Coupling (physics)

In physics, two objects are said to be coupled when they are interacting with each other. In classical mechanics, coupling is a connection between two oscillating systems, such as pendulums connected by a spring. The connection affects the oscillatory pattern of both objects. In particle physics, two particles are coupled if they are connected by one of the four fundamental forces.

Wave mechanics

Coupled harmonic oscillator

Coupled pendulums connected by a spring

If two waves are able to transmit energy to 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 a spring. If the pendulums are identical, then their equations of motion are given by $${\displaystyle m{\ddot {x}}=-mg{\frac {x}{l_{1}}}-k(x-y)}$$ $${\displaystyle m{\ddot {y}}=-mg{\frac {y}{l_{2}}}+k(x-y)}$$

These equations represent the simple harmonic motion of the pendulum with an added coupling factor of the spring.^[1] This behavior is also seen in certain molecules (such as CO₂ and 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.

In LC circuits, charge oscillates between the capacitor and the inductor and can therefore be modeled as a simple harmonic oscillator. When the magnetic flux from one inductor is able to affect the inductance of an inductor in an unconnected LC circuit, the circuits are said to be coupled.^[1] The coefficient of coupling k defines how closely the two circuits are coupled and is given by the equation $${\displaystyle {\frac {M}{\sqrt {L_{p}L_{s}}}}=k}$$

where M is the mutual inductance of the circuits and L_p and L_s are 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 coupling occurs when the magnetic field of one atom affects the magnetic field of another nearby atom. This is very common in NMR imaging. If the atoms are not coupled, then there will be two 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 to either side. This occurs due to the spins of 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's gravity. 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 are binary systems, two objects gravitationally coupled to each other. Examples of this are binary stars which circle each other. Multiple objects may also be coupled to each other simultaneously, such as with globular clusters and galaxy groups. When smaller particles, such as dust, which are coupled together over time accumulate into much larger objects, accretion is occurring. This is the major process by which stars and planets form.^[3]

Plasma

Main article: Frequency classification of plasmas

The coupling constant of a plasma is given by the ratio of its average Coulomb-interaction energy to its average kinetic 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 the solar corona, are weakly coupled, while the plasma in a white dwarf star is an example of a strongly coupled plasma.^[4]

Quantum mechanics

Two coupled quantum systems can be modeled by a Hamiltonian of the form

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

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

Angular momentum coupling

Main article: Angular momentum coupling

When angular momenta from two separate sources interact with each other, they are said to be coupled.^[6] For example, two electrons orbiting around the same nucleus may have coupled angular momenta. Due to the conservation of angular momentum and the nature of the angular momentum operator, the total angular momentum is always the sum of the individual angular momenta of the electrons, or^[6] $${\displaystyle \mathbf {J} =\mathbf {J_{1}} +\mathbf {J_{2}} }$$ Spin-Orbit interaction (also known as spin-orbit coupling) is a special case of angular momentum coupling. Specifically, it is the interaction between the intrinsic spin of 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, ${\displaystyle \mathbf {J} =\mathbf {L} +\mathbf {S} }$.^[6]

Particle physics and quantum field theory

Examples of gluon coupling

Particles which 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 dimensionless coupling constant. In quantum electrodynamics, this value is known as the fine-structure constant α, approximately equal to 1/137. For quantum chromodynamics, the constant changes with respect to the distance between the particles. This phenomenon is known as asymptotic 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

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. Retrieved 13 Apr 2017.
3. Kaufmann, William (1988). Universe, Second Edition. W.H. Freeman and Company. ISBN 978-0-7167-1927-4.
4. Ichimaru, Setsuo (1986). Plasma Physics. Menlo Park, California: Benjamin/Cumming Publishing Company. ISBN 978-0-8053-8754-4.
5. Hagelstein, Peter; Senturia, Stephen; Orlando, Terry (2004). Introductory Applied Quantum and Statistical Mechanics. Hoboken, New Jersey: Wiley. ISBN 978-0-471-20276-9.
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.