Compton wavelength - Citizendia

The Compton wavelength $\lambda \$ of a particle is given by

$\lambda = \frac{h}{m c} = 2 \pi \frac{\hbar}{m c} \$ ,

where

$h \$ is the Planck constant,

$m \$ is the particle's rest mass,

$c \$ is the speed of light. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object

The CODATA 2002 value for the Compton wavelength of the electron is 2. CODATA ( Committee on Data for Science and Technology) was established in 1966 as an interdisciplinary committee of the International Council of Science (ICSU formerly 4263102175×10^-12 meters with a standard uncertainty of 0. 0000000033×10^-12 m. ^[1] Other particles have different Compton wavelengths.

The Compton wavelength can be thought of as a fundamental limitation on measuring the position of a particle, taking quantum mechanics and special relativity into account. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial This depends on the mass $m \$ of the particle. To see this, note that we can measure the position of a particle by bouncing light off it - but measuring the position accurately requires light of short wavelength. Light with a short wavelength consists of photons of high energy. If the energy of these photons exceeds $mc^2 \$ , when one hits the particle whose position is being measured the collision may have enough energy to create a new particle of the same type. This renders moot the question of the original particle's location.

This argument also shows that the Compton wavelength is the cutoff below which quantum field theory– which can describe particle creation and annihilation – becomes important. In quantum field theory (QFT the forces between particles are mediated by other particles

We can make the above argument a bit more precise as follows. Suppose we wish to measure the position of a particle to within an accuracy $\Delta x \$ . Then the uncertainty relation for position and momentum says that

$\Delta x\,\Delta p\ge \hbar/2$

so the uncertainty in the particle's momentum satisfies

$\Delta p \ge \frac{\hbar}{2\Delta x}$

Using the relativistic relation between momentum and energy, when $Δ p$ exceeds $m c$ then the uncertainty in energy is greater than $mc^2 \$ , which is enough energy to create another particle of the same type. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The term Mass in Special relativity usually refers to the Rest mass of the object which is the Newtonian mass as measured by an observer moving along with So, with a little algebra, we see there is a fundamental limitation

$\Delta x \ge \frac{\hbar}{2mc}$

So, at least to within an order of magnitude, the uncertainty in position must be greater than the Compton wavelength $h/mc \$ .

The Compton wavelength can be contrasted with the de Broglie wavelength, which depends on the momentum of a particle and determines the cutoff between particle and wave behavior in quantum mechanics. In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

For fermions, the Compton wavelength sets the cross-section of interactions. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. For example, the cross-section for Thomson scattering of a photon from an electron is equal to

$(8\pi/3)\alpha^2\lambda_e^2$ ,

where $\alpha \$ is the fine-structure constant and $\lambda_e \$ is the Compton wavelength of the electron. In Physics, Thomson scattering is the scattering of Electromagnetic radiation by acharged particle The fine-structure constant or Sommerfeld fine-structure constant, usually denoted \alpha \ is the Fundamental physical constant characterizing For gauge bosons, the Compton wavelength sets the effective range of the Yukawa interaction: since the photon has no rest mass, electromagnetism has infinite range. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field \phi and a Dirac field In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena

The Compton wavelength of the electron is one of a trio of related units of length, the other two being the Bohr radius $a 0$ and the classical electron radius $r e$ . In the Bohr model of the structure of an Atom, put forward by Niels Bohr in 1913 Electrons orbit a central nucleus. The classical electron radius, also known as the Lorentz radius or the Thomson scattering length is based on a classical (i The Compton wavelength is built from the electron mass $m e$ , Planck's constant $h$ and the speed of light $c$ . The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. The Bohr radius is built from $m e$ , $h$ and the electron charge $e$ . In the Bohr model of the structure of an Atom, put forward by Niels Bohr in 1913 Electrons orbit a central nucleus. The elementary charge, usually denoted e, is the Electric charge carried by a single Proton, or equivalently the negative of the electric charge carried The classical electron radius is built from $m e$ , $c$ and $e$ . The classical electron radius, also known as the Lorentz radius or the Thomson scattering length is based on a classical (i Any one of these three lengths can be written in terms of any other using the fine structure constant $α$ :

$r_e = {\alpha \lambda_e \over 2\pi} = \alpha^2 a_0$

The Planck mass is special because ignoring factors of $2π$ and the like, the Compton wavelength for this mass is equal to its Schwarzschild radius. The Planck mass is the unit of Mass, denoted by m P in the system of Natural units known as Planck units. The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic Radius associated with every Mass. This special distance is called the Planck length. The Planck length, denoted by \scriptstyle\ell_P \, is the unit of Length approximately 1 This is a simple case of dimensional analysis: the Schwarzschild radius is proportional to the mass, whereas the Compton wavelength is proportional to the inverse of the mass. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand

References

^ CODATA 2002 value for Compton wavelength for the electron from NIST

External links

Length Scales in Physics: the Compton Wavelength

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