WKB approximation

I was scared by this term once before. in fact, don't panic, it is easy. Let me explain.

i just copy what written in Introduction to Quantum Mechanics by David Griffiths (1995) Chapter 8.

The approx. can be applied when the potential is varies slowly compare the wavelength of the wave function. when it expressed in Exp[ i k x], wavelength = 2 pi / k, when it expressed in Exp[ - kapper x ], wavelength = 1/kapper.

in general, the wavefunction can be expressed as amplitude and phase:

A[x] Exp[ i phase[x] ], A[x] and phase[x] are real function.

sub this into the Schrödinger equation ( Psi ''[x] = - { 2 m / hb (E - V[x])} Psi[x] ). and separate the imaginary part and real part.

The imaginary part is can be simplified as:

D[ A^2 phase' ] = 0 => A = Const. /Sqrt[ phase'[x] ]

The real part is

A''[x] = ( phase'[x]^2 -  2 m / hb (E - V[x]) ) A[x]

we use the approx. that A''[x] = 0 since it varies slowly.

Thus,

phase'[x] =  Sqrt[2 m / hb (E - V[x])]
=>
phase[x] = Integrate[ Sqrt[2 m / hb (E - V[x])], x]

and the solution is, ( if we use p[x] = Sqrt[2 m / hb (E - V[x])] )

Psi [x] = Const. /Sqrt[ p[x] ] Exp[ i Integrate[ p[x] ] ]

Simple! but one thing should keep in mind that, the WKB approx is not OK when Energy = potential.

This tell you, the phase part of the wave function is equal the square of the area of the different of Energy and the Potential.

when the energy is smaller then the potential, than, the wavefunction is under decay.

one direct application of WKB approxi is on the Tunneling effect.
if the potential is large enough, so, the transmittance is dominated by the decay, Thus, the probability of the tunneling is equal to Exp[- 2 Sqrt[ 2m / hb] Sqrt[area between V[x] and Energy] ]. Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.