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| In [[theoretical physics]], the '''eikonal approximation''' ([[Greek language|Greek]] εἰκών for likeness, icon or image) is an approximative method useful in wave scattering equations which occur in [[optics]], [[quantum mechanics]], [[quantum electrodynamics]], and [[Scattering amplitude#Partial wave expansion|partial wave expansion]].
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| ==Informal description==
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| The main advantage the eikonal approximation offers is that the equations reduce to a [[differential equation]] in a single variable. This reduction into a single variable is the result of the straight line approximation or the eikonal approximation which allows us to choose the straight line as a special direction.
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| ==Relation to the WKB approximation==
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| The early steps involved in the eikonal approximation in quantum mechanics are very closely related to the [[WKB approximation]]. It, like the eikonal approximation, reduces the equations into a differential equation in a single variable. But the difficulty with the WKB approximation is that this variable is described by the trajectory of the particle which, in general, is complicated.
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| ==Formal description==
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| Making use of WKB approximation we can write the wave function of the scattered system in term of [[action (physics)|action]] ''S'':
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| :<math>\Psi=e^{iS/{\hbar}} </math> | |
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| Inserting the [[wavefunction]] Ψ in the [[Schrödinger equation]] we obtain
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| :<math> -\frac{{\hbar}^2}{2m} {\nabla}^2 \Psi= (E-V) \Psi</math>
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| :<math> -\frac{{\hbar}^2}{2m} {\nabla}^2 {e^{iS/{\hbar}}}=(E-V) e^{iS/{\hbar}}</math>
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| :<math>\frac{1}{2m} {(\nabla S)}^2 - \frac{i\hbar}{2m}{\nabla}^2 S= E-V</math>
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| We write ''S'' as a ''ħ'' [[power series]]
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| :<math>S= S_0 + \frac {\hbar}{i} S_1 + ...</math>
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| For the zero-th order:
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| :<math>{(\nabla S_0)}^2= E-V</math>
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| If we consider the one-dimensional case then <math>{\nabla}^2 \rightarrow {\delta_z}^2</math>.
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| We obtain a [[differential equation]] with the [[Boundary value problem|boundary condition]]:
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| :<math>\frac{S(z=z_0)}{\hbar}= k z_0</math>
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| for ''V'' → 0, ''z'' → -∞.
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| :<math>\frac{d}{dz}\frac{S_0}{\hbar}= \sqrt{k^2 - 2mV/{\hbar}^2}</math>
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| :<math>\frac{S_0(z)}{\hbar}= kz - \frac{m}{{\hbar}^2 k} \int_{-\infty}^{Z}{V dz'} </math>
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| ==See also==
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| * [[Eikonal equation]]
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| * [[Correspondence principle]]
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| * [[Principle of least action]]
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| ==References==
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| ===Notes===
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| * [http://www.nhn.ou.edu/~shajesh/eikonal/sp.pdf]''Eikonal Approximation'' K. V. Shajesh Department of Physics and Astronomy, University of Oklahoma
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| ===Further reading===
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| * {{cite book|title=Comparison of exact solution with Eikonal approximation for elastic heavy ion scattering|edition=3rd|author=R.R. Dubey|location=|publisher=NASA|year=1995 |isbn=|url = http://books.google.co.uk/books?id=NwgVAQAAIAAJ&q=Eikonal+approximation&dq=Eikonal+approximation&hl=en&sa=X&ei=LCnkUOP8HfDa0QW-34GIBA&ved=0CDwQ6AEwAQ}}
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| * {{cite book|title=Eikonal approximation in partial wave version|edition=3rd|author=W. Qian, H. Narumi, N. Daigaku. P. Kenkyūjo|location=Nagoya|publisher=|year=1989|isbn=|url = http://books.google.co.uk/books?id=5RdRAAAAMAAJ&q=Eikonal+approximation&dq=Eikonal+approximation&hl=en&sa=X&ei=LCnkUOP8HfDa0QW-34GIBA&ved=0CDYQ6AEwAA}}
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| *{{cite article
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| | author = M. Lévy, J. Sucher
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| | year = 1969
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| | location = Maryland, USA
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| | publisher =
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| | journal = Phys. Rev
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| | title = Eikonal Approximation in Quantum Field Theory
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| | arxiv =
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| | url = http://prola.aps.org/abstract/PR/v186/i5/p1656_1
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| | doi = 10.1103/PhysRev.186.1656
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| }}
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| *{{cite article
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| | author = I. T. Todorov
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| | year = 1970
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| | location = New Jersey, USA
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| | publisher =
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| | journal = Phys. Rev D
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| | title = Quasipotential Equation Corresponding to the Relativistic Eikonal Approximation
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| | arxiv =
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| | url = http://prd.aps.org/abstract/PRD/v3/i10/p2351_1
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| | doi = 10.1103/PhysRevD.3.2351
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| }}
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| *{{cite article
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| | author = D.R. Harrington
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| | year = 1969
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| | location = New Jersey, USA
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| | publisher =
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| | journal = Phys. Rev
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| | title = Multiple Scattering, the Glauber Approximation, and the Off-Shell Eikonal Approximation
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| | arxiv =
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| | url = http://prola.aps.org/abstract/PR/v184/i5/p1745_1
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| | doi = 10.1103/PhysRev.184.1745
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| }}
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| [[Category:Theoretical physics]]
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| [[Category:Mathematical analysis]]
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| {{applied-math-stub}}
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| {{Quantum-stub}}
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