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: 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. •
Doppler and
range: the more we know about how far away a
radar 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;
A = surface area). • Elastic stretching:
F d
L (
F = elastic force;
L length stretched). • Energy and time: Units \Delta E \times \Delta t being kg⋅m2/s.
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 of a particle at a certain
event is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the
time of the event. • The
linear momentum of a particle is the derivative of its action with respect to its
position. • The
angular momentum of a particle is the derivative of its action with respect to its
orientation (angular position). • The
mass-moment (\mathbf{N}=t\mathbf{p}-E\mathbf{r}) of a particle is the negative of the derivative of its action with respect to its
rapidity. • The
electric potential (φ,
voltage) and
electric charge in a
quantum LC circuit. • 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 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 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 at that event.
Quantum theory In
quantum mechanics, 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 \left (x \right) and momentum \left (p \right) . In the quantum-mechanical formalism, the two observables x and p correspond to operators \widehat{x} and \widehat{p\,} , which necessarily satisfy the
canonical commutation relation: [\widehat{x},\widehat{p\,}]=\widehat{x}\widehat{p\,}-\widehat{p\,}\widehat{x}=i \hbar For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form: \Delta x \, \Delta p \geq \hbar/2 In this ill-defined notation, \Delta x and \Delta p denote "uncertainty" in the simultaneous specification of x and p . A more precise, and statistically complete, statement involving the standard deviation \sigma reads: \sigma_x \sigma_p \geq \hbar/2 More generally, for any two observables A and B corresponding to operators \widehat{A} and \widehat{B} , the
generalized uncertainty principle is given by: {\sigma_A}^2 {\sigma_B}^2 \geq \left (\frac{1}{2i} \left \langle \left [ \widehat{A},\widehat{B} \right ] \right \rangle \right)^2 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 \mathfrak h_3, with a corresponding group called the Heisenberg group H_3 .
Fluid mechanics In
Hamiltonian fluid mechanics and
quantum hydrodynamics, the
action itself (or
velocity potential) is the conjugate variable of the
density (or
probability density). ==See also==