An illuminating alternative derivation of the WKB formula (Equation 1) is based on an expa...

An illuminating alternative derivation of the WKB formula (Equation 1) 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. (Note that there is no loss of generality here—any nonzero function can be written in this way.)

(a) Put this into Schrödinger's equation (in the form of Equation 8.1), and shov that

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

and, collecting like powers of ћ, show that

 etc.

(c) Solve for f0(x) and f1(x), and show that—to first order in ћ—you recover Equation 1.

Note: The logarithm of a negative number is defined by ln(‒z) = ln(z) + inπ, where n is an odd integer. If this formula is new to you, try exponentiating both sides, and you'll see where it comes from.

Equation 1