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The Compton wavelength
Compton wavelength
is a quantum mechanical property of a particle. It was introduced by Arthur Compton
Arthur Compton
in his explanation of the scattering of photons by electrons (a process known as Compton scattering). The Compton wavelength
Compton wavelength
of a particle is equal to the wavelength of a photon whose energy is the same as the mass (see mass-energy equivalence) of that particle. The standard Compton wavelength, λ, of a particle is given by

λ =

h

m c

,  

displaystyle lambda = frac h mc ,

where h is the Planck constant, m is the particle's mass, and c is the speed of light. The significance of this formula is shown in the derivation of the Compton shift formula. The CODATA
CODATA
2014 value for the Compton wavelength
Compton wavelength
of the electron is 6988242631023670000♠2.4263102367(11)×10−12 m.[1] Other particles have different Compton wavelengths.

Contents

1 Reduced Compton wavelength 2 Role in equations for massive particles 3 Relationship between the reduced and non-reduced Compton wavelength 4 Limitation on measurement 5 Relationship to other constants 6 References 7 External links

Reduced Compton wavelength[edit] When the Compton wavelength
Compton wavelength
is divided by 2π, one obtains the "reduced" Compton wavelength
Compton wavelength
ƛ (barred lambda), i.e. the Compton wavelength for 1 radian instead of 2π radians:

ƛ = λ/2π = ħ/mc,

where ħ is the "reduced" Planck constant. Role in equations for massive particles[edit]

The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties. However, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object.

The Schwarzschild radius
Schwarzschild radius
(rs) represents the ability of mass to cause curvature in space and time. The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies. Inertial mass (m) represents the Newtonian response of mass to forces. Rest energy (E0) represents the ability of mass to be converted into other forms of energy. The Compton wavelength
Compton wavelength
(λ) represents the quantum response of mass to local geometry.

The reduced Compton wavelength
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
Compton wavelength
appears in the relativistic Klein–Gordon equation for a free particle:

2

ψ −

1

c

2

2

t

2

ψ =

(

m c

)

2

ψ .

displaystyle mathbf nabla ^ 2 psi - frac 1 c^ 2 frac partial ^ 2 partial t^ 2 psi =left( frac mc hbar right)^ 2 psi .

It appears in the Dirac equation
Dirac equation
(the following is an explicitly covariant form employing the Einstein summation convention):

− i

γ

μ

μ

ψ +

(

m c

)

ψ = 0.

displaystyle -igamma ^ mu partial _ mu psi +left( frac mc hbar right)psi =0.

The reduced Compton wavelength
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
Schrödinger's equation
for an electron in a hydrogen-like atom:

i ℏ

∂ t

ψ = −

2

2 m

2

ψ −

1

4 π

ϵ

0

Z

e

2

r

ψ .

displaystyle ihbar frac partial partial t psi =- frac hbar ^ 2 2m nabla ^ 2 psi - frac 1 4pi epsilon _ 0 frac Ze^ 2 r psi .

Dividing through by

ℏ c

displaystyle hbar c

, and rewriting in terms of the fine structure constant, one obtains:

i c

∂ t

ψ = −

1 2

(

m c

)

2

ψ −

α Z

r

ψ .

displaystyle frac i c frac partial partial t psi =- frac 1 2 left( frac hbar mc right)nabla ^ 2 psi - frac alpha Z r psi .

Relationship between the reduced and non-reduced Compton wavelength[edit] The reduced Compton wavelength
Compton wavelength
is a natural representation for mass on the quantum scale. Equations that pertain to inertial mass like Klein-Gordon and Schrödinger's, use the reduced Compton wavelength. The non-reduced Compton wavelength
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 mass m has a rest energy of E = mc2. The non-reduced Compton wavelength
Compton wavelength
for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by

E = h f =

h c

λ

= m

c

2

,  

displaystyle E=hf= frac hc lambda =mc^ 2 ,

which yields the non-reduced or standard Compton wavelength
Compton wavelength
formula if solved for λ. Limitation on measurement[edit] The Compton wavelength
Compton wavelength
expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity.[2] This limitation depends on the mass m of the particle. To see how, 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 mc2, when one hits the particle whose position is being measured the collision may yield enough energy to create a new particle of the same type.[citation needed] This renders moot the question of the original particle's location. This argument also shows that the reduced Compton wavelength
Compton wavelength
is the cutoff below which quantum field theory – which can describe particle creation and annihilation – becomes important. The above argument can be made a bit more precise as follows. Suppose we wish to measure the position of a particle to within an accuracy Δx. Then the uncertainty relation for position and momentum says that

Δ x

Δ p ≥

ℏ 2

,

displaystyle Delta x,Delta pgeq frac hbar 2 ,

so the uncertainty in the particle's momentum satisfies

Δ p ≥

2 Δ x

.

displaystyle Delta pgeq frac hbar 2Delta x .

Using the relativistic relation between momentum and energy E2 = (pc)2 + (mc2)2, when Δp exceeds mc then the uncertainty in energy is greater than mc2, which is enough energy to create another particle of the same type. But we must exclude this. In particular the minimum uncertainty is when the scattered photon has limit energy equal to the incident observing energy. It follows that there is a fundamental minimum for Δx:

Δ x ≥

1 2

(

m c

)

.

displaystyle Delta xgeq frac 1 2 left( frac hbar mc right).

Thus the uncertainty in position must be greater than half of the reduced Compton wavelength
Compton wavelength
ħ/mc. The Compton wavelength
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[edit] Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength
Compton wavelength
for the electron (

λ ¯

e

λ

e

2 π

≃ 386  

fm

displaystyle bar lambda _ e equiv tfrac lambda _ e 2pi simeq 386~ textrm fm

) and the electromagnetic fine structure constant (

α ≃

1 137

displaystyle alpha simeq tfrac 1 137

). The Bohr radius is related to the Compton wavelength
Compton wavelength
by:

a

0

=

1 α

(

λ

e

2 π

)

≃ 137 ×

λ ¯

e

≃ 5.29 ×

10

4

 

fm

displaystyle a_ 0 = frac 1 alpha left( frac lambda _ e 2pi right)simeq 137times bar lambda _ e simeq 5.29times 10^ 4 ~ textrm fm

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

r

e

= α

(

λ

e

2 π

)

λ ¯

e

137

≃ 2.82  

fm

displaystyle r_ e =alpha left( frac lambda _ e 2pi right)simeq frac bar lambda _ e 137 simeq 2.82~ textrm fm

The Rydberg constant is written:

R

=

α

2

2

λ

e

displaystyle R_ infty = frac alpha ^ 2 2lambda _ e

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

σ

T

=

8 π

3

α

2

λ ¯

e

2

≃ 66.5  

fm

2

displaystyle sigma _ T = frac 8pi 3 alpha ^ 2 bar lambda _ e ^ 2 simeq 66.5~ textrm fm ^ 2

which is roughly the same as the cross-sectional area of an iron-56 nucleus. For gauge bosons, the Compton wavelength
Compton wavelength
sets the effective range of the Yukawa interaction: since the photon has no mass, electromagnetism has infinite range. Typical lengths and areas in gravitational physics can be related to the Compton wavelength
Compton wavelength
and the gravitational coupling constant

α

G

=

G

m

e

2

ℏ c

=

m

e

2

/

m

P

2

displaystyle alpha _ text G = frac Gm_ text e ^ 2 hbar c =m_ text e ^ 2 /m_ text P ^ 2

, which is the gravitational analog of the fine structure constant. The Planck mass

m

P

=

ℏ c

/

G

displaystyle m_ rm P = sqrt hbar c/G

is special because the Compton wavelength
Compton wavelength
and the Schwarzschild radius

r

S

= 2 G M

/

c

2

displaystyle r_ rm S =2GM/c^ 2

for this mass are of the same order. Their value is close to the Planck length (

l

P

displaystyle l_ rm P

). This is a simple case of dimensional analysis: the Schwarzschild radius is proportional to the mass, whereas the Compton wavelength
Compton wavelength
is proportional to the inverse of the mass. The Planck length is written:

l

P

=

ℏ G

/

c

3

=

λ

e

α

G

2 π

displaystyle l_ rm P = sqrt hbar G/c^ 3 =lambda _ e , frac sqrt alpha _ G 2pi

References[edit]

^ CODATA
CODATA
20104 value for Compton wavelength
Compton wavelength
for the electron from NIST ^ Garay, Luis J. (1995). "Quantum Gravity And Minimum Length". International Journal of Modern Physics A. 10 (2): 145–65. arXiv:gr-qc/9403008 . Bibcode:1995IJMPA..10..145G. doi:10.1142/S0217751X95000085. 

External links[edit]

Length Scales in Physics: the Compton Wavelength B.G. Sidharth, Planck scale to Compton scale, International Institute for Applicable Mathematics, Hyderabad (India) & Udine (Italy

.