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but with a remarkable feature: "the time runs backwards"
This article summarizes equations in the theory of quantum mechanics.
A fundamental physical constant occurring in quantum mechanics is the Planck constant, "h". A common abbreviation is , also known as the "reduced Planck constant" or "Dirac constant".
The general form of wavefunction for a system of particles, each with position r"i" and z-component of spin "sz i". Sums are over the discrete variable "sz", integrals over continuous positions r.
For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.
Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation.
Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media.
In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity.
Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. The speed of a plane wave, formula_1, is a function of the wave's wavelength formula_2:
The wave's speed, wavelength, and frequency, "f", are related by the identity
The function formula_5 expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the angular frequency formula_6 and wavenumber formula_7. Rewriting the relation above in these variables gives
where we now view "f" as a function of "k". The use of ω("k") to describe the dispersion relation has become standard because both the phase velocity ω/"k" and the group velocity dω/d"k" have convenient representations via this function.
The plane waves being considered can be described by
Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.
For electromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber:
This is a "linear" dispersion relation. In this case, the phase velocity and the group velocity are the same:
they are given by "c", the speed of light in vacuum, a frequency-independent constant.
Total energy, momentum, and mass of particles are connected through the relativistic dispersion relation:
where formula_15 is the invariant mass. In the nonrelativistic limit, formula_16 is a constant, and formula_17 is the familiar kinetic energy expressed in terms of the momentum formula_18.
The transition from ultrarelativistic to nonrelativistic behaviour shows up as a slope change from "p" to "p"2 as shown in the log–log dispersion plot of "E" vs. "p".
Elementary particles, atomic nuclei, atoms, and even molecules behave in some contexts as matter waves. According to the de Broglie relations, their kinetic energy "E" can be expressed as a frequency "ω", and their momentum "p" as a wavenumber "k", using the reduced Planck constant "ħ":
Accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads
As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the refractive index—it is common to refer to the functional dependence of angular frequency on wavenumber as the "dispersion relation". For particles, this translates to a knowledge of energy as a function of momentum.
The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e., be dispersed. In these materials, formula_21 is known as the group velocity and corresponds to the speed at which the peak of the pulse propagates, a value different from the phase velocity.
The dispersion relation for deep water waves is often written as
where "g" is the acceleration due to gravity. Deep water, in this respect, is commonly denoted as the case where the water depth is larger than half the wavelength. In this case the phase velocity is
For an ideal string, the dispersion relation can be written as
where "T" is the tension force in the string, and "μ" is the string's mass per unit length. As for the case of electromagnetic waves in vacuum, ideal strings are thus a non-dispersive medium, i.e. the phase and group velocities are equal and independent (to first order) of vibration frequency.
For a nonideal string, where stiffness is taken into account, the dispersion relation is written as
where formula_27 is a constant that depends on the string.
In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that many levels of energy are possible for a given momentum and that some energies might not be available at any momentum. The collection of all possible energies and momenta is known as the band structure of a material. Properties of the band structure define whether the material is an insulator, semiconductor or conductor.
Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation of phonons is also non-trivial and important, being directly related to the acoustic and thermal properties of a material. For most systems, the phonons can be categorized into two main types: those whose bands become zero at the center of the Brillouin zone are called acoustic phonons, since they correspond to classical sound in the limit of long wavelengths. The others are optical phonons, since they can be excited by electromagnetic radiation.
With high-energy (e.g., ) electrons in a transmission electron microscope, the energy dependence of higher-order Laue zone (HOLZ) lines in convergent beam electron diffraction (CBED) patterns allows one, in effect, to "directly image" cross-sections of a crystal's three-dimensional dispersion surface. This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.
Isaac Newton studied refraction in prisms but failed to recognize the material dependence of the dispersion relation, dismissing the work of another researcher whose measurement of a prism's dispersion did not match Newton's own.
Dispersion of waves on water was studied by Pierre-Simon Laplace in 1776.
The universality of the Kramers–Kronig relations (1926–27) became apparent with subsequent papers on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles.
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a non relativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
When any charged particle (such as an electron, a proton, or an ion) accelerates, it radiates away energy in the form of electromagnetic waves. For velocities that are small relative to the speed of light, the total power radiated is given by the Larmor formula:
where formula_3 or formula_4 is the proper acceleration, formula_5 is the charge, and formula_6 is the speed of light. A relativistic generalization is given by the Liénard–Wiechert potentials.
In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as:
One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory was introduced.
Derivation 1: Mathematical approach (using CGS units).
We first need to find the form of the electric and magnetic fields. The fields can be written (for a fuller derivation see Liénard–Wiechert potential)
where formula_10 is the charge's velocity divided by formula_11, formula_12 is the charge's acceleration divided by "c", formula_13 is a unit vector in the formula_14 direction, formula_15 is the magnitude of formula_16, formula_17 is the charge's location, and formula_18. The terms on the right are evaluated at the retarded time formula_19.
The right-hand side is the sum of the electric fields associated with the velocity and the acceleration of the charged particle. The velocity field depends only upon formula_10 while the acceleration field depends on both formula_10 and formula_12 and the angular relationship between the two. Since the velocity field is proportional to formula_23, it falls off very quickly with distance. On the other hand, the acceleration field is proportional to formula_24, which means that it falls much more slowly with distance. Because of this, the acceleration field is representative of the radiation field and is responsible for carrying most of the energy away from the charge.
We can find the energy flux density of the radiation field by computing its Poynting vector:
where the 'a' subscripts emphasize that we are taking only the acceleration field. Substituting in the relation between the magnetic and electric fields while assuming that the particle instantaneously at rest at time formula_26 and simplifying gives
If we let the angle between the acceleration and the observation vector be equal to formula_28, and we introduce the acceleration formula_29, then the power radiated per unit solid angle is
The total power radiated is found by integrating this quantity over all solid angles (that is, over formula_28 and formula_32). This gives
which is the Larmor result for a non-relativistic accelerated charge. It relates the power radiated by the particle to its acceleration. It clearly shows that the faster the charge accelerates the greater the radiation will be. We would expect this since the radiation field is dependent upon acceleration.
The full derivation can be found here.
Here is an explanation which can help understanding the above page.
This approach is based on the finite speed of light. A charge moving with
constant velocity has a radial electric field formula_34
from the charge), always emerging from the future position of the charge,
and there is no tangential component of the electric field formula_36.
This future position is completely deterministic as long as the velocity
is constant. When the velocity of the charge changes, (say it bounces back
during a short time) the future position "jumps", so from this moment and
on, the radial electric field formula_34 emerges from a new
position. Given the fact that the electric field must be continuous, a
non-zero tangential component of the electric field formula_38 appears,
which decreases like formula_24 (unlike the radial component which
Hence, at large distances from the charge, the radial component is negligible
relative to the tangential component, and in addition to that, fields which
behave like formula_23 cannot radiate, because the Poynting vector
associated with them will behave like formula_42.
The tangential component comes out (SI units):
And to obtain the Larmour formula, one has to integrate over all angles, at
large distance formula_15 from the charge, the
Poynting vector associated with formula_38, which is:
Since formula_49, we recover the result quoted at the top of the article, namely
Written in terms of momentum, , the non-relativistic Larmor formula is (in CGS units)
The power can be shown to be Lorentz invariant. Any relativistic generalization of the Larmor formula must therefore relate to some other Lorentz invariant quantity. The quantity formula_52 appearing in the non-relativistic formula suggests that the relativistically correct formula should include the Lorentz scalar found by taking the inner product of the four-acceleration with itself [here is the four-momentum]. The correct relativistic generalization of the Larmor formula is (in CGS units)
It can be shown that this inner product is given by
and so in the limit , it reduces to formula_54, thus reproducing the nonrelativistic case.
The above inner product can also be written in terms of and its time derivative. Then the relativistic generalization of the Larmor formula is (in CGS units)
This is the Liénard result, which was first obtained in 1898. The formula_55 means that when the Lorentz factor formula_56 is very close to one (i.e. formula_57) the radiation emitted by the particle is likely to be negligible. However, as formula_58 the radiation grows like formula_55 as the particle tries to lose its energy in the form of EM waves. Also, when the acceleration and velocity are orthogonal the power is reduced by a factor of formula_60, i.e. the factor formula_55 becomes formula_62. The faster the motion becomes the greater this reduction gets.
We can use Liénard's result to predict what sort of radiation losses to expect in different kinds of motion.
The angular distribution of radiated power is given by a general formula, applicable whether or not the particle is relativistic. In CGS units, this formula is
where formula_64 is a unit vector pointing from the particle towards the observer. In the case of linear motion (velocity parallel to acceleration), this simplifies to
where formula_28 is the angle between the observer and the particle's motion.
The radiation from a charged particle carries energy and momentum. In order to satisfy energy and momentum conservation, the charged particle must experience a recoil at the time of emission. The radiation must exert an additional force on the charged particle. This force is known as the Abraham–Lorentz force in the nonrelativistic limit and the Abraham–Lorentz–Dirac force in the relativistic setting.
A classical electron in the Bohr model orbiting a nucleus experiences acceleration and should radiate. Consequently, the electron loses energy and the electron should eventually spiral into the nucleus. Atoms, according to classical mechanics, are consequently unstable. This classical prediction is violated by the observation of stable electron orbits. The problem is resolved with a quantum mechanical description of atomic physics, initially provided by the Bohr model. Classical solutions to the stability of electron orbitals can be demonstrated using Non-radiation conditions and in accordance with known physical laws.
In fluid dynamics the Morison equation is a semi-empirical equation for the inline force on a body in oscillatory flow. It is sometimes called the MOJS equation after all four authors—Morison, O'Brien, Johnson and Schaaf—of the 1950 paper in which the equation was introduced. The Morison equation is used to estimate the wave loads in the design of oil platforms and other offshore structures.
The Morison equation is the sum of two force components: an inertia force in phase with the local flow acceleration and a drag force proportional to the (signed) square of the instantaneous flow velocity. The inertia force is of the functional form as found in potential flow theory, while the drag force has the form as found for a body placed in a steady flow. In the heuristic approach of Morison, O'Brien, Johnson and Schaaf these two force components, inertia and drag, are simply added to describe the inline force in an oscillatory flow. The transverse force—perpendicular to the flow direction, due to vortex shedding—has to be addressed separately.
The Morison equation contains two empirical hydrodynamic coefficients—an inertia coefficient and a drag coefficient—which are determined from experimental data. As shown by dimensional analysis and in experiments by Sarpkaya, these coefficients depend in general on the Keulegan–Carpenter number, Reynolds number and surface roughness.
The descriptions given below of the Morison equation are for uni-directional onflow conditions as well as body motion.
In an oscillatory flow with flow velocity formula_1, the Morison equation gives the inline force parallel to the flow direction:
For instance for a circular cylinder of diameter "D" in oscillatory flow, the reference area per unit cylinder length is formula_12 and the cylinder volume per unit cylinder length is formula_13. As a result, formula_3 is the total force per unit cylinder length:
Besides the inline force, there are also oscillatory lift forces perpendicular to the flow direction, due to vortex shedding. These are not covered by the Morison equation, which is only for the inline forces.
In case the body moves as well, with velocity formula_16, the Morison equation becomes:
Note that the added mass coefficient formula_18 is related to the inertia coefficient formula_19 as formula_10.
Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. formula_1. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
The equations of motion for Stokes flow, called the Stokes equations, are a linearization of the Navier–Stokes equations, and thus can be solved by a number of well-known methods for linear differential equations. The primary Green's function of Stokes flow is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives, other fundamental solutions can be obtained. The Stokeslet was first derived by the Nobel Laureate Hendrik Lorentz, as far back as 1896. Despite its name, Stokes never knew about the Stokeslet; the name was coined by Hancock in 1953. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids.
The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier-Stokes equations. The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations:
where formula_3 is the stress (sum of viscous and pressure stresses), and formula_4 an applied body force. The full Stokes equations also include an equation for the conservation of mass, commonly written in the form:
where formula_6 is the fluid density and formula_7 the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, formula_6, is a constant.
Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term formula_9 is added to the left hand side of the momentum balance equation.