1. Consider the Sturm-Liouville equation dz (p(x)du) — q(x)u + \p(x)u = 0 dx on a < x 0 and q > 0 for a < x satisfies sin λ = p(t) p(t) n²π² p(t) (Sa √ de with n an integer. Ideally, you use the WKB ansatz. 2' dt 7

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.5: Systems Of Linear Equations In More Than Two Variables
Problem 43E
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1. Consider the Sturm-Liouville equation
dz (p(x)du) — q(x)u + \p(x)u = 0
dx
on a < x <b, with p > 0 and q > 0 for a < x <b. The boundary conditions are
u(a) = u(b) = 0.
(a) Show that the change of variables u = w/√p leads to the equation
2
1 d
+ [ - ²2 (1) - () ² + x² - 1 - = 0
(²
w=0
dx
P P
d² w
dx²
with w(a) = w(b) = 0.
(b) Show that for A large, the solution of the problem in part (a) is
(VTS)
-1/4
W =
()
where the eigenvalue > satisfies
sin
λ =
p(t)
p(t)
n²π²
p(t)
(Sa √ de
with n an integer. Ideally, you use the WKB ansatz.
2'
dt 7
Transcribed Image Text:1. Consider the Sturm-Liouville equation dz (p(x)du) — q(x)u + \p(x)u = 0 dx on a < x <b, with p > 0 and q > 0 for a < x <b. The boundary conditions are u(a) = u(b) = 0. (a) Show that the change of variables u = w/√p leads to the equation 2 1 d + [ - ²2 (1) - () ² + x² - 1 - = 0 (² w=0 dx P P d² w dx² with w(a) = w(b) = 0. (b) Show that for A large, the solution of the problem in part (a) is (VTS) -1/4 W = () where the eigenvalue > satisfies sin λ = p(t) p(t) n²π² p(t) (Sa √ de with n an integer. Ideally, you use the WKB ansatz. 2' dt 7
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