Coupling (physics)

Inphysics,two objects are saidto be coupled when they are interacting with each other. Inclassical mechanics,coupling is a connection between twooscillatingsystems, such aspendulumsconnected by a spring. The connection affects the oscillatory pattern of both objects. Inparticle physics,two particles are coupledif they are connected byone of the fourfundamental forces.

Wave mechanics

Coupled harmonic oscillator

Coupled pendulums connected by a spring

If twowavesare able to transmitenergyto each other, then these waves are said to be "coupled." This normally occurs when the waves share a common component. An example of this is two pendulums connected by aspring.If the pendulums are identical, then their equations of motion are given by $m{\ddot {x}}=-mg{\frac {x}{l_{1}}}-k(x-y)$ $m{\ddot {y}}=-mg{\frac {y}{l_{2}}}+k(x-y)$

These equations represent thesimple harmonic motionof the pendulum with an added coupling factor of the spring.^[1]This behavior is also seen in certain molecules (such asCO₂and H₂O), wherein two of the atoms will vibrate around a central one in a similar manner.^[1]

Coupled LC circuits

Two LC circuits coupled together.

InLC circuits,charge oscillates between thecapacitorand theinductorand can therefore be modeled as a simple harmonic oscillator. When themagnetic fluxfromone inductor is ableto affect theinductanceof an inductor in an unconnected LC circuit, the circuits are said to be coupled.^[1]The coefficient of coupling k defines how closely thetwo circuits are coupledand is given by the equation ${\frac {M}{\sqrt {L_{p}L_{s}}}}=k$

where M is themutual inductanceof the circuits and L_pand L_sare the inductances of the primary and secondary circuits, respectively. If the flux lines of the primary inductor thread every line of the secondary one, then the coefficient of coupling is 1 and ${\textstyle M={\sqrt {L_{p}L_{s}}}}$ In practice, however, there is often leakage,so most systems are not perfectly coupled.^[1]

Peaks in an NMR image of Ethyl Acetate.

Chemistry

Spin-spin coupling

Spin-spin couplingoccurs when themagnetic fieldofone atom affects themagnetic field of another nearby atom. This is very common inNMR imaging.If the atoms are not coupled, then there will betwo individual peaks,known as a doublet, representing the individual atoms. If coupling is present, then there will be a triplet,one larger peak with two smaller ones toeither side. This occurs due to thespinsof the individual atoms oscillating in tandem.^[2]

Astrophysics

Objects in space which are coupled to each other are under the mutual influence of each other'sgravity.For instance, the Earth is coupled to both the Sun and the Moon, as it is under the gravitational influence of both. Common in space arebinary systems,two objects gravitationally coupled to each other. Examples of this arebinary starswhich circle each other. Multiple objects may also be coupled to each other simultaneously, such as withglobular clustersandgalaxy groups.When smaller particles, such as dust, which are coupled together over time accumulate into much larger objects,accretionis occurring. This is the major process by which stars and planets form.^[3]

Plasma

The coupling constant of aplasmais given by the ratio of its averageCoulomb-interactionenergy to its averagekinetic energy—or how strongly the electric force of each atom holds the plasma together.^[4]Plasmas can therefore be categorized into weakly- and strongly-coupled plasmas depending upon the value of this ratio. Many of the typical classical plasmas, such as the plasma in thesolar corona,are weakly coupled, while the plasma in awhite dwarfstar is an example of a strongly coupled plasma.^[4]

Quantum mechanics

Two coupled quantum systems can be modeled by aHamiltonianof the form

Dispersion relations for non-coupled, weakly-coupled, and strongly-coupled particles

${\hat {H}}={\hat {H}}_{a}+{\hat {H}}_{b}+{\hat {V}}_{ab}$ which is the addition of the two Hamiltonians in isolation with an added interaction factor. In most simple systems, ${\hat {H}}_{a}$ and ${\hat {H}}_{b}$ can be solved exactly while ${\hat {V}}_{ab}$ can be solved throughperturbation theory.^[5]If the two systems have similar total energy, then the system may undergoRabi oscillation.^[5]

Angular momentum coupling

Whenangular momentafrom two separate sources interact with each other, they are said to be coupled.^[6]For example, twoelectronsorbiting around the samenucleusmay have coupled angular momenta. Due to theconservation of angular momentumand the nature of theangular momentum operator,the total angular momentum is always the sum of the individual angular momenta of the electrons, or^[6] $\mathbf {J} =\mathbf {J_{1}} +\mathbf {J_{2}}$ Spin-Orbit interaction(also known as spin-orbit coupling) is a special case of angular momentum coupling. Specifically, it is the interaction between theintrinsic spinof a particle,S,and its orbital angular momentum,L.As they are both forms of angular momentum, they must be conserved. Even if energy is transferred between the two, the total angular momentum,J,of the system must be constant, $\mathbf {J} =\mathbf {L} +\mathbf {S}$ .^[6]

Particle physics and quantum field theory

Examples of gluon coupling

Particleswhich interact with each other are said to be coupled. This interaction is caused by one of the fundamental forces, whose strengths are usually given by a dimensionlesscoupling constant.Inquantum electrodynamics,this value is known as thefine-structure constantα, approximately equal to 1/137. Forquantum chromodynamics,the constant changes with respect to the distance between the particles. This phenomenon is known asasymptotic freedom.Forces which have a coupling constant greater than 1 are said to be "strongly coupled" while those with constants less than 1 are said to be "weakly coupled."^[7]

References

^^a ^b ^c ^dPain, H.J. (1993).The Physics of Vibrations and Waves(Fourth ed.). West Sussex, England: Wiley.ISBN 0-471-93742-8.
^"5.5 Spin-Spin Coupling".Chemistry Libretexts.2015-07-21.Retrieved13 Apr2017.
^Kaufmann, William (1988).Universe, Second Edition.W.H. Freeman and Company.ISBN 978-0-7167-1927-4.
^^a ^bIchimaru, Setsuo (1986).Plasma Physics.Menlo Park, California: Benjamin/Cumming Publishing Company.ISBN 978-0-8053-8754-4.
^^a ^bHagelstein, Peter; Senturia, Stephen; Orlando, Terry (2004).Introductory Applied Quantum and Statistical Mechanics.Hoboken, New Jersey: Wiley.ISBN 978-0-471-20276-9.
^^a ^b ^cMerzbacher, Eugene (1998).Quantum Mechanics(Third ed.). Wiley.ISBN 978-0-471--88702-7.
^Griffiths, David (2010).Elementary Particle-Second, Revised Edition.Wiley-VCH.ISBN 978-3-527-40601-2.

[:0-1] Pain, H.J. (1993).The Physics of Vibrations and Waves(Fourth ed.). West Sussex, England: Wiley.ISBN 0-471-93742-8.

[2] "5.5 Spin-Spin Coupling".Chemistry Libretexts.2015-07-21.Retrieved13 Apr2017.

[3] Kaufmann, William (1988).Universe, Second Edition.W.H. Freeman and Company.ISBN 978-0-7167-1927-4.

[:1-4] Ichimaru, Setsuo (1986).Plasma Physics.Menlo Park, California: Benjamin/Cumming Publishing Company.ISBN 978-0-8053-8754-4.

[:2-5] Hagelstein, Peter; Senturia, Stephen; Orlando, Terry (2004).Introductory Applied Quantum and Statistical Mechanics.Hoboken, New Jersey: Wiley.ISBN 978-0-471-20276-9.

[:3-6] Merzbacher, Eugene (1998).Quantum Mechanics(Third ed.). Wiley.ISBN 978-0-471--88702-7.

[7] Griffiths, David (2010).Elementary Particle-Second, Revised Edition.Wiley-VCH.ISBN 978-3-527-40601-2.

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