WKB Homework #1 for Math 605
Due: March 3, 2005
It may be helpful to read Bender and Orszag, Chapter 10, on reserve.
1. The position of a particle in a potential V (x) with −V 0 (x) = −x3 +x and driven by Brownian
motion with variance 22 1, has a probability density function p(x) which satisfies
2 pxx + ((x3 − x)p)x = 0,
−∞ < x < ∞,
Use a WKB expansion to give the leading order approximation
to the steady-state probability
R∞
density p(x). Note that p(x) must be normalized so that −∞ p(x)dx = 1. Give a physical
interpretation of your result, keeping in mind that p(x) is a probability density.
2. Find the leading order contribution to the solution of
utt + ω 2 (t)u = −(ut )3 ,
1
Give the amplitude in terms of an equation (you don’t have to solve it).
3. Consider Schrödinger’s equation with potential V (x) = 1 + e−x and energy level E 6= 1
−2 ψxx + (V (x) − E)ψ(x) = 0
a) Determine any turning points
b) Find the leading order approximation to the solution on either side of any turning point(s),
as well as in the vicinity of the turning point(s).