Phys. Dept/IITK D.
Chakrabarti Date: November 10, 2023
PHY 421A ( Mathematical Methods - I)
Problem set-8
1. A quantum mechanical analysis of the Stark effect (in parabolic coordinates)
leads to the differential equation
d du 1 m2 1 2
z + Ez + α − − F z u = 0.
dz dz 2 4z 4
Here α is a constant, E is the total energy, and F is a constant such that F x
is the potential energy added to the system by the introduction of the electric
field. Using the larger root of the indicial equation, develop a power-series
solution about z = 0. Evaluate the first three coefficients in terms of a0 .
2. For the special case of no azimuthal dependence the quantum mechanical anal-
ysis of the hydrogen molecular ion leads to the equation
dh du i
(1 − η 2 ) + αu + βη 2 u = 0.
dη dη
Develop a power series solution for u(η). Evaluate the first three non-vanishing
coefficients in terms of a0 .
3. Given that one solution of
1 m2
R00 + R0 − 2 R = 0
r r
is R = rm , show that the Wronskian method predicts a second solution, R =
r−m .
4. One solution of the Chebyshev equation
(1 − x2 )y 00 − xy 0 + n2 y = 0
for n = 0 is y1 = 1.
(a) Using Wronskian, develop a second, linearly independent solution.
(b) Find a second solution by direct integration of the Chebyshev equation.
(c) For n = 1, one solution is given by y1 (x) = x, find the second solution by
Wronskian double integral.
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Phys. Dept/IITK D. Chakrabarti Date: November 10, 2023
5. The Laguerre’s equation is given by
xy 00 + (1 − x)y 0 + ny = 0.
Show that for n = 0, only one series solution is possible. Find the second
solution by Wronskian double integral method.
6. Consider the 3D Helmholtz equation in spherical polar coordinates:
[∇2 + k02 ]ψ(r, θ, φ) = 0. (k0 = constant)
using the separation of variables, construct the equations for r, θ and φ. Find a
series solution for the differential equation in θ when ψ has no φ dependence.[Set
the radial and angular parts of the equations are equal to l(l + 1) with l =
integer.]
7. Find a particular solution for
(a) y 00 + y 0 − 4y = e−x (1 − 8x2 ),
d
(b) (D − 1)2 y = ex . (D = )
dx
d2 x dx
(c) M 2 + c + kx = F0 cos(ωt)
dt dt
8. Find the general solution to
y 00 + 9y = 3 tan(3t).
9. Find the Green’s function for the 2D Laplace operator bounded in a region of
radius R.
10. Find the ranges of integration that guarantee that the Hermitian operator
boundary conditions are satisfied for (a) Legendre equation (b) Hermite equa-
tion and (c) Laguerre equantion.
11. A set of functions un (x) satisfies the Sturm-Liouville equation
dh d i
p(x) un (x) + λn w(x)un (x) = 0.
dx dx
The functions um (x) and un (x) satisfy boundary conditions that lead to orthog-
onality. The corresponding eigenvalues λm and λn are distinct. Prove that for
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Phys. Dept/IITK D. Chakrabarti Date: November 10, 2023
appropriate boundary conditions, u0m (x) and u0n (x) are orthogonal with p(x) as
a weighting function.
12. The ultraspherical polynomials Cnα (x) are solutions of the differential equation
n d2 d o
(1 − x ) 2 − (2α + 1)x + n(n + 2α) Cnα (x) = 0.
2
dx dx
(a) Transform this differential equation into self-adjoint form.
(b) Find an interval of integration and weighting factor that make Cnα (x) of the
same α but different n orthogonal.
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