Mathematical Methods MT2017: Problems 4
John Magorrian, john.magorrian@physics.ox.ac.uk
(mostly recycled from Fabian Essler’s MT2009 problems)
1. Orthogonality
Suppose that the functions Ψ0 (x), Ψ1 (x), Ψ2 (x), ... are orthogonal over the interval [a, b] with respect
to the weight function w(x). Show that the Ψn (x) are linearly indepdendent.
2. Orthogonal, normalised eigenfunctions
The real functions un (x) (n = 1 to ∞) are an orthogonal, normalised set on the interval (a, b) with
weight function w(x) = 1. The function f (x) is expressed as a linear combination of the un (x) via
∞
X
f (x) = an un (x). (Q2.1)
n=1
Show that
(i)
Z b
an = un (x)f (x) dx; (Q2.2)
a
(ii)
Z b ∞
X
[f (x)]2 dx = a2n . (Q2.3)
a n=1
[Hint for part (ii): writing out the left-hand side in long-hand notation gives
Z b
(a1 u1 (x) + a2 u2 (x) + · · ·)(a1 u1 (x) + a2 u2 (x) + · · ·) dx
a
Z b
(Q2.4)
2
a1 [u1 (x)]2 + a22 [u2 (x)]2 + · · · + 2a1 a2 u1 (x)u2 (x) + · · · dx.
=
a
2
R R
Why do the [un (x)] dx terms each give 1? Why do the un (x)um (x) dx terms with n 6= m each give
0?]
3. Eigenvalues and eigenfunctions
By substituting x = et , find the normalized eigenfunctions yn (x) and the eigenvalues λn of the operator
L̂ defined by
1
L̂y = x2 y 00 + 2xy 0 + y, 1 ≤ x ≤ e, (Q3.1)
4
with boundary conditions y(1) = y(e) = 0.
4. Hermiticity
Consider the set of functions {f (x)} of the real variable x defined on the interval −∞ < x < ∞ that go
to zero faster than 1/x for x → ±∞, i.e.,
lim xf (x) = 0. (Q4.1)
x→±∞
For unit weight function, determine which of the following linear operators is Hermitian when acting
d d d d3
upon {f (x)}: (a) dx + x (b) −i dx + x2 (c) ix dx (d) i dx 3.
1
5. More Hermiticity
Recall that an operator A is Hermitian if hu| A |vi = hv| A |ui? , or, equivalently,
"Z #?
Z b b Z b
? ? ?
u (x) [Av(x)] w(x)dx = v (x) [Au(x)] w(x)dx = [Au(x)] v(x) w(x)dx. (Q5.1)
a a a
The dual A† of the operator A is defined such that hu| A† |vi = hv| A |ui? , or, equivalently
Z b Z b
?
u (x) A† v(x) w(x)dx =
?
[Au(x)] v(x) w(x)dx. (Q5.2)
a a
(a) Let A be a non-Hermitian operator. Show that A + A† and i(A − A† ) are Hermitian operators.
(b) Using the preceding result, show that every non-Hermitian operator may be written as a linear
combination of two Hermitian operators.
6. Sturm–Liouville Problem
The equation
L̂y(x) = λy(x) (Q6.1)
is a Sturm–Liouville equation for the operator
1 d d
L̂ = p(x) + q(x) , (Q6.2)
w(x) dx dx
where p(x), q(x) and w(x) are real functions with w(x) > 0. Any two real solutions yn (x), ym (x) with
distinct eigenvalues λn , λm satisfy the boundary condition
dyn dyn
ym p = y m p . (Q6.3)
dx x=a dx x=b
Without assuming any results proved in lectures, show directly from equations (Q6.1) to (Q6.3) that
Z b
yn (x)ym (x) w(x)dx = 0 (Q6.4)
a
when n 6= m.
7. Express the differential equation
xy 00 + (k + 1 − x)y 0 = λy, (Q7.1)
where k is a constant, as a Sturm–Liouville equation. What are the natural limits (a, b) to place on x
to satisfy the Sturm–Liouville boundary conditions?
8. Quantum harmonic oscillator
Consider the time-independent Schrödinger equation for the quantum harmonic oscillator
Hψ(x) = Eψ(x),
~2 d2 1 (Q8.1)
H=− 2
+ mω 2 x2 .
2m dx 2
2
p mω
(a) Using the substitutions y = x ~ and = E/~ω reduce the Schrödinger equation to
d2
Ψ(y) + (2 − y 2 )Ψ(y) = 0. (Q8.2)
dy 2
(b) Consider the limit y → ∞ and verifty that in this limit
2
Ψ(y) → Ay k e−y /2
. (Q8.3)
Hint: you can neglect compared to y 2 in this limit.
(c) Separate off the exponential factor and define
2
Ψ(y) = u(y)e−y /2
. (Q8.4)
Show that u(y) fulfils the ODE
u00 − 2yu0 + (2 − 1)u = 0. (Q8.5)
(d) Show that this differential equation can be converted to Sturm–Liouville form by multiplying
2
both sides of the equation by e−y . What is the weight function w(y) of the Sturm–Liouville
problem?
(e) Solve (Q8.5) by the ansatz
X∞
u(y) = an y n (Q8.6)
n=0
by deriving a recurrence relation for the coefficients an . You should get
(2n + 1 − 2)
an+2 = an . (Q8.7)
(n + 2)(n + 1)
(f) We know from (b) that for y → ∞ the function u(y) must go to Ay k . This means that the
recurrence relation must terminate, i.e., we must have an = 0. This quantizes the allowed values
of = E/~ω:
1
n = n + , n = 0, 1, 2, . . . . (Q8.8)
2
Find the polynomial solutions Hn (x) corresponding to these values of for n = 0, 1, 2, 3. These
polynomials are called Hermite polynomials.
(g) Show that your results for n = 0, 1, 2, 3 agree with Rodrigues’ formula
2 dn −x2
Hn (x) = (−1)n ex e . (Q8.9)
dxn
(h) Show that the Hn can be normalized such that
Z ∞ √
2
dy e−y Hn (y)Hl (y) = δnl π2n n!. (Q8.10)
−∞
3
9. Generating function
Hermite polynomials can be defined by the generating function
∞
2 2 2 X tn
G(x, t) = ex e−(t−x) = e2tx−t = Hn (x) . (Q9.1)
n=0
n!
(a) Find H0 (x), H1 (x), H2 (x) by expanding this generating function as a power law in t.
(b) By differentiating G(x, t) with respect to t, show that
∞ ∞
X tn X tn−1
2(x − t) Hn (x) = Hn (x)n (Q9.2)
n=0
n! n=0 n!
and hence that the Hn (x) satisfy the recurrence relation
Hn+1 (x) = 2xHn (x) − 2nHn−1 (x). (Q9.3)
(c) By differentiating G(x, t) with respect to x, show that
Hn0 (x) = 2nHn−1 (x). (Q9.4)
(d) Using the results from (b) and (c), show that the Hn defined in this way satisfy Hermite’s
differential equation
Hn00 − 2xHn0 + 2nHn = 0. (Q9.5)
10. Legendre polynomials
Position vectors r1 and r2 are such that r2 r1 , where r1 = |r1 | and r2 = |r2 |. Show that
( 2 )
1 1 r1 r1
= 1+ P1 (cos θ12 ) + P2 (cos θ12 ) + · · · , (Q10.1)
|r2 − r1 | r2 r2 r2
where θ12 is the angle between r1 and r2 , and P1 (cos θ) = cos θ, P2 (cos θ) = 21 (3 cos2 θ − 1).
An electric quadrupole is formed by charges Q and coordinates (0, ±a, 0) and charges −Q at coordi-
nates (±a, 0, 0). Show that the potential V in the (x, y) plane at a distance r large compared to a is
approximately
−3Qa2 cos 2θ
V = , (Q10.2)
4π0 r3
where θ is the angle between r and the x-axis.
Derive an expression for the couple exerted on the quadrupole by a positive point charge Q at a position
r in the (x, y) plane, where r a.
Deduce the angles θ for which this couple is zero. If the charges of the quadrupole are rigidly connected
and free to rotate about the z-axis, determine whether the equilibrium is stable or unstable in each case.
