainly occupied with two topics: the mind—consciousness, in particular—and free will. Anti-mechanists argue that anthropic mechanism be incompatible with our commonsense intuitions: in philosophy of mind they argue that if matter is devoid of mental properties, then the phenomenon of consciousness cannot be explained by mechanistic principles acting on matter. In metaphysics anti-mechanists argue that anthropic mechanism implies determinism about human action, which is incompatible with our experience of free will. Contemporary philosophers who have argued for this position include Norman Malcolm and David Chalmers.

Anthropic mechanists typically respond in one of two ways. In the first, they agree with anti-mechanists that mechanism conflicts with some of our commonsense intuitions, but go on to argue that our commonsense intuitions are simply mistaken and need to be revised. Down this path lies eliminative materialism in philosophy of mind, and hard determinism on the question of free will. This option is accepted by the eliminative materialist philosopher Paul Churchland. Some have questioned how eliminative materialism is compatible with the freedom of will apparently required for anyone (including its adherents) to make truth claims. The second option, common amongst philosophers who adopt anthropic mechanism, is to argue that the arguments given for incompatibility are specious: whatever it is we mean by "consciousness" and "free will," be fully compatible with a mechanistic understanding of the human mind and will. As a result, they tend to argue for one or another non-eliminativist physicalist theories of mind, and for compatibilism on the question of free will. Contemporary philosophers who have argued for this sort of account include J. J. C. Smart and Daniel Dennett.

Gödelian arguments

Some scholars have debated over what, if anything, Gödel's incompleteness theorems imply about anthropic mechanism. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.

Gödelian arguments claim that a system of human mathematicians (or some idealization of human mathematicians) is both consistent and powerful enough to recognize its own consistency. Since this is impossible for a Turing machine, the Gödelian concludes that human reasoning must be non-mechanical.

However, the modern consensus in the scientific and mathematical community is that actual human reasoning is inconsistent: any consistent "idealized version" H of human reasoning would logically be forced to adopt a healthy but counter-intuitive open-minded skepticism about the consistency of H (otherwise H is provably inconsistent); and that Gödel's theorems do not lead to any valid argument against mechanism. This consensus that Gödelian anti-mechanist arguments are doomed to failure is laid out strongly in Artificial Intelligence: "any attempt to utilize [Gödel's incompleteness results] to attack the computationalist thesis is bound to be illegitimate, since these results are quite consistent with the computationalist thesis."

History

One of the earliest attempts to use incompleteness to reason about human intelligence was by Gödel himself in his 1951 Gibbs Lecture entitled "Some basic theorems on the foundations of mathematics and their philosophical implications". In this lecture, Gödel uses the incompleteness theorem to arrive at the following disjunction: (a) the human mind is not a consistent finite machine, or (b) there exist Diophantine equations for which it cannot decide whether solutions exist. Gödel finds (b) implausible, and thus seems to have believed the human mind was not equivalent to a finite machine, i.e., its power exceeded that of any finite machine. He recognized that this was only a conjecture, since one could never disprove (b). Yet he considered the disjunctive conclusion to be a "certain fact".

In subsequent years, more direct anti-mechanist lines of reasoning were apparently floating around the intellectual atmosphere. In 1960, Hilary Putnam published a paper entitled "Minds and Machines," in which he points out the flaws of a typical anti-mechanist argument. Informally, this is the argument that the (alleged) difference between "what can be mechanically proven" and "what can be seen to be true by humans" shows that human intelligence is not mechanical in nature. Or, as Putnam puts it:

Let T be a Turing machine which "represents" me in the sense that T can prove just the mathematical statements I prove. Then using Gödel's technique I can discover a proposition that T cannot prove, and moreover I can prove this proposition. This refutes the assumption that T "represents" me, hence I am not a Turing machine.

Hilary Putnam objects that this argument ignores the issue of consistency. Gödel's technique can only be applied to consistent systems. It is conceivable, argues Putnam, that the human mind is inconsistent. If one is to use Gödel's technique to prove the proposition that T cannot prove, one must first prove (the mathematical statement representing) the consistency of T, a daunting and perhaps impossible task. Later Putnam suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. If we are to believe that it is consistent, then either we cannot prove its consistency, or it cannot be represented by a Turing machine.

J. R. Lucas in Minds, Machines and Gödel (1961), and later in his book The Freedom of the Will (1970), lays out an anti-mechanist argument closely following the one described by Putnam, including reasons for why the human mind can be considered consistent. Lucas admits that, by Gödel's second theorem, a human mind cannot formally prove its own consistency, and even says (perhaps facetiously) that women and politicians are inconsistent. Nevertheless, he sets out arguments for why a male non-politician can be considered consistent.

Another work was done by Judson Webb in his 1968 paper "Metamathematics and the Philosophy of Mind". Webb claims that previous attempts have glossed over whether one truly can see that the Gödelian statement p pertaining to oneself, is true. Using a different formulation of Gödel's theorems, namely, that of Raymond Smullyan and Emil Post, Webb shows one can derive convincing arguments for oneself of both the truth and falsity of p. He furthermore argues that all arguments about the philosophical implications of Gödel's theorems are really arguments about whether the Church-Turing thesis is true.

Later, Roger Penrose entered the fray, providing somewhat novel anti-mechanist arguments in his books, The Emperor's New Mind (1989) [ENM] and Shadows of the Mind (1994) [SM]. These books have proved highly controversial. Martin Davis responded to ENM in his paper "Is Mathematical Insight Algorithmic?" (ps), where he argues that Penrose ignores the issue of consistency. Solomon Feferman gives a critical examination of SM in his paper "Penrose's Gödelian argument." The response of the scientific community to Penrose's arguments has been negative, with one group of scholars calling Penrose's repeated attempts to form a persuasive Gödelian argument "a kind of intellectual shell game, in which a precisely defined notion to which a mathematical result applies ... is switched for a vaguer notion".

A Gödel-based anti-mechanism argument can be found in Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid, though Hofstadter is widely viewed as a known skeptic of such arguments:

Looked at this way, Gödel's proof suggests – though by no means does it prove! – that there could be some high-level way of viewing the mind/brain, involving concepts which do not appear on lower levels, and that this level might have explanatory power that does not exist – not even in principle – on lower levels. It would mean that some facts could be explained on the high level quite easily, but not on lower levels at all. No matter how long and cumbersome a low-level statement were made, it would not explain the phenomena in question. It is analogous to the fact that, if you make derivation after derivation in Peano arithmetic, no matter how long and cumbersome you make them, you will never come up with one for G – despite the fact that on a higher level, you can see that the Gödel sentence is true.

What might such high-level concepts be? It has been proposed for eons, by various holistically or "soulistically" inclined scientists and humanists that consciousness is a phenomenon that escapes explanation in terms of brain components; so here is a candidate at least. There is also the ever-puzzling notion of free will. So perhaps these qualities could be "emergent" in the sense of requiring explanations which cannot be furnished by the physiology alone.

Bra–ket notation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation

Bra–ket notation, also called Dirac 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.

Bra-ket notation was created by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions. The name comes from the English word "Bracket".

Quantum mechanics

In quantum mechanics, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, ${\displaystyle ⟨}$ and ${\displaystyle ⟩}$, and a vertical bar ${\displaystyle |}$, to construct "bras" and "kets".

A ket is of the form ${\displaystyle |v⟩}$. Mathematically it denotes a vector, ${\displaystyle \mathit{v}}$, in an abstract (complex) vector space ${\displaystyle V}$, and physically it represents a state of some quantum system.

A bra is of the form ${\displaystyle ⟨f|}$. Mathematically it denotes a linear form ${\displaystyle f:V\to \mathbb{C}}$, i.e. a linear map that maps each vector in ${\displaystyle V}$ to a number in the complex plane ${\displaystyle \mathbb{C}}$. Letting the linear functional ${\displaystyle ⟨f|}$ act on a vector ${\displaystyle |v⟩}$ is written as ${\displaystyle ⟨f|v⟩\in \mathbb{C}}$.

Assume that on ${\displaystyle V}$ there exists an inner product ${\displaystyle (\cdot ,\cdot )}$ with antilinear first argument, which makes ${\displaystyle V}$ an inner product space. Then with this inner product each vector ${\displaystyle \mathit{\varphi }\equiv |\varphi ⟩}$ can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: ${\displaystyle (\mathit{\varphi },\cdot )\equiv ⟨\varphi |}$. The correspondence between these notations is then ${\displaystyle (\mathit{\varphi },\mathit{\psi })\equiv ⟨\varphi |\psi ⟩}$. The linear form ${\displaystyle ⟨\varphi |}$ is a covector to ${\displaystyle |\varphi ⟩}$, and the set of all covectors form a subspace of the dual vector space ${\displaystyle V^{\vee }}$, to the initial vector space ${\displaystyle V}$. The purpose of this linear form ${\displaystyle ⟨\varphi |}$ can now be understood in terms of making projections on the state ${\displaystyle \mathit{\varphi }}$, to find how linearly dependent two states are, etc.

For the vector space ${\displaystyle {\mathbb{C}}^n}$, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication. If ${\displaystyle {\mathbb{C}}^n}$ has the standard Hermitian inner product ${\displaystyle (\mathit{v},\mathit{w})=v^{†}w}$, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted ${\displaystyle †}$).

It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator ${\displaystyle {\hat{\sigma }}_z}$ on a two dimensional space ${\displaystyle \mathrm{\Delta }}$ of spinors, has eigenvalues $\pm \frac{1}{2}$ with eigenspinors ${\displaystyle {\mathit{\psi }}_{+},{\mathit{\psi }}_{-}\in \mathrm{\Delta }}$. In bra–ket notation, this is typically denoted as ${\displaystyle {\mathit{\psi }}_{+}=|+⟩}$, and ${\displaystyle {\mathit{\psi }}_{-}=|-⟩}$. 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 effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of ${\displaystyle [\varphi \mid \psi ]}$ for inner products nearly 100 years earlier.

Vector spaces

Vectors vs kets

In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer 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 spacetime. Such vectors are typically denoted with over arrows (${\displaystyle \overrightarrow{r}}$), boldface (${\displaystyle \mathbf{p}}$) or indices (${\displaystyle v^{\mu }}$).

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. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element ${\displaystyle \varphi }$ of an abstract complex vector space as a ket ${\displaystyle |\varphi ⟩}$, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-${\displaystyle \varphi }$" 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 ${\displaystyle |⟩}$ making clear that the label indicates a vector in vector space. In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example, |1⟩ + |2⟩ is not necessarily 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. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as ${\displaystyle \hat{x}}$, ${\displaystyle \hat{p}}$, ${\displaystyle {\hat{L}}_z}$, etc.

Notation

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

${\displaystyle \begin{array}{rl}|A⟩& =|B⟩+|C⟩\\ |C⟩& =(-1+2i)|D⟩\\ |D⟩& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}|x⟩\mathrm{d}x.\end{array}}$

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

Since the ket is an element of a vector space, a bra ${\displaystyle ⟨A|}$ 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 ${\displaystyle ⟨\varphi |}$ and a ket ${\displaystyle |\psi ⟩}$ (i.e. a functional and a vector), can be combined to an operator ${\displaystyle |\psi ⟩⟨\varphi |}$ of rank one with outer product

${\displaystyle |\psi ⟩⟨\varphi |:|\xi ⟩\mapsto |\psi ⟩⟨\varphi |\xi ⟩.}$

Inner product and bra–ket identification on Hilbert space

Main article: Inner product

The bra–ket notation is particularly useful in Hilbert spaces which have an inner product that allows Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem). The inner product on Hilbert space ${\displaystyle (,)}$ (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 ${\displaystyle \varphi =|\varphi ⟩}$ define a functional (i.e. bra) ${\displaystyle f_{\varphi }=⟨\varphi |}$ by

${\displaystyle (\varphi ,\psi )=(|\varphi ⟩,|\psi ⟩)=:f_{\varphi }(\psi )=⟨\varphi |(|\psi ⟩)=:⟨\varphi \mid \psi ⟩}$

Bras and kets as row and column vectors

In the simple case where we consider the vector space ${\displaystyle {\mathbb{C}}^n}$, a ket can be identified with a column vector, and a bra as a row vector. If moreover we use the standard Hermitian inner product on ${\displaystyle {\mathbb{C}}^n}$, 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 that writing bras, kets, and linear operators next to each other simply imply matrix multiplication.^[7] In particular the outer product ${\displaystyle |\psi ⟩⟨\varphi |}$ 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:

$${\displaystyle ⟨A|B⟩\doteq A_1^{\ast }{B}_1+A_2^{\ast }{B}_2+\cdots +A_N^{\ast }{B}_N=\left(\begin{array}{cccc}{A}_1^{\ast }& A_2^{\ast }& \cdots & A_N^{\ast }\end{array}\right)\left(\begin{array}{c}{B}_1\\ B_2\\ ⋮\\ B_N\end{array}\right)}$$

Based on this, the bras and kets can be defined as:

$${\displaystyle \begin{array}{rl}⟨A|& \doteq \left(\begin{array}{cccc}{A}_1^{\ast }& A_2^{\ast }& \cdots & A_N^{\ast }\end{array}\right)\\ |B⟩& \doteq \left(\begin{array}{c}{B}_1\\ B_2\\ ⋮\\ B_N\end{array}\right)\end{array}}$$

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:

$${\displaystyle ⟨A{|}^{†}=|A⟩,|A{⟩}^{†}=⟨A|}$$

because if one starts with the bra

$${\displaystyle \left(\begin{array}{cccc}{A}_1^{\ast }& A_2^{\ast }& \cdots & A_N^{\ast }\end{array}\right),}$$

then performs a complex conjugation, and then a matrix transpose, one ends up with the ket

$${\displaystyle \left(\begin{array}{c}{A}_1\\ A_2\\ ⋮\\ A_N\end{array}\right)}$$

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 spaces

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 mechanics

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/√2|1⟩ + i/√2|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 function

Discrete components A_k of a complex vector |A⟩ = Σ_k A_k |e_k⟩, which belongs to a countably infinite-dimensional Hilbert space; there are countably infinitely many k values and basis vectors |e_k⟩.

 

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, ${\displaystyle \hat{\mathbf{r}}|\mathbf{r}⟩=\mathbf{r}|\mathbf{r}⟩}$. 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,

$${\displaystyle \mathrm{\Psi }(\mathbf{r})\stackrel{\text{def}}{=}⟨\mathbf{r}|\mathrm{\Psi }⟩.}$$

On the left-hand side, Ψ(r) is a function mapping any point in space to a complex number; on the right-hand side,

$${\displaystyle |\mathrm{\Psi }⟩=\int d^3\mathbf{r}\mathrm{\Psi }(\mathbf{r})|\mathbf{r}⟩}$$

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

$${\displaystyle \hat{A}(\mathbf{r})\mathrm{\Psi }(\mathbf{r})\stackrel{\text{def}}{=}⟨\mathbf{r}|\hat{A}|\mathrm{\Psi }⟩.}$$

For instance, the momentum operator ${\displaystyle \hat{\mathbf{p}}}$ has the following coordinate representation:

$${\displaystyle \hat{\mathbf{p}}(\mathbf{r})\mathrm{\Psi }(\mathbf{r})\stackrel{\text{def}}{=}⟨\mathbf{r}|\hat{\mathbf{p}}|\mathrm{\Psi }⟩=-i\hslash \mathrm{\nabla }\mathrm{\Psi }(\mathbf{r}).}$$

One occasionally even encounters a expressions such as ${\displaystyle \mathrm{\nabla }|\mathrm{\Psi }⟩}$, 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, ${\displaystyle \mathrm{\nabla }⟨\mathbf{r}|\mathrm{\Psi }⟩,}$ even though, in the momentum basis, this operator amounts to a mere multiplication operator (by iħp). That is, to say,

$${\displaystyle ⟨\mathbf{r}|\hat{\mathbf{p}}=-i\hslash \mathrm{\nabla }⟨\mathbf{r}|,}$$

or

$${\displaystyle \hat{\mathbf{p}}=\int d^3\mathbf{r}|\mathbf{r}⟩(-i\hslash \mathrm{\nabla })⟨\mathbf{r}|.}$$

Overlap of states

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 particle

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

$${\displaystyle |{\uparrow }_z⟩,|{\downarrow }_z⟩}$$

where |↑_z⟩ is the state with a definite value of the spin operator S_z equal to +1⁄2 and |↓_z⟩ is the state with a definite value of the spin operator S_z 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:

$${\displaystyle |\psi ⟩=a_{\psi }|{\uparrow }_z⟩+b_{\psi }|{\downarrow }_z⟩}$$

where a_ψ and b_ψ are complex numbers.

A different basis for the same Hilbert space is:

$${\displaystyle |{\uparrow }_x⟩,|{\downarrow }_x⟩}$$

defined in terms of S_x rather than S_z.

Again, any state of the particle can be expressed as a linear combination of these two:

$${\displaystyle |\psi ⟩=c_{\psi }|{\uparrow }_x⟩+d_{\psi }|{\downarrow }_x⟩}$$

In vector form, you might write

$${\displaystyle |\psi ⟩\doteq \left(\begin{array}{c}{a}_{\psi }\\ b_{\psi }\end{array}\right)\text{or}|\psi ⟩\doteq \left(\begin{array}{c}{c}_{\psi }\\ d_{\psi }\end{array}\right)}$$

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 ${\displaystyle a_{\psi }}$, ${\displaystyle b_{\psi }}$, ${\displaystyle c_{\psi }}$ and ${\displaystyle d_{\psi }}$; see change of basis.

Pitfalls and ambiguous uses

There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.

Separation of inner product and vectors

A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as ${\displaystyle \mathit{\psi }}$, and ${\displaystyle (\cdot ,\cdot )}$ for the inner product. Consider the following dual space bra-vector in the basis ${\displaystyle \{|e_n⟩\}}$:

$${\displaystyle ⟨\psi |=\sum _n⟨e_n|{\psi }_n}$$

It has to be determined by convention if the complex numbers ${\displaystyle \{{\psi }_n\}}$ are inside or outside of the inner product, and each convention gives different results.

$${\displaystyle ⟨\psi |\equiv (\mathit{\psi },\cdot )=\sum _n({\mathit{e}}_n,\cdot ){\psi }_n}$$

$${\displaystyle ⟨\psi |\equiv (\mathit{\psi },\cdot )=\sum _n({\mathit{e}}_n{\psi }_n,\cdot )=\sum _n({\mathit{e}}_n,\cdot ){\psi }_n^{\ast }}$$

Reuse of symbols

It is common to use the same symbol for labels and constants. For example, ${\displaystyle \hat{\alpha }|\alpha ⟩=\alpha |\alpha ⟩}$, where the symbol ${\displaystyle \alpha }$ is used simultaneously as the name of the operator ${\displaystyle \hat{\alpha }}$, its eigenvector ${\displaystyle |\alpha ⟩}$ and the associated eigenvalue ${\displaystyle \alpha }$. Sometimes the hat is also dropped for operators, and one can see notation such as ${\displaystyle A|a⟩=a|a⟩}$.

Hermitian conjugate of kets

It is common to see the usage ${\displaystyle |\psi {⟩}^{†}=⟨\psi |}$, where the dagger (${\displaystyle †}$) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, ${\displaystyle |\psi ⟩}$, represents a vector in a complex Hilbert-space ${\displaystyle \mathcal{H}}$, and the bra, ${\displaystyle ⟨\psi |}$, is a linear functional on vectors in ${\displaystyle \mathcal{H}}$. In other words, ${\displaystyle |\psi ⟩}$ is just a vector, while ${\displaystyle ⟨\psi |}$ is the combination of a vector and an inner product.

Operations inside bras and kets

This is done for a fast notation of scaling vectors. For instance, if the vector ${\displaystyle |\alpha ⟩}$ is scaled by ${\displaystyle 1/\sqrt{2}}$, it may be denoted ${\displaystyle |\alpha /\sqrt{2}⟩}$. This can be ambiguous since ${\displaystyle \alpha }$ is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. ${\displaystyle |\alpha ⟩=|\alpha /\sqrt{2}{⟩}_1\otimes |\alpha /\sqrt{2}{⟩}_2}$.

Linear operators

See also: Linear operator

Linear operators acting on kets

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 ${\displaystyle \hat{A}}$ is a linear operator and ${\displaystyle |\psi ⟩}$ is a ket-vector, then ${\displaystyle \hat{A}|\psi ⟩}$ is another ket-vector.

In an ${\displaystyle N}$-dimensional Hilbert space, we can impose a basis on the space and represent ${\displaystyle |\psi ⟩}$ in terms of its coordinates as a ${\displaystyle N\times 1}$ column vector. Using the same basis for ${\displaystyle \hat{A}}$, it is represented by an ${\displaystyle N\times N}$ complex matrix. The ket-vector ${\displaystyle \hat{A}|\psi ⟩}$ can now be computed by 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 bras

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

$${\displaystyle (⟨\varphi |\mathit{A})|\psi ⟩=⟨\varphi |(\mathit{A}|\psi ⟩),}$$

(in other words, a function composition). This expression is commonly written as (cf. energy inner product)

$${\displaystyle ⟨\varphi |\mathit{A}|\psi ⟩.}$$

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,

$${\displaystyle ⟨\psi |\mathit{A}|\psi ⟩,}$$

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 products

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

$${\displaystyle |\varphi ⟩⟨\psi |}$$

denotes the rank-one operator with the rule

$${\displaystyle (|\varphi ⟩⟨\psi |)(x)=⟨\psi |x⟩|\varphi ⟩.}$$

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:

$${\displaystyle |\varphi ⟩⟨\psi |\doteq \left(\begin{array}{c}{\varphi }_1\\ {\varphi }_2\\ ⋮\\ {\varphi }_N\end{array}\right)\left(\begin{array}{cccc}{\psi }_1^{\ast }& {\psi }_2^{\ast }& \cdots & {\psi }_N^{\ast }\end{array}\right)=\left(\begin{array}{cccc}{\varphi }_1{\psi }_1^{\ast }& {\varphi }_1{\psi }_2^{\ast }& \cdots & {\varphi }_1{\psi }_N^{\ast }\\ {\varphi }_2{\psi }_1^{\ast }& {\varphi }_2{\psi }_2^{\ast }& \cdots & {\varphi }_2{\psi }_N^{\ast }\\ ⋮& ⋮& \ddots & ⋮\\ {\varphi }_N{\psi }_1^{\ast }& {\varphi }_N{\psi }_2^{\ast }& \cdots & {\varphi }_N{\psi }_N^{\ast }\end{array}\right)}$$

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

$${\displaystyle |\psi ⟩⟨\psi |.}$$

This is an idempotent in the algebra of observables that acts on the Hilbert space.

Hermitian conjugate operator

Main article: Hermitian conjugate

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,

$${\displaystyle |\varphi ⟩=A|\psi ⟩\text{if and only if}⟨\varphi |=⟨\psi |A^{†}.}$$

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.

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.

Linearity

• Since bras are linear functionals,
  $${\displaystyle ⟨\varphi |(c_1|{\psi }_1⟩+c_2|{\psi }_2⟩)=c_1⟨\varphi |{\psi }_1⟩+c_2⟨\varphi |{\psi }_2⟩.}$$

  
• By the definition of addition and scalar multiplication of linear functionals in the dual space,
  $${\displaystyle (c_1⟨{\varphi }_1|+c_2⟨{\varphi }_2|)|\psi ⟩=c_1⟨{\varphi }_1|\psi ⟩+c_2⟨{\varphi }_2|\psi ⟩.}$$

  

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.e., the associative property holds). For example:

${\displaystyle \begin{array}{rl}⟨\psi |(A|\varphi ⟩)=(⟨\psi |A)|\varphi ⟩& \stackrel{\text{def}}{=}⟨\psi |A|\varphi ⟩\\ (A|\psi ⟩)⟨\varphi |=A(|\psi ⟩⟨\varphi |)& \stackrel{\text{def}}{=}A|\psi ⟩⟨\varphi |\end{array}}$

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 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.,
  $${\displaystyle {\left(x^{†}\right)}^{†}=x.}$$

  
• 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:
  $${\displaystyle (c_1|{\psi }_1⟩+c_2|{\psi }_2⟩{)}^{†}=c_1^{\ast }⟨{\psi }_1|+c_2^{\ast }⟨{\psi }_2|.}$$

  
• Inner products:
  $${\displaystyle ⟨\varphi |\psi {⟩}^{\ast }=⟨\psi |\varphi ⟩.}$$

  
  Note that ⟨φ|ψ⟩ is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e.,
  $${\displaystyle (⟨\varphi |\psi ⟩{)}^{†}=⟨\varphi |\psi {⟩}^{\ast }}$$

  
• Matrix elements:
  $${\displaystyle \begin{array}{rl}⟨\varphi |A|\psi {⟩}^{†}& =⟨\psi \left|A^{†}\right|\varphi ⟩\\ {⟨\varphi \left|A^{†}{B}^{†}\right|\psi ⟩}^{†}& =⟨\psi |BA|\varphi ⟩.\end{array}}$$

  
• Outer products:
  $${\displaystyle ((c_1|{\varphi }_1⟩⟨{\psi }_1|)+(c_2|{\varphi }_2⟩⟨{\psi }_2|){)}^{†}=(c_1^{\ast }|{\psi }_1⟩⟨{\varphi }_1|)+(c_2^{\ast }|{\psi }_2⟩⟨{\varphi }_2|).}$$

  

Composite bras and kets

Two Hilbert spaces V and W may form a third space V ⊗ W 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 tensor product of the two kets is a ket in V ⊗ W. This is written in various notations:

${\displaystyle |\psi ⟩|\varphi ⟩,|\psi ⟩\otimes |\varphi ⟩,|\psi \varphi ⟩,|\psi ,\varphi ⟩.}$

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

The unit operator

Consider a complete orthonormal system (basis),

$${\displaystyle \{e_i|i\in \mathbb{N}\},}$$

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

From basic functional analysis, it is known that any ket ${\displaystyle |\psi ⟩}$ can also be written as

$${\displaystyle |\psi ⟩=\sum _{i\in \mathbb{N}}⟨e_i|\psi ⟩|e_i⟩,}$$

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

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

$${\displaystyle \sum _{i\in \mathbb{N}}|e_i⟩⟨e_i|=\mathbb{I}}$$

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

$${\displaystyle \begin{array}{rl}⟨v|w⟩& =⟨v|\left(\sum _{i\in \mathbb{N}}|e_i⟩⟨e_i|\right)|w⟩\\ & =⟨v|\left(\sum _{i\in \mathbb{N}}|e_i⟩⟨e_i|\right)\left(\sum _{j\in \mathbb{N}}|e_j⟩⟨e_j|\right)|w⟩\\ & =⟨v|e_i⟩⟨e_i|e_j⟩⟨e_j|w⟩,\end{array}}$$

where, in the last line, the Einstein summation convention has been used to avoid clutter.

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|e_i⟩ = ⟨e_i|ψ⟩* and ⟨e_i|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,

$${\displaystyle \mathbb{I}=\int dx|x⟩⟨x|=\int dp|p⟩⟨p|,}$$

where

$${\displaystyle |p⟩=\int dx\frac{e^{ixp/\hslash }|x⟩}{\sqrt{2\pi \hslash }}.}$$

Since ⟨x′|x⟩ = δ(x − x′), plane waves follow,

$${\displaystyle ⟨x|p⟩=\frac{e^{ixp/\hslash }}{\sqrt{2\pi \hslash }}.}$$

In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate $|\varpi ⟩=\underset{p\to 0}{lim}|p⟩$ in the momentum representation, i.e., ${\displaystyle \hat{p}|\varpi ⟩=0}$. Consequently, the corresponding wavefunction is a constant, ${\displaystyle ⟨x|\varpi ⟩\sqrt{2\pi \hslash }=1}$, and

$${\displaystyle |x⟩=\delta (\hat{x}-x)|\varpi ⟩\sqrt{2\pi \hslash },}$$

as well as

$${\displaystyle |p⟩=\exp (ip\hat{x}/\hslash )|\varpi ⟩.}$$

Typically, when all matrix elements of an operator such as

$${\displaystyle ⟨x|A|y⟩}$$

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

$${\displaystyle \int dxdy|x⟩⟨x|A|y⟩⟨y|=A.}$$

Notation used by mathematicians

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

Let ${\displaystyle (\mathcal{H},⟨\cdot ,\cdot ⟩)}$ be a Hilbert space and h ∈ H a vector in H. What physicists would denote by |h⟩ is the vector itself. That is,

$${\displaystyle |h⟩\in \mathcal{H}.}$$

Let H* be the dual space of H. This is the space of linear functionals on H. The embedding ${\displaystyle \mathrm{\Phi }:\mathcal{H}\hookrightarrow {\mathcal{H}}^{\ast }}$ is defined by ${\displaystyle \mathrm{\Phi }(h)={\phi }_h}$, where for every h ∈ H the linear functional ${\displaystyle {\phi }_h:\mathcal{H}\to \mathbb{C}}$ satisfies for every g ∈ H the functional equation ${\displaystyle {\phi }_h(g)=⟨h,g⟩=⟨h\mid g⟩}$. Notational confusion arises when identifying φ_h and g with ⟨h| and |g⟩ respectively. This is because of literal symbolic substitutions. Let ${\displaystyle {\phi }_h=H=⟨h\mid }$ and let g = G = |g⟩. This gives

$${\displaystyle {\phi }_h(g)=H(g)=H(G)=⟨h|(G)=⟨h|(|g⟩).}$$

One ignores the parentheses and removes the double bars.

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

$${\displaystyle ⟨\varphi ,\psi ⟩=\int \varphi (x)\cdot \overline{\psi (x)}\mathrm{d}x,}$$

whereas physicists would write for the same quantity

$${\displaystyle ⟨\psi |\varphi ⟩=\int dx{\psi }^{\ast }(x)\varphi (x).}$$