Wenzel-Kramers-Brillouin Method

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

2 -ℏ--∂2xψ(x)+ (E - V(x))ψ(x) = 0 (A.14) 2me

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 $ℏ∕i$

2 S (x) = S0(x) + (iℏ) S1(x) + (iℏ) S2(x) + ... , (A.17)

one obtains

( ∫x ) S0(x) = exp (± i- p(x′)dx′) , (A.18) ℏ 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

1 S1(x) = 2ln|p(x)|+ c (A.21)

which yields

( x ) c i ∫ ′ ′ ψ(x) = ∘-|p(x)| exp (± ℏ p(x)dx ) , (A.22) x0

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 $V(x)$ .

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