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ACM 40690 Survey of Applied and Computational Mathematics
WKB Assignment
1. In the lecture we assumed that our differential equation was in the ‘standard form’
d2 y
− V (t)y = 0.
dt2
This question shows how we may do that: starting from the differential equation
dz
d2 z
+ p(x) + q(x)z = 0,
2
dx
dx
we need to change both the independent and dependent variables.
(a) Show that a change of independent variable to x = x(t) leads to the equation
x00 dz
d2 z
0
+ p.x − 0
+ q.(x0 )2 y = 0,
dt2
x
dt
where . denotes that we mean multiplication not function argument.
(b) Show that a subsequent change of dependent variable to
Z
1
0
00
0
y = z exp
(p.x − x /x ) dt ,
2
leads to a differential equation of the standard form with
V (t) = 14 p2 − q (x0 )2 + 21 f 0 x0 + 34 (x00 /x0 )2 − 12 x000 /x0
where 0 always denotes differentiation with respect to t.
2.
(a) Show that the large eigenvalues of
y 00 + λ2 q(t)y = 0,
are given by λn = nπ
y(−1) = 0,
y(1) = 0
y(−1) = 0,
y 0 (1) = 0
i−1
hR p
1
q(s)
ds
.
−1
(b) Show that the large eigenvalues of
y 00 + λ2 q(t)y = 0,
are given by λn = (n + 21 )π
hR p
i−1
1
.
q(s)
ds
−1
(c) Find the large eigenvalues of
y 00 (t) + λ2 e2t y = 0,
using
y(−1) = 0,
y(1) = 0
(i) the WKB approximation to terms of order λ0
(ii) the higher order WKB approximation to terms of order λ−1
Compute the relative error for λn given the ‘exact’ values are
n
1
2
3
λn
1.282308147
2.637191061
3.983144275
n
10
20
30
λn
13.35704083
26.72773278
40.09546303
4
5
5.325363969 6.665693559
40
50
53.46242832 66.82908492
3. The Airy differential equation is
d2 y
− xy(x) = 0.
dx2
(†)
Note: that this just corresponds to Schr¨odinger equation a particle in a uniform gravitational field
with V (x) ∝ x for a particle with total energy 0; the region x > 0 is then classically disallowed so
quantum mechanically we expect exponential decay while the region x < 0 is classically allowed
so quantum mechanically we expect oscillatory solution.
(a) Conduct the rescaling x = λ2/3 t (where λ ∈ R+ ) and show that Equation (†) becomes
y 00 (t) − λ2 ty(t) = 0.
(b) Show that the corresponding WKB approximation to order λ−1 yields
y ∼ t−1/4 exp ±
5 −3/2
λt3/2 + λ−1 48
t
3
2
,
λ → ∞.
(c) Note that for fixed t as λ → ∞ so |x| → ∞ to find the asymptotic solution to the Airy
equation as |x| → ∞.
(d) Identify the Stokes lines for these solutions (identified informally in this course as the lines
along which the ±-solutions are of equal magnitude).
4. Find the WKB approximation to the eigenvalues of normalisable functions satisfying
y 00 (x) + (E − |x|)y(x) = 0.
(a) Using the results of Section 15.5 compute the WKB approximation to the allowed energy
levels:
2/3
3π
1
En =
n+ 2
.
4
(b) Letting z = E − x for x > 0 and the corresponding transformation for x < 0 show that the
exact solution is given by

α Ai(−x − E) x < 0
y(x) =
β Ai(x − E)
x > 0.
(c) Deduce that the exact eigenvalues fall into two classes:
(i) α = β and Ai0 (−E) = 0,
(ii) α = −β and Ai(−E) = 0.
(d) Show that using the following asymptotic expression, derived from Question 3 to order λ0 ,
in Part (c) reproduces the results of Part (a).:
2 3/2 π
−1/4
E +
,
E→∞
Ai(−E) ∼ E
sin
3
4
4pm Thursday, April 23rd 2015