Tru Physics

Problem 9.2 – Griffith’s Intro to QM

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Problem 9.2

An alternative derivation of the WKB formula (Equation 9.10) is based on an expansion in powers of \hbar. Motivated by the free-particle wave function, \psi = Ae^{\pm ipx/\hbar}, we write

\psi(x) = e^{if(x)/\hbar},

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 9.1), and
show that

i\hbar f''-(f')^2+p^2=0.

(b) Write as a power series in \hbar :

f(x) = f_0(x) + \hbar f_1(x) + \hbar^2 f_2(x) +...,

and, collecting like powers of \hbar , show that

(f_0')^2=p^2 \text{, } if_0''=2f_0'f_1' \text{, } if_1'' = 2f_0'f_2' + (f_1')^2 \text{, etc.}

(c) Solve for f_0(x) and f_1(x), and show that—to first order in \hbar—you recover Equation 9.10.

Note: The logarithm of a negative number is defined by \ln(-z)=\ln(z) + in\pi , 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.

Solution:

Problem 9.2 - Griffith's Intro to QM 1 of 2
Problem 9.2 - Griffith's Intro to QM 2 of 2

Problem 9.2 Solution (Download)

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