The
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
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
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
The reduced
∇ 2 ψ − 1 c 2 ∂ 2 ∂ t 2 ψ = ( m c ℏ ) 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
− i γ μ ∂ μ ψ + ( m c ℏ ) ψ = 0. displaystyle -igamma ^ mu partial _ mu psi +left( frac mc hbar right)psi =0. The reduced
i ℏ ∂ ∂ t ψ = − ℏ 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 ( ℏ 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
E = h f = h c λ = m c 2 , displaystyle E=hf= frac hc lambda =mc^ 2 , which yields the non-reduced or standard
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 ≥ ℏ 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 ( ℏ 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
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
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
R ∞ = α 2 2 λ e displaystyle R_ infty = frac alpha ^ 2 2lambda _ e For fermions, the reduced
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
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
m P = ℏ c / G displaystyle m_ rm P = sqrt hbar c/G is special because the
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
l P displaystyle l_ rm P ). This is a simple case of dimensional analysis: the Schwarzschild
radius is proportional to the mass, whereas the
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] ^
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 |