SCIENCE ONE: MATHEMATICS ASSIGNMENT 11 (Due Feb. 10, 2012)
There are two parts to this assignment. The first part is online and is due by 12:00 noon on Fri. Feb. 10;
you will need to login at www.mathxl.com. The second part consists of the questions on this page and is also
due by 12:00 noon on Fri. Feb. 10. You are expected to provide full solutions with complete arguments and
justifications. You will be graded primarily on the correctness, clarity and elegance of your solutions. Your
answers must be typeset or very neatly written. They must be stapled, with your name and student number
at the top of the first page.
1. An electric point charge of 10 nC is fixed at the origin in 3-dimensional space. Use Coulomb’s Law and
integration to find the work done to move a point charge of −2 nC from Pi = (xi , yi , zi ), where xi = 3
cm, yi = 4 cm, zi = 0 cm, to Pf = (xf , yf , zf ), where xf = 12 cm, yf = 0 cm, zf = 5 cm. Use the
work done to give the change in the electric potential V (Pf ) − V (Pi ) of the 10 nC charge at the origin
(consider the −2 nC charge as a ‘probe’ charge).
2. Water is drawn from a well 12 m deep, using a bucket that scoops up 36 kg of water. The bucket itself
has a mass of 10 kg, and is raised at constant speed with a rope that has a linear mass density of 0.8
kg/m. The bucket leaks water at a constant rate and finishes draining just as the bucket reaches the
top of the well. Determine how much work is done. (Neglect the dimensions of the bucket and treat the
leaking bucket as a point with a changing mass.)
3. If u and v are two continuous functions defined on a closed interval [a, b], we define the inner product of
the functions u and v to be the definite integral
Z
hu, vi =
b
u(x)v(x) dx.
a
A function defined on an interval I is C 2 (or twice continuously differentiable) on I if its second derivative
is a continuous function on I (recall that this implies that its first derivative and the function itself are
also continuous on I).
(a) Use integration by parts to show that if u and v are any two functions that are C 2 on [a, b], with
u(a) = 0, u(b) = 0, v(a) = 0, v(b) = 0, then hu00 , vi = −hu0 , v 0 i.
(b) Let Ĥ be the Hamiltonian differential operator defined by
Ĥψ = −
~2 00
ψ
2m
where ψ is any function that is C 2 on [0, L], and ~ and m are positive constants. Use integration
by parts, twice, to show that if ψ and φ are any two functions that are C 2 on [0, L], with ψ(0) = 0,
ψ(L) = 0, φ(0) = 0, φ(L) = 0, then hĤψ, φi = hψ, Ĥφi. (This property of Ĥ is called self-adjointness.)
(c) Prove that if ψ is any function that is C 2 on [0, L], with ψ(0) = 0, ψ(L) = 0, ψ not identically zero
(i.e. there is at least one point c in [0, L] such that ψ(c) 6= 0) then hĤψ, ψi > 0. (This property of Ĥ
is called positive definiteness.)
(d) Prove that if Ĥ is positive definite (i.e. satisfies the result of part (c)) and if there is (i) a function ψ
that is C 2 on [0, L], with ψ(0) = 0, ψ(L) = 0, ψ not identically zero, and (ii) a constant E such that
Ĥψ = Eψ, then E > 0 (all energy levels must be positive).