Entropy in thermodynamics and information theory

From Wikipedia, the free encyclopedia

There are close parallels between the mathematical expressions for the thermodynamic entropy, usually denoted by S, of a physical system in the statistical thermodynamics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, and the information-theoretic entropy, usually expressed as H, of Claude Shannon and Ralph Hartley developed in the 1940s. Shannon commented on the similarity upon publicizing information theory in A Mathematical Theory of Communication.

This article explores what links there are between the two concepts, and how far they can be regarded as connected.

Equivalence of form of the defining expressions

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

The defining expression for entropy in the theory of statistical mechanics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, is of the form:

${\displaystyle S=-k_{\text{B}}\sum _{i}p_{i}\ln p_{i},}$

where ${\displaystyle p_{i}}$ is the probability of the microstate i taken from an equilibrium ensemble.

The defining expression for entropy in the theory of information established by Claude E. Shannon in 1948 is of the form:

${\displaystyle H=-\sum _{i}p_{i}\log _{b}p_{i},}$

where ${\displaystyle p_{i}}$ is the probability of the message ${\displaystyle m_{i}}$ taken from the message space M, and b is the base of the logarithm used. Common values of b are 2, Euler's number e, and 10, and the unit of entropy is shannon (or bit) for b = 2, nat for b = e, and hartley for b = 10.^[1]

Mathematically H may also be seen as an average information, taken over the message space, because when a certain message occurs with probability p_i, the information quantity −log(p_i) will be obtained.

If all the microstates are equiprobable (a microcanonical ensemble), the statistical thermodynamic entropy reduces to the form, as given by Boltzmann,

${\displaystyle S=k_{\text{B}}\ln W,}$

where W is the number of microstates that corresponds to the macroscopic thermodynamic state. Therefore S depends on temperature.

If all the messages are equiprobable, the information entropy reduces to the Hartley entropy

${\displaystyle H=\log _{b}|M|\ ,}$

where ${\displaystyle |M|}$ is the cardinality of the message space M.

The logarithm in the thermodynamic definition is the natural logarithm. It can be shown that the Gibbs entropy formula, with the natural logarithm, reproduces all of the properties of the macroscopic classical thermodynamics of Rudolf Clausius. (See article: Entropy (statistical views)).

The logarithm can also be taken to the natural base in the case of information entropy. This is equivalent to choosing to measure information in nats instead of the usual bits (or more formally, shannons). In practice, information entropy is almost always calculated using base 2 logarithms, but this distinction amounts to nothing other than a change in units. One nat is about 1.44 bits.

For a simple compressible system that can only perform volume work, the first law of thermodynamics becomes

${\displaystyle dE=-pdV+TdS.}$

But one can equally well write this equation in terms of what physicists and chemists sometimes call the 'reduced' or dimensionless entropy, σ = S/k, so that

${\displaystyle dE=-pdV+k_{\text{B}}Td\sigma .}$

Just as S is conjugate to T, so σ is conjugate to k_BT (the energy that is characteristic of T on a molecular scale).

Theoretical relationship

Despite the foregoing, there is a difference between the two quantities. The information entropy H can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p_i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p_i specifically. The difference is more theoretical than actual, however, because any probability distribution can be approximated arbitrarily closely by some thermodynamic system.^{[citation needed]}

Moreover, a direct connection can be made between the two. If the probabilities in question are the thermodynamic probabilities p_i: the (reduced) Gibbs entropy σ can then be seen as simply the amount of Shannon information needed to define the detailed microscopic state of the system, given its macroscopic description. Or, in the words of G. N. Lewis writing about chemical entropy in 1930, "Gain in entropy always means loss of information, and nothing more". To be more concrete, in the discrete case using base two logarithms, the reduced Gibbs entropy is equal to the minimum number of yes–no questions needed to be answered in order to fully specify the microstate, given that we know the macrostate.

Furthermore, the prescription to find the equilibrium distributions of statistical mechanics—such as the Boltzmann distribution—by maximising the Gibbs entropy subject to appropriate constraints (the Gibbs algorithm) can be seen as something not unique to thermodynamics, but as a principle of general relevance in statistical inference, if it is desired to find a maximally uninformative probability distribution, subject to certain constraints on its averages. (These perspectives are explored further in the article Maximum entropy thermodynamics.)

The Shannon entropy in information theory is sometimes expressed in units of bits per symbol. The physical entropy may be on a "per quantity" basis (h) which is called "intensive" entropy instead of the usual total entropy which is called "extensive" entropy. The "shannons" of a message (H) are its total "extensive" information entropy and is h times the number of bits in the message.

A direct and physically real relationship between h and S can be found by assigning a symbol to each microstate that occurs per mole, kilogram, volume, or particle of a homogeneous substance, then calculating the 'h' of these symbols. By theory or by observation, the symbols (microstates) will occur with different probabilities and this will determine h. If there are N moles, kilograms, volumes, or particles of the unit substance, the relationship between h (in bits per unit substance) and physical extensive entropy in nats is:

${\displaystyle S=k_{\mathrm {B} }\ln(2)Nh}$

where ln(2) is the conversion factor from base 2 of Shannon entropy to the natural base e of physical entropy. N h is the amount of information in bits needed to describe the state of a physical system with entropy S. Landauer's principle demonstrates the reality of this by stating the minimum energy E required (and therefore heat Q generated) by an ideally efficient memory change or logic operation by irreversibly erasing or merging N h bits of information will be S times the temperature which is

${\displaystyle E=Q=Tk_{\mathrm {B} }\ln(2)Nh}$

where h is in informational bits and E and Q are in physical Joules. This has been experimentally confirmed.^[2]

Temperature is a measure of the average kinetic energy per particle in an ideal gas (Kelvins = 2/3*Joules/k_b) so the J/K units of k_b is fundamentally unitless (Joules/Joules). k_b is the conversion factor from energy in 3/2*Kelvins to Joules for an ideal gas. If kinetic energy measurements per particle of an ideal gas were expressed as Joules instead of Kelvins, k_b in the above equations would be replaced by 3/2. This shows that S is a true statistical measure of microstates that does not have a fundamental physical unit other than the units of information, in this case "nats", which is just a statement of which logarithm base was chosen by convention.

Information is physical

Szilard's engine

N-atom engine schematic

A physical thought experiment demonstrating how just the possession of information might in principle have thermodynamic consequences was established in 1929 by Leó Szilárd, in a refinement of the famous Maxwell's demon scenario.

Consider Maxwell's set-up, but with only a single gas particle in a box. If the supernatural demon knows which half of the box the particle is in (equivalent to a single bit of information), it can close a shutter between the two halves of the box, close a piston unopposed into the empty half of the box, and then extract ${\displaystyle k_{B}T\ln 2}$ joules of useful work if the shutter is opened again. The particle can then be left to isothermally expand back to its original equilibrium occupied volume. In just the right circumstances therefore, the possession of a single bit of Shannon information (a single bit of negentropy in Brillouin's term) really does correspond to a reduction in the entropy of the physical system. The global entropy is not decreased, but information to energy conversion is possible.

Using a phase-contrast microscope equipped with a high speed camera connected to a computer, as demon, the principle has been actually demonstrated.^[3] In this experiment, information to energy conversion is performed on a Brownian particle by means of feedback control; that is, synchronizing the work given to the particle with the information obtained on its position. Computing energy balances for different feedback protocols, has confirmed that the Jarzynski equality requires a generalization that accounts for the amount of information involved in the feedback.

Landauer's principle

In fact one can generalise: any information that has a physical representation must somehow be embedded in the statistical mechanical degrees of freedom of a physical system.

Thus, Rolf Landauer argued in 1961, if one were to imagine starting with those degrees of freedom in a thermalised state, there would be a real reduction in thermodynamic entropy if they were then re-set to a known state. This can only be achieved under information-preserving microscopically deterministic dynamics if the uncertainty is somehow dumped somewhere else – i.e. if the entropy of the environment (or the non information-bearing degrees of freedom) is increased by at least an equivalent amount, as required by the Second Law, by gaining an appropriate quantity of heat: specifically kT ln 2 of heat for every 1 bit of randomness erased.

On the other hand, Landauer argued, there is no thermodynamic objection to a logically reversible operation potentially being achieved in a physically reversible way in the system. It is only logically irreversible operations – for example, the erasing of a bit to a known state, or the merging of two computation paths – which must be accompanied by a corresponding entropy increase. When information is physical, all processing of its representations, i.e. generation, encoding, transmission, decoding and interpretation, are natural processes where entropy increases by consumption of free energy.^[4]

Applied to the Maxwell's demon/Szilard engine scenario, this suggests that it might be possible to "read" the state of the particle into a computing apparatus with no entropy cost; but only if the apparatus has already been SET into a known state, rather than being in a thermalised state of uncertainty. To SET (or RESET) the apparatus into this state will cost all the entropy that can be saved by knowing the state of Szilard's particle.

Negentropy

Shannon entropy has been related by physicist Léon Brillouin to a concept sometimes called negentropy. In 1953, Brillouin derived a general equation^[5] stating that the changing of an information bit value requires at least kT ln(2) energy. This is the same energy as the work Leo Szilard's engine produces in the idealistic case, which in turn equals to the same quantity found by Landauer. In his book,^[6] he further explored this problem concluding that any cause of a bit value change (measurement, decision about a yes/no question, erasure, display, etc.) will require the same amount, kT ln(2), of energy. Consequently, acquiring information about a system’s microstates is associated with an entropy production, while erasure yields entropy production only when the bit value is changing. Setting up a bit of information in a sub-system originally in thermal equilibrium results in a local entropy reduction. However, there is no violation of the second law of thermodynamics, according to Brillouin, since a reduction in any local system’s thermodynamic entropy results in an increase in thermodynamic entropy elsewhere. In this way, Brillouin clarified the meaning of negentropy which was considered as controversial because its earlier understanding can yield Carnot efficiency higher than one. Additionally, the relationship between energy and information formulated by Brillouin has been proposed as a connection between the amount of bits that the brain processes and the energy it consumes.^[7]

In 2009, Mahulikar & Herwig redefined thermodynamic negentropy as the specific entropy deficit of the dynamically ordered sub-system relative to its surroundings.^[8] This definition enabled the formulation of the Negentropy Principle, which is mathematically shown to follow from the 2nd Law of Thermodynamics, during order existence.

Black holes

Stephen Hawking often speaks of the thermodynamic entropy of black holes in terms of their information content.^[9] Do black holes destroy information? It appears that there are deep relations between the entropy of a black hole and information loss^[10] See Black hole thermodynamics and Black hole information paradox.

Quantum theory

Hirschman showed,^[11] cf. Hirschman uncertainty, that Heisenberg's uncertainty principle can be expressed as a particular lower bound on the sum of the classical distribution entropies of the quantum observable probability distributions of a quantum mechanical state, the square of the wave-function, in coordinate, and also momentum space, when expressed in Planck units. The resulting inequalities provide a tighter bound on the uncertainty relations of Heisenberg.

It is meaningful to assign a "joint entropy", because positions and momenta are quantum conjugate variables and are therefore not jointly observable. Mathematically, they have to be treated as joint distribution. Note that this joint entropy is not equivalent to the Von Neumann entropy, −Tr ρ lnρ = −⟨lnρ⟩. Hirschman's entropy is said to account for the full information content of a mixture of quantum states.^[12]

(Dissatisfaction with the Von Neumann entropy from quantum information points of view has been expressed by Stotland, Pomeransky, Bachmat and Cohen, who have introduced a yet different definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This definition allows distinction between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.^[13])

The fluctuation theorem

The fluctuation theorem provides a mathematical justification of the second law of thermodynamics under these principles, and precisely defines the limitations of the applicability of that law for systems away from thermodynamic equilibrium.

Criticism

There exists criticisms of the link between thermodynamic entropy and information entropy.

The most common criticism is that information entropy cannot be related to thermodynamic entropy because there is no concept of temperature, energy, or the second law, in the discipline of information entropy.^{[14][15][16][17][18]} This can best be discussed by considering the fundamental equation of thermodynamics:

${\displaystyle dU=\sum F_{i}\,dx_{i}}$

where the F_i are "generalized forces" and the dx_i are "generalized displacements". This is analogous to the mechanical equation dE = F dx where dE is the change in the kinetic energy of an object having been displaced by distance dx under the influence of force F. For example, for a simple gas, we have:

${\displaystyle dU=TdS-PdV+\mu dN}$

where the temperature (T ), pressure (P ), and chemical potential (µ ) are generalized forces which, when imbalanced, result in a generalized displacement in entropy (S ), volume (-V ) and quantity (N ) respectively, and the products of the forces and displacements yield the change in the internal energy (dU ) of the gas.

In the mechanical example, to declare that dx is not a geometric displacement because it ignores the dynamic relationship between displacement, force, and energy is not correct. Displacement, as a concept in geometry, does not require the concepts of energy and force for its definition, and so one might expect that entropy may not require the concepts of energy and temperature for its definition. The situation is not that simple, however. In classical thermodynamics, which is the study of thermodynamics from a purely empirical, or measurement point of view, thermodynamic entropy can only be measured by considering energy and temperature. Clausius' statement dS= δQ/T, or, equivalently, when all other effective displacements are zero, dS=dU/T, is the only way to actually measure thermodynamic entropy. It is only with the introduction of statistical mechanics, the viewpoint that a thermodynamic system consists of a collection of particles and which explains classical thermodynamics in terms of probability distributions, that the entropy can be considered separately from temperature and energy. This is expressed in Boltzmann's famous entropy formula S=k_B ln(W). Here k_B is Boltzmann's constant, and W is the number of equally probable microstates which yield a particular thermodynamic state, or macrostate.

Boltzmann's equation is presumed to provide a link between thermodynamic entropy S and information entropy H = −Σi pi ln pi = ln(W) where p_i=1/W are the equal probabilities of a given microstate. This interpretation has been criticized also. While some say that the equation is merely a unit conversion equation between thermodynamic and information entropy, this is not completely correct.^[19] A unit conversion equation will, e.g., change inches to centimeters, and yield two measurements in different units of the same physical quantity (length). Since thermodynamic and information entropy are dimensionally unequal (energy/unit temperature vs. units of information), Boltzmann's equation is more akin to x = c t where x is the distance travelled by a light beam in time t, c being the speed of light. While we cannot say that length x and time t represent the same physical quantity, we can say that, in the case of a light beam, since c is a universal constant, they will provide perfectly accurate measures of each other. (For example, the light-year is used as a measure of distance). Likewise, in the case of Boltzmann's equation, while we cannot say that thermodynamic entropy S and information entropy H represent the same physical quantity, we can say that, in the case of a thermodynamic system, since k_B is a universal constant, they will provide perfectly accurate measures of each other.

The question then remains whether ln(W) is an information-theoretic quantity. If it is measured in bits, one can say that, given the macrostate, it represents the number of yes/no questions one must ask to determine the microstate, clearly an information-theoretic concept. Objectors point out that such a process is purely conceptual, and has nothing to do with the measurement of entropy. Then again, the whole of statistical mechanics is purely conceptual, serving only to provide an explanation of the "pure" science of thermodynamics.

Ultimately, the criticism of the link between thermodynamic entropy and information entropy is a matter of terminology, rather than substance. Neither side in the controversy will disagree on the solution to a particular thermodynamic or information-theoretic problem.

Topics of recent research

Is information quantized?

In 1995, Tim Palmer signalled^{[citation needed]} two unwritten assumptions about Shannon's definition of information that may make it inapplicable as such to quantum mechanics:

• The supposition that there is such a thing as an observable state (for instance the upper face of a dice or a coin) before the observation begins
• The fact that knowing this state does not depend on the order in which observations are made (commutativity)

Anton Zeilinger's and Caslav Brukner's article^[20] synthesized and developed these remarks. The so-called Zeilinger's principle suggests that the quantization observed in QM could be bound to information quantization (one cannot observe less than one bit, and what is not observed is by definition "random"). Nevertheless, these claims remain quite controversial. Detailed discussions of the applicability of the Shannon information in quantum mechanics and an argument that Zeilinger's principle cannot explain quantization have been published,^[21][22][23] that show that Brukner and Zeilinger change, in the middle of the calculation in their article, the numerical values of the probabilities needed to compute the Shannon entropy, so that the calculation makes little sense.

Extracting work from quantum information in a Szilárd engine

In 2013, a description was published^[24] of a two atom version of a Szilárd engine using Quantum discord to generate work from purely quantum information.^[25] Refinements in the lower temperature limit were suggested.^[26]

Algorithmic cooling

Algorithmic cooling is an algorithmic method for transferring heat (or entropy) from some qubits to others or outside the system and into the environment, thus resulting in a cooling effect. This cooling effect may have usages in initializing cold (highly pure) qubits for quantum computation and in increasing polarization of certain spins in nuclear magnetic resonance.

Planck length

From Wikipedia, the free encyclopedia

Planck length
Unit system	Planck units
Unit of	length
Symbol	ℓP
Unit conversions
1 ℓP in ...	... is equal to ...
SI units	1.616229(38)×10−35 m
natural units	11.706 ℓS  3.0542×10−25 a0
imperial/US units	6.3631×10−34 in

In physics, the Planck length, denoted ℓ_P, is a unit of length, equal to 1.616229(38)×10⁻³⁵ metres. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant.

Value

The Planck length ℓ_P is defined as:

${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}$

Solving the above will show the approximate equivalent value of this unit with respect to the meter:
${\displaystyle 1\ \ell _{\mathrm {P} }\approx 1.616\;229(38)\times 10^{-35}\ \mathrm {m} }$, where ${\displaystyle c}$ is the speed of light in a vacuum, G is the gravitational constant, and ħ is the reduced Planck constant. The two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value.^[1][2]

The Planck length is about 10⁻²⁰ times the diameter of a proton. It can be defined using the radius of Planck particle.

Measuring the Planck length

In 2017 it was suggested by E. Haug^[3] that the Planck length can be indirectly measured independent of any knowledge of Newton's gravitational constant with for example the use of a Cavendish apparatus. Further, it seems like the error in the Planck length measures must be exactly half of that in the measurement errors of the Newton's gravitational constant. That is the error as measured in percentage term, also known as the relative standard uncertainty. This is in line with the relative standard uncertainty reported by NIST, which for the gravitational constant is ${\displaystyle 4.7\times 10^{-5}}$ and for the Planck length is ${\displaystyle 2.3\times 10^{-5}}$.

History

In 1899 Max Planck^[4] suggested that there existed some fundamental natural units for length, mass, time and energy. These he derived using dimensional analysis only using the Newton gravitational constant, the speed of light and the Planck constant. The natural units he derived has later been known as: the Planck length, the Planck mass, the Planck time and the Planck energy.

Theoretical significance

The Planck length is the scale at which quantum gravitational effects are believed to begin to be apparent, where interactions require a working theory of quantum gravity to be analyzed.^[5] The Planck area is the area by which the surface of a spherical black hole increases when the black hole swallows one bit of information.^[6]

The Planck length is sometimes misconceived as the minimum length of spacetime, but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry.^[5] However, certain theories of loop quantum gravity do attempt to establish a minimum length on the scale of the Planck length, though not necessarily the Planck length itself^[5], or attempt to establish the Planck length as observer-invariant, known as doubly special relativity.^{[citation needed]}

The strings of string theory are modelled to be on the order of the Planck length.^[5][7] In theories of large extra dimensions, the Planck length has no fundamental physical significance, and quantum gravitational effects appear at other scales.^{[citation needed]}

Planck length and Euclidean geometry

The gravitational field performs zero-point oscillations, and the geometry associated with it also oscillates. The ratio of the circumference to the radius varies near the Euclidean value. The smaller the scale, the greater the deviations from the Euclidean geometry. Let us estimate the order of the wavelength of zero gravitational oscillations, at which the geometry becomes completely unlike the Euclidean geometry. The degree of deviation ${\displaystyle \zeta }$ of geometry from Euclidean geometry in the gravitational field is determined by the ratio of the gravitational potential ${\displaystyle \varphi }$ and the square of the speed of light ${\displaystyle c}$: ${\displaystyle \zeta =\varphi /c^{2}}$. When ${\displaystyle \zeta \ll 1}$, the geometry is close to Euclidean geometry; for ${\displaystyle \zeta \sim 1}$, all similarities disappear. The energy of the oscillation of scale ${\displaystyle l}$ is equal to ${\displaystyle E=\hbar \nu \sim \hbar c/l}$ (where ${\displaystyle c/l}$ is the order of the oscillation frequency). The gravitational potential created by the mass ${\displaystyle m}$, at this length is ${\displaystyle \varphi =Gm/l}$, where ${\displaystyle G}$ is the constant of universal gravitation. Instead of ${\displaystyle m}$, we must substitute a mass, which, according to Einstein's formula, corresponds to the energy ${\displaystyle E}$ (where ${\displaystyle m=E/c^{2}}$). We get ${\displaystyle \varphi =GE/l\,c^{2}=G\hbar /l^{2}c}$. Dividing this expression by ${\displaystyle c^{2}}$, we obtain the value of the deviation ${\displaystyle \zeta =G\hbar /c^{3}l^{2}=\ell _{P}^{2}/l^{2}}$. Equating ${\displaystyle \zeta =1}$, we find the length at which the Euclidean geometry is completely distorted. It is equal to Planck length ${\displaystyle \ell _{P}={\sqrt {G\hbar /c^{3}}}\approx 10^{-35}}$m. Here there is a quantum foam.

Visualization

The size of the Planck length can be visualized as follows: if a particle or dot about 0.005 mm in size (which is the same size as a small grain of silt) were magnified in size to be as large as the observable universe, then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.005 mm dot. In other words, a 0.005 mm dot is halfway between the Planck length and the size of the observable universe on a logarithmic scale.^[8] All said, the attempt to visualize to an arbitrary scale of a 0.005 mm dot is only for a hinge point. With no fixed frame of reference for time or space, where the spatial units shrink toward infinitesimally small spatial sections and time stretches toward infinity, scale breaks down. Inverted, where space is stretched and time is shrunk, the scale adjusts the other way according to the ratio V-squared/C-squared (Lorentz transformation).^{[clarification needed]}

Bra–ket notation

From Wikipedia, the free encyclopedia

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation begins with using angle brackets, ⟨ and ⟩, and a vertical bar, |, to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space. The scalar product or action is written as

${\displaystyle \langle \phi {\mid }\psi \rangle .}$

The right part is called the ket /kɛt/; it is a vector, typically represented as a column vector and written

${\displaystyle |\psi \rangle .}$

The left part is called the bra, /brɑː/; it is the Hermitian conjugate of the ket with the same label, typically represented as a row vector and is written

${\displaystyle \langle \phi |.}$

A combination of bras, kets, and operators is interpreted using matrix multiplication. A bra and a ket with the same label are Hermitian conjugates of each other.

Bra-ket notation was introduced in 1939 by Paul Dirac^[1][2] and is also known as the Dirac notation.

The bra-ket notation has a precursor in Hermann Grassmann's use of the notation ${\displaystyle [\phi {\mid }\psi ]}$ for his inner products nearly 100 years earlier.^[3]

Introduction

Bra–ket notation is a notation for linear algebra, particularly focused on vectors, inner products, linear operators, Hermitian conjugation, and the dual space, for both finite-dimensional and infinite-dimensional complex vector spaces. 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. Many phenomena that are explained using quantum mechanics are usually explained using bra–ket notation.

In simple cases, a ket |m⟩ can be described as a column vector, a bra with the same label ⟨m| is its conjugate transpose (which is a row vector), and writing bras, kets, and linear operators next to each other implies matrix multiplication.^[4] However, kets may also exist in uncountably-infinite-dimensional vector spaces, such that they cannot be literally written as a column vector. Also, writing a column vector as a list of numbers requires picking a basis, whereas one can write "|m⟩" without committing to any particular basis. This is helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis, etc.), so it is better to have the basis vectors (if any) written out explicitly. In some situations involving two important basis vectors they will be referred to simply as "|-⟩" and "|+⟩".

The standard mathematical notation for the inner product, preferred as well by some physicists, expresses exactly the same thing as the bra–ket notation,

${\displaystyle (\phi ,\psi )=\langle \phi {\mid }\psi \rangle ={\bigl (}\langle \phi |{\bigr )}\,{\bigl (}|\psi \rangle {\bigr )},}$

Bras and kets can also be configured in other ways, such as the outer product

${\displaystyle |\psi \rangle \langle \phi |}$

which can also be represented as a matrix multiplication (i.e., a column vector times a row vector equals a matrix).

If the ket is an element of a vector space, the bra is technically an element of its dual space—see Riesz representation theorem.

Vector spaces

Vectors vs kets

In mathematics, the term "vector" is used to refer generally to any element of any vector space. In physics, however, the term "vector" is much more specific: "Vector" refers almost exclusively to quantities like displacement or velocity, which have three components that relate directly to the three dimensions of the real world. Such vectors are typically denoted with over arrows (r→) or boldface (r).

In quantum mechanics, a quantum state is typically represented as an element of an abstract complex vector space—for example the infinite-dimensional vector space of all possible wavefunctions (functions mapping each point of 3D space to a complex number). Since the term "vector" is already used for something else (see previous paragraph), it is very common to refer to these elements of abstract complex vector spaces as "kets", and to write them using ket notation.

Ket notation

Ket notation, invented by Dirac, uses vertical bars and angular brackets: |A⟩. When this notation is used, these quantities are called "kets", and |A⟩ is read as "ket-A".^[5] These kets can be manipulated using the usual rules of linear algebra, for example:

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

Note how any symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket. For example, the last line above involves infinitely many different kets, one for each real number x. In other words, the symbol "|A⟩" has a specific and universal mathematical meaning, while just the "A" by itself does not. For example, |1⟩ + |2⟩ might or might not be 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.

Inner products and bras

An inner product is a generalization of the dot product. The inner product of two vectors is a scalar. In neutral notation (notation dedicated to the inner product only), this might be written (A, B), where A and B are elements of the abstract vector space, i.e. both are kets.

Bra–ket notation uses a specific notation for inner products:

${\displaystyle (A,B)=\langle A|B\rangle ={\text{the inner product of ket }}|A\rangle {\text{ with ket }}|B\rangle }$

Bra–ket notation splits this inner product (also called a "bracket") into two pieces, the "bra" and the "ket":

${\displaystyle \langle A|B\rangle ={\bigl (}\langle A|{\bigr )}\,{\bigl (}|B\rangle {\bigr )}}$

where ⟨A| is called a bra, read as "bra-A", and |B⟩ is a ket as above.

The purpose of "splitting" the inner product into a bra and a ket is that both the bra ⟨A| and the ket |B⟩ are meaningful on their own, and can be used in other contexts besides within an inner product. There are two main ways to think about the meanings of separate bras and kets. Accordingly, the interpretation of the expression ⟨A|B⟩ has a second interpretation, namely that of the action of a linear functional per below.

Bras and kets as row and column vectors

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:

${\displaystyle \langle A|B\rangle \doteq A_{1}^{*}B_{1}+A_{2}^{*}B_{2}+\cdots +A_{N}^{*}B_{N}={\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}}$

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

${\displaystyle {\begin{aligned}\langle A|&\doteq {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\\|B\rangle &\doteq {\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}\end{aligned}}}$

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:

${\displaystyle \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|}$

because if one starts with the bra

${\displaystyle {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,}$

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

${\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}$

Bras as linear functionals

A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say that bras are linear functionals on the space of kets, i.e. linear transformations that input a ket and output a complex number. The bra linear functionals are defined to be consistent with the inner product. Thus, if ⟨B| is the linear functional corresponding to |B⟩ under the Riesz representation theorem, then

${\displaystyle \langle A|B\rangle =\langle B|{\bigl (}|A\rangle {\bigr )}\,,}$

i.e. it produces the same complex number as the inner product does. The terminology for the right hand side is though not inner product, which always involves two kets. Confusing this is harmless, since the same number is produced in the end.

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

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⟩ + i |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⟩ = ∑_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, ${\displaystyle {\hat {\mathbf {r} }}|\mathbf {r} \rangle =\mathbf {r} |\mathbf {r} \rangle }$. 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, we can define a complex scalar function of r, known as a wavefunction:

${\displaystyle \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\Psi \rangle \,.}$

On the left side, Ψ(r) is a function mapping any point in space to a complex number; on the right side, |Ψ⟩ = ∫ d³r Ψ(r) |r⟩ is a ket.

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

${\displaystyle A\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |A|\Psi \rangle \,.}$

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

${\displaystyle \mathbf {p} \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\mathbf {p} |\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.}$

One occasionally encounters an expression such as

${\displaystyle \nabla |\Psi \rangle \,,}$

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 into the position basis,

${\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}$

even though, in the momentum basis, the operator amounts to a mere multiplication operator (by iħp).

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:

${\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }$

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:

${\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }$

where a_ψ and b_ψ are complex numbers.

A different basis for the same Hilbert space is:

${\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }$

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:

${\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }$

In vector form, you might write

${\displaystyle |\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad |\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}}$

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 a_ψ, b_ψ, c_ψ and d_ψ; see change of basis.

Misleading uses

There are a few conventions and abuses of notation that are generally accepted by the physics community, but which might confuse the non-initiated.

It is common to use the same symbol for labels and constants in the same equation. For example, α̂ |α⟩ = α |α⟩, where the symbol α is used simultaneously as the name of the operator α̂, its eigenvector |α⟩ and the associated eigenvalue α.

Something similar occurs in component notation of vectors. While Ψ (uppercase) is traditionally associated with wavefunctions, ψ (lowercase) may be used to denote a label, a wave function or complex constant in the same context, usually differentiated only by a subscript.

The main abuses are including operations inside the vector labels. This is done for a fast notation of scaling vectors. E.g. if the vector |α⟩ is scaled by √2, it might be denoted by |α/√2⟩, which makes no sense since α is a label, not a function or a number, so you can't perform operations on it.

This is especially common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. |α⟩ = |α/√2₁⟩ ⊗ |α/√2₂⟩. Here part of the labeling that should state that all three vectors are different was moved outside the kets, as subscripts 1 and 2. And a further abuse occurs, since α is meant to refer to the norm of the first vector—which is a label denoting a value.

Linear operators

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 A is a linear operator and |ψ⟩ is a ket, then A|ψ⟩ is another ket.

In an N-dimensional Hilbert space, |ψ⟩ can be written as an N × 1 column vector, and then A is an N × N matrix with complex entries. The ket A|ψ⟩ can be computed by normal 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

${\displaystyle {\bigl (}\langle \phi |{\boldsymbol {A}}{\bigr )}|\psi \rangle =\langle \phi |{\bigl (}{\boldsymbol {A}}|\psi \rangle {\bigr )}\,,}$

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

${\displaystyle \langle \phi |{\boldsymbol {A}}|\psi \rangle \,.}$

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,

${\displaystyle \langle \psi |{\boldsymbol {A}}|\psi \rangle \,,}$

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

${\displaystyle |\phi \rangle \,\langle \psi |}$

denotes the rank-one operator with the rule

${\displaystyle {\bigl (}|\phi \rangle \langle \psi |{\bigr )}(x)=\langle \psi |x\rangle |\phi \rangle }$.

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

${\displaystyle |\phi \rangle \,\langle \psi |\doteq {\begin{pmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{pmatrix}}{\begin{pmatrix}\psi _{1}^{*}&\psi _{2}^{*}&\cdots &\psi _{N}^{*}\end{pmatrix}}={\begin{pmatrix}\phi _{1}\psi _{1}^{*}&\phi _{1}\psi _{2}^{*}&\cdots &\phi _{1}\psi _{N}^{*}\\\phi _{2}\psi _{1}^{*}&\phi _{2}\psi _{2}^{*}&\cdots &\phi _{2}\psi _{N}^{*}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{*}&\phi _{N}\psi _{2}^{*}&\cdots &\phi _{N}\psi _{N}^{*}\end{pmatrix}}}$

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

${\displaystyle |\psi \rangle \,\langle \psi |\,.}$

Hermitian conjugate operator

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,

${\displaystyle |\phi \rangle =A|\psi \rangle \quad {\text{if and only if}}\quad \langle \phi |=\langle \psi |A^{\dagger }\,.}$

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,

${\displaystyle \langle \phi |{\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}=c_{1}\langle \phi |\psi _{1}\rangle +c_{2}\langle \phi |\psi _{2}\rangle \,.}$

• By the definition of addition and scalar multiplication of linear functionals in the dual space,^[6]

${\displaystyle {\bigl (}c_{1}\langle \phi _{1}|+c_{2}\langle \phi _{2}|{\bigr )}|\psi \rangle =c_{1}\langle \phi _{1}|\psi \rangle +c_{2}\langle \phi _{2}|\psi \rangle \,.}$

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:

${\displaystyle {\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}}}$

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.,

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

• Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows:

• Kets:

${\displaystyle {\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}^{\dagger }=c_{1}^{*}\langle \psi _{1}|+c_{2}^{*}\langle \psi _{2}|\,.}$

• Inner products:

${\displaystyle \langle \phi |\psi \rangle ^{*}=\langle \psi |\phi \rangle \,.}$

Note that ⟨φ|ψ⟩ is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e.

${\displaystyle {\bigl (}\langle \phi |\psi \rangle {\bigr )}^{\dagger }=\langle \phi |\psi \rangle ^{*}}$

• Matrix elements:

${\displaystyle {\begin{aligned}\langle \phi |A|\psi \rangle ^{*}&=\left\langle \psi \left|A^{\dagger }\right|\phi \right\rangle \\\left\langle \phi \left|A^{\dagger }B^{\dagger }\right|\psi \right\rangle ^{*}&=\langle \psi |BA|\phi \rangle \,.\end{aligned}}}$

• Outer products:

${\displaystyle {\Big (}{\bigl (}c_{1}|\phi _{1}\rangle \langle \psi _{1}|{\bigr )}+{\bigl (}c_{2}|\phi _{2}\rangle \langle \psi _{2}|{\bigr )}{\Big )}^{\dagger }={\bigl (}c_{1}^{*}|\psi _{1}\rangle \langle \phi _{1}|{\bigr )}+{\bigl (}c_{2}^{*}|\psi _{2}\rangle \langle \phi _{2}|{\bigr )}\,.}$

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 direct product of the two kets is a ket in V ⊗ W. This is written in various notations:

${\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.}$

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 |ψ⟩ can also be written as

${\displaystyle |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,}$

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

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

${\displaystyle \sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=1\!\!1}$

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

${\displaystyle {\begin{aligned}\langle v|w\rangle &=\left\langle v\left|\sum _{i\in \mathbb {N} }\right|e_{i}\right\rangle \langle e_{i}|w\rangle \\&=\left\langle v\left|\sum _{i\in \mathbb {N} }\right|e_{i}\right\rangle \left\langle e_{i}\left|\sum _{j\in \mathbb {N} }\right|e_{j}\right\rangle \langle e_{j}|w\rangle \\&=\langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle \,,\end{aligned}}}$

where, in the last identity, the Einstein summation convention has been used.

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,

•     1 = ∫ dx |x⟩⟨x| = ∫ dp |p⟩⟨p|, where |p⟩ = ∫ dx e^ixp/ħ|x⟩/√2πħ.

Since ⟨x′|x⟩ = δ(x − x′), plane waves follow, ⟨x|p⟩ = e^ixp/ħ/√2πħ.^[7]

Notation used by mathematicians

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

Let H be a Hilbert space and h ∈ H a vector in H. What physicists would denote by |h⟩ is the vector itself. That is,

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

Let H* be the dual space of H. This is the space of linear functionals on H. The isomorphism Φ : H → H* is defined by Φ(h) = φ_h, where for every g ∈ H we define

${\displaystyle \phi _{h}(g)={\mbox{IP}}(h,g)=(h,g)=\langle h,g\rangle =\langle h|g\rangle }$,

where IP(·,·), (·,·), ⟨·,·⟩ and ⟨·|·⟩ are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises when identifying φ_h and g with ⟨h| and |g⟩ respectively. This is because of literal symbolic substitutions. Let φ_h = H = ⟨h| and let g = G = |g⟩. This gives

${\displaystyle \phi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}$

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.

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

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

whereas physicists would write for the same quantity

${\displaystyle \langle \psi |\phi \rangle =\int \mathrm {d} x\,\psi ^{*}(x)\cdot \phi (x)\,.}$