Conjugate variables are pairs of variables mathematically defined in such a way that they become
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...
, or more generally are related through
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
. The duality relations lead naturally to an uncertainty relation—in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
called the
Heisenberg uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
—between them. In mathematical terms, conjugate variables are part of a
symplectic basis In linear algebra, a standard symplectic basis is a basis _i, _i of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega, such that \omega(_i, _j) = 0 = \omega(_i, _j), \omega(_i, _j) = \delta_. A ...
, and the uncertainty relation corresponds to the
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument s ...
. Also, conjugate variables are related by
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
, which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved).
Examples
There are many types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). Examples of canonically conjugate variables include the following:
* Time and
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
: the longer a musical note is sustained, the more precisely we know its frequency, but it spans a longer duration and is thus a more-distributed event or 'instant' in time. Conversely, a very short musical note becomes just a click, and so is more temporally-localized, but one can't determine its frequency very accurately.
"The Chirplet Transform", IEEE Transactions on Signal Processing, 43(11), November 1995, pp 2745–2761
/ref>
* Doppler and range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
: the more we know about how far away a radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
target is, the less we can know about the exact velocity of approach or retreat, and vice versa. In this case, the two dimensional function of doppler and range is known as a radar ambiguity function or radar ambiguity diagram.
* Surface energy: ''γ'' d''A'' (''γ'' = surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
; ''A'' = surface area).
* Elastic stretching: ''F'' d''L'' (''F'' = elastic force; ''L'' length stretched).
Derivatives of action
In classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, the derivatives of action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
are conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberg uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
.
* The ''energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
'' of a particle at a certain event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of eve ...
is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the ''time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
'' of the event.
* The ''linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
'' of a particle is the derivative of its action with respect to its ''position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
''.
* The ''angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
'' of a particle is the derivative of its action with respect to its ''orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
'' (angular position).
* The ''mass-moment'' () of a particle is the negative of the derivative of its action with respect to its ''rapidity
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...
''.
* The ''electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
'' (φ, voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
) at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of (free) ''electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
'' at that event.
* The ''magnetic potential Magnetic potential may refer to:
* Magnetic vector potential, the vector whose curl is equal to the magnetic B field
* Magnetic scalar potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electr ...
'' (A) at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) ''electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
'' at that event.
* The ''electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
'' (E) at an event is the derivative of the action of the electromagnetic field with respect to the ''electric polarization density
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
'' at that event.
* The '' magnetic induction'' (B) at an event is the derivative of the action of the electromagnetic field with respect to the ''magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Di ...
'' at that event.
* The Newtonian ''gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric po ...
'' at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the ''mass density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
'' at that event.
Quantum theory
In quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, conjugate variables are realized as pairs of observables whose operators do not commute. In conventional terminology, they are said to be ''incompatible observables''. Consider, as an example, the measurable quantities given by position and momentum . In the quantum-mechanical formalism, the two observables and correspond to operators and , which necessarily satisfy the canonical commutation relation
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p ...
:
For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form:
In this ill-defined notation, and denote "uncertainty" in the simultaneous specification of and . A more precise, and statistically complete, statement involving the standard deviation reads:
More generally, for any two observables and corresponding to operators and , the generalized uncertainty principle is given by:
Now suppose we were to explicitly define two particular operators, assigning each a ''specific'' mathematical form, such that the pair satisfies the aforementioned commutation relation. It's important to remember that our particular "choice" of operators would merely reflect one of many equivalent, or isomorphic, representations of the general algebraic structure that fundamentally characterizes quantum mechanics. The generalization is provided formally by the Heisenberg Lie algebra , with a corresponding group called the Heisenberg group .
Fluid mechanics
In Hamiltonian fluid mechanics Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to nondissipative fluids.
Irrotational barotropic flow
Take the simple example of a barotropic, inviscid vorticity-f ...
and quantum hydrodynamics
In condensed matter physics, quantum hydrodynamics is most generally the study of hydrodynamic-like systems which demonstrate quantum mechanical behavior. They arise in semiclassical mechanics in the study of metal and semiconductor devices, in wh ...
, the ''action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
'' itself (or '' velocity potential'') is the conjugate variable of the ''density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
'' (or ''probability density
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
).
See also
*Canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
Notes
{{DEFAULTSORT:Conjugate Variables
Classical mechanics
Quantum mechanics