An alternative derivation of the WKB formula (Equation 9.10) is based on an expansion in powers of ћ. Motivated by the free-particle wave function, Ψ = A exp (±ipx/ћ), we write

where f(x) is some complex function.
(a) Put this into Schrödinger’s equation (in the form of Equation 9.1), and show that

(b) Write f(x) as a power series in ћ:

and, collecting like powers of ћ, show that

(c) Solve for f₀(x) and f₁(x), and show that—to first order in ћ —you recover Equation 9.10.

Transcribed Image Text:

(x) = f(x)/h