A.2 Wenzel-Kramers-Brillouin Method

The WKB method [102] is an approximative semiclassical approach to compute the stationary solution of the Schrödinger equation without struggling with the difficulties of a second order differential equation. Taking the time-independent, one-dimensional Schrödinger equation

-ℏ--∂2xψ(x)+ (E - V(x))ψ(x) = 0              (A.14)
as a starting point and inserting the ansatz
          ( S(x))
ψ(x) = exp i-ℏ--                      (A.15)
leads to
       2                      2
(∂xS(x)) = 2me (E - V (x ))+ iℏ ∂xS(x) .          (A.16)
Substituting the action S(x)  by its expansion in power series of ℏ∕i
S (x) = S0(x) + (iℏ) S1(x) + (iℏ)  S2(x) + ... ,      (A.17)
one obtains
            (    ∫x       )
S0(x)  = exp (± i-  p(x′)dx′) ,            (A.18)
    2           x0
p(x)   ≡ 2me (E - V(x))                   (A.19)
for terms of the order ~ ℏ0  and
         1∂2S0(x)   1∂x (p(x)∕ℏ)
∂xS1(x) = 2∂xS-(x) = 2-(p(x)∕ℏ)-.             (A.20)
           x 0
for terms of the order    1
~ ℏ  . Integrating (A.20), one obtains
S1(x) = 2ln|p(x)|+ c                   (A.21)
which yields
                 (     x       )
         c          i ∫    ′  ′
ψ(x) = ∘-|p(x)| exp (± ℏ  p(x)dx ) ,            (A.22)
where the integration spans from x0  to an arbitrary point x  . x0  is also referred to as the classical turning point, where the particle energy E  equals the potential energy V (x)  . Note that close to this point, the WKB approximation breaks down and the expression for the wavefunction diverges since p(x)  in the denominator approaches zero. As a result, the wavefunction left and right to this point cannot be adjusted, which is the case at the discontinuity of the semiconductor-dielectric interface for instance. One way to overcome this problem is to apply Langer’s procedure [102], which is not presented here. The above formula also applies to classical forbidden regions where the particle energy E  lies below the potential barrier V(x)  .