Eikonal approximation

Relation to the WKB approximation

The early steps involved in the eikonal approximation in quantum mechanics are very closely related to the WKB approximation for one-dimensional waves. The WKB method, 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.

Formal description

Making use of WKB approximation we can write the wave function of the scattered system in terms of action S:

${\displaystyle \mathrm{\Psi }=e^{iS/\hslash }}$

Inserting the wavefunction Ψ in the Schrödinger equation without the presence of a magnetic field we obtain

${\displaystyle -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }=(E-V)\mathrm{\Psi }}$

${\displaystyle -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2{e}^{iS/\hslash }=(E-V)e^{iS/\hslash }}$

${\displaystyle \frac{1}{2m}{(\mathrm{\nabla }S)}^2-\frac{i\hslash }{2m}{\mathrm{\nabla }}^{2}S=E-V}$

We write S as a power series in ħ

${\displaystyle S=S_0+\frac{\hslash }{i}{S}_1+...}$

For the zero-th order:

${\displaystyle \frac{1}{2m}{(\mathrm{\nabla }{S}_0)}^2=E-V}$

If we consider the one-dimensional case then ${\displaystyle {\mathrm{\nabla }}^2\to {{\mathrm{\partial }}_z}^2}$.

We obtain a differential equation with the boundary condition:

${\displaystyle \frac{S(z=z_0)}{\hslash }=k{z}_0}$

for ${\displaystyle V\to 0}$, ${\displaystyle z\to -\mathrm{\infty }}$.

${\displaystyle \frac{d}{dz}\frac{S_0}{\hslash }=\sqrt{k^2-2mV/{\hslash }^2}}$

${\displaystyle \frac{S_0(z)}{\hslash }=kz-\frac{m}{{\hslash }^{2}k}{\int }_{-\mathrm{\infty }}^{z}Vd{z}^{\prime }}$