en>Josve05a: clean up per WP:TPL using AWB (9783)

2013-12-16T23:26:56Z

clean up per WP:TPL using AWB (9783)

← Older revision		Revision as of 01:26, 17 December 2013
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	~~Nice~~ to ~~meet you~~, ~~I am Marvella Shryock. I am~~ a ~~meter reader. Minnesota~~ is ~~exactly~~ where he's ~~been living~~ for many ~~years~~. The ~~favorite hobby~~ for ~~my children~~ and me is to ~~play baseball~~ but ~~I haven~~'~~t produced~~ a ~~dime~~ with it.<br><br>~~Here~~ is ~~my blog post~~ ... [~~http~~://~~conex360~~.~~com~~/~~members~~/~~bzrvanit~~/~~activity~~/~~80536~~/ at home ~~std testing~~]		The '''Compton wavelength''' is a [[quantum mechanics\|quantum mechanical]] property of a particle. It was introduced by [[Arthur Compton]] in his explanation of the scattering of [[photon]]s by [[electron]]s (a process known as [[Compton scattering]]). The Compton wavelength of a particle is equivalent to the [[wavelength]] of a photon whose energy is the same as the [[Mass–energy equivalence\|rest-mass energy]] of the particle.

			The Compton wavelength, ''λ'', of a particle is given by
			: <math> \lambda = \frac{h}{m c} \ </math>
			where ''h'' is the [[Planck constant]], ''m'' is the particle's rest [[mass]], and ''c'' is the [[speed of light]]. The significance of this formula is shown in the [[Compton scattering#The Compton shift formula\|derivation of the Compton shift formula]].

			The [[CODATA]] 2010 value for the Compton wavelength of the electron is {{val\|fmt=commas\|2.4263102389\|(16)\|e=-12\|u=m}}.<ref>CODATA 2010 value for [http://physics.nist.gov/cgi-bin/cuu/Value?ecomwl\|search_for=Compton+wavelength Compton wavelength] for the electron from [[NIST]]</ref> Other particles have different Compton wavelengths.

			==Significance==
			{{Properties_of_mass}}

			===Reduced Compton wavelength===
			When the Compton wavelength is divided by <math>{2 \pi}</math>, one obtains a smaller or “reduced” Compton wavelength:

			: <math> \frac{\lambda}{2 \pi} = \frac{\hbar}{m c} \ </math>

			The reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics. The reduced Compton wavelength appears in the relativistic [[Klein–Gordon equation]] for a free particle:

			:<math> \mathbf{\nabla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi </math>

			It appears in the [[Dirac equation]] (the following is an explicitly [[Covariance and contravariance of vectors\|covariant]] form employing the [[Einstein notation\|Einstein summation convention]]):

			:<math>-i \gamma^\mu \partial_\mu \psi + \left( \frac{m c}{\hbar} \right) \psi = 0 \,</math>

			The reduced Compton wavelength also appears in [[Schrödinger's equation]], although its presence is obscured in traditional representations of the equation. The following is the traditional representation of Schrödinger's equation for an electron in a [[hydrogen-like atom]]:

			:<math> i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\nabla^2\psi -\frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r} \psi</math>

			Dividing through by <math>\hbar c</math>, and rewriting in terms of the [[fine structure constant]], one obtains:

			: <math>\frac{i}{c}\frac{\partial}{\partial t}\psi=-\frac{1}{2} \left(\frac{\hbar}{m c} \right) \nabla^2\psi - \frac{\alpha Z}{r} \psi</math>

			===Relationship between the reduced and non-reduced Compton wavelength===

			The reduced Compton wavelength is a natural representation for mass on the quantum scale. Equations that pertain to mass in the form of mass, like Klein-Gordon and Schrödinger's, use the reduced Compton wavelength. The non-reduced Compton wavelength is a natural representation for mass that has been converted into energy. Equations that pertain to the conversion of mass into energy, or to the wavelengths of photons interacting with mass, use the non-reduced Compton wavelength.

			A particle of rest mass ''m'' has a rest energy of {{nowrap\|''E'' {{=}} ''mc''<sup>2</sup>}}.
			The non-reduced Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of [[frequency]] ''f'', energy is given by
			:<math> E = h f = \frac{h c}{\lambda} = m c^2 \ </math>
			which yields the non-reduced Compton wavelength formula if solved for ''λ''.

			===Limitation on measurement===
			The reduced 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.{{citation needed\|date=June 2013}}
			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''<sup>2</sup>, 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.{{citation needed\|date=October 2013}} This
			renders moot the question of the original particle's location.

			This argument also shows that the reduced Compton wavelength is the cutoff below which [[quantum field theory]] – which can describe particle creation and annihilation – becomes important.

			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 Δ''x''.
			Then the [[Uncertainty principle\|uncertainty relation]] for position and [[momentum]] says that
			:<math>\Delta x\,\Delta p\ge \frac{\hbar}{2},</math>
			so the uncertainty in the particle's momentum satisfies
			:<math>\Delta p \ge \frac{\hbar}{2\Delta x}.</math>
			Using the [[Momentum#Momentum in relativistic mechanics\|relativistic relation between momentum and energy]] {{nowrap\|'''p''' {{=}} ''γm''<sub>0</sub>'''v'''}}, when Δ''p'' exceeds ''mc'' then the uncertainty in energy is greater than ''mc''<sup>2</sup>, which is enough [[Mass in special relativity\|energy]] to create another particle of the same type. It follows that there is a fundamental limitation on Δ''x'':
			:<math>\Delta x \ge \frac{1}{2} \left(\frac{\hbar}{mc} \right).</math>
			Thus the uncertainty in position must be greater than half of the reduced Compton wavelength ''ħ''/''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]].

			=== Relationship to other constants ===

			Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength (<math>\bar{\lambda}_e \equiv \tfrac{\lambda_e}{2\pi}\simeq 386~\textrm{fm}</math>) and the electromagnetic [[fine structure constant]] (<math>\alpha\simeq\tfrac{1}{137}</math>)

			The [[Bohr radius]] is related to the Compton wavelength by:

			:<math>a_0 = \frac{1}{\alpha}\left(\frac{\lambda_e}{2\pi}\right)\simeq 137\times\bar{\lambda}_e\simeq 5.29\times 10^4~\textrm{fm} </math>

			The [[classical electron radius]] is about 3 times larger than the proton radius, and is written:

			:<math>r_e = \alpha\left(\frac{\lambda_e}{2\pi}\right)\simeq\frac{\bar{\lambda}_e}{137}\simeq 2.82~\textrm{fm}</math>

			The [[Rydberg constant]] is written:

			:<math>R_\infty=\frac{\alpha^2}{2\lambda_e}</math>

			For [[fermion]]s, the reduced Compton wavelength sets the cross-section of interactions. For example, the cross-section for [[Thomson scattering]] of a photon from an electron is equal to

			:<math>\sigma_T = \frac{8\pi}{3}\alpha^2\bar{\lambda}_e^2 \simeq 66.5~\textrm{fm}^2 </math>

			which is roughly the same as the cross-sectional area of an iron-56 nucleus. For [[gauge theory\|gauge]] [[boson]]s, the Compton wavelength sets the effective range of the [[Yukawa interaction]]: since the [[photon]] has no rest mass, electromagnetism has infinite range.

			Typical lengths and areas in gravitational physics can be related to the Compton wavelength and the [[gravitational coupling constant]] (<math>\alpha_G</math> which is the gravitational analog of the fine structure constant):

			The [[Planck mass]] is special because the reduced Compton wavelength for this mass is equal to half of the [[Schwarzschild radius]]. This special distance is called the [[Planck length]] (<math>\ell_P</math>). 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. The Planck length is written:

			:<math>\ell_P=\lambda_e\,\frac{\sqrt{\alpha_G}}{2\pi}</math>

			==References==
			<references />

			== External links ==
			* [http://math.ucr.edu/home/baez/lengths.html#compton_wavelength Length Scales in Physics: the Compton Wavelength]

			[[Category:Atomic physics]]
			[[Category:Foundational quantum physics]]

			[[de:Compton-Effekt#Compton-Wellenlänge]]

en>777sms at 02:58, 14 April 2012

2012-04-14T02:58:02Z

New page

Nice to meet you, I am Marvella Shryock. I am a meter reader. Minnesota is exactly where he's been living for many years. The favorite hobby for my children and me is to play baseball but I haven't produced a dime with it.<br><br>Here is my blog post ... [http://conex360.com/members/bzrvanit/activity/80536/ at home std testing]

Range problem - Revision history

en>Josve05a: clean up per WP:TPL using AWB (9783)

en>777sms at 02:58, 14 April 2012