PHYS 201 Mathematical Physics, Fall 2015, Final
Due date: Friday, December 11th, 2015
Rules: Open book but without help from another person.
1. (5pts) Consider a Schrodinger’s equation
2
d2 ψ
k − 1 2n + k + 1 1
+
ψ(x) = 0
−
−
dx2
4x2
2x
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where n, k are non-negative integers with the boundary conditions ψ(0) = ψ(∞) = 0.
Find the asymptotic behavior as x → 0+ using methods you learned in class. You may
leave the coefficients of the asymptotic series as a recurrence relation. (Optional: Find
the leading behavior as x → ∞ (+4pts))
2. (7pts) We can find the full asymptotic behavior of certain integrals by finding the
differential equation that they satisfy and then perform an asymptotic analysis of the
differential equation. Find the differential equation satisfied by
Z ∞
e−xt−1/t dt
y(x) =
0
and then find the leading behavior of y(x) as x → ∞ (you may ignore constant prefactors). (Optional: Find the full asymptotic behavior as x → ∞ (+4pts)) (Hint: The
differential equation is second order; integration by parts will be helpful in obtaining it.
Use the tricks from Homework 6 (13,14) to analyze the behavior of differential equations as x → ∞. You can check the answer with the leading behavior obtained using
Laplace’s method.)
3. (8pts) Consider the one-turning-point problem
2 y 00 (x) = Q(x)y(x)
where Q(x) vanishes just once at x = 0 and Q(x) ∼ a2 x2 , a > 0 when |x| 1. Find
the WKB solution y(x) as → 0 up to a constant normalization factor such that
y(±∞) = 0. Just like when Q(x) is linear near the origin, the solution can be split into
three regions - a region close to the turning point x = 0 and two symmetric (in this
case) regions away from x = 0. You may follow these steps:
i. Find the exponentially decaying physical optics WKB approximation in the regions
away from x = 0. Establish more precisely (in terms of ) the region where this
approximation is valid.
ii. Find, in terms of parabolic cylinder functions, the solution close to x = 0. Again
establish the region of validity for this approximation.
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iii. Match the prefactors in the region where both approximations are valid and find
the solution up to a constant normalization factor.
(Hint: The asymptotic form of the parabolic cylinder functions was discussed in one
of the previous homeworks. You may also look up the derivation for the standard case
of Q(x) ∼ ax, |x| 1 from BO for hints on how to do this)
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