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, or more generally are related through
Pontryagin duality. 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 ...
called the
Heisenberg uncertainty principle—between them. In mathematical terms, conjugate variables are part of a
symplectic basis, and the uncertainty relation corresponds to the
symplectic form. Also, conjugate variables are related by
Noether's theorem, 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: 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, Marine radar, ships, spacecraft, guided missiles, motor v ...
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 In pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of propagation delay \tau and Doppler frequency f, \chi(\tau,f). It represents the distortion of a returned pulse due to the receiver matched filter ( ...
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) t ...
; ''A'' = surface area).
* Elastic stretching: ''F'' d''L'' (''F'' = elastic force; ''L'' length stretched).
Derivatives of action
In classical physics, the derivatives of action 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.
* 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 ...
'' 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 ev ...
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 event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
'' of the event.
* The '' linear momentum'' of a particle is the derivative of its action with respect to its '' position''.
* 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'' (angular position).
* The ''mass-moment'' () of a particle is the negative of the derivative of its action with respect to its '' rapidity''.
* 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 ...
) 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 res ...
'' at that event.
* The '' magnetic potential'' (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 movi ...
'' at that event.
* The '' electric field'' (E) at an event is the derivative of the action of the electromagnetic field with respect to the ''electric polarization density'' 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 D ...
'' at that event.
* The Newtonian '' gravitational potential'' 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. Mathematicall ...
'' 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:
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 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'' 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. Mathematicall ...
'' (or '' probability density).
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 cl ...
Notes
{{DEFAULTSORT:Conjugate Variables
Classical mechanics
Quantum mechanics