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NATURAL SCIENCES TRIPOS Part IB & II (General)
Tuesday, 29 May, 2012 9:00 am to 12:00 pm
MATHEMATICS (1)
Before you begin read these instructions carefully:
You may submit answers to no more than six questions. All questions carry the same number of marks.
The approximate number of marks allocated to a part of a question is indicated in the left hand margin.
Write on one side of the paper only and begin each answer on a separate sheet.
At the end of the examination:
Each question has a number and a letter (for example, 6A).
Answers must be tied up in separate bundles, marked A, B or C according to the letter affixed to each question.
Do not join the bundles together.
For each bundle, a blue cover sheet must be completed and attached to the bundle.
A separate green master cover sheet listing all the questions attempted must also be completed.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
6 blue cover sheets and treasury tags None Green master cover sheet Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
1C
Let F be a vector field and a be an arbitrary constant vector. Show that ∇ × (a × F) = a (∇ · F) − (a · ∇)F. [4] State Stokes’ theorem. [2] By applying Stokes’ theorem to the above identity, show that ∫
C
dl × F =
S
(dS × ∇) × F
for any closed curve C that bounds a surface S. [8]
Verify this identity for the case where C is a square path starting at (0, 0 , 0), then progressing in a straight line to (0, 1 , 0), then to (1, 1 , 0), then to (1, 0 , 0) and finally back to the origin, with F = r and r = (x, y, z). [6]
2C
The number density of neutrons n(r, t) in a lump of uranium is determined by the partial differential equation
∇^2 n = ∂n ∂t
− λn
where ∇^2 is the Laplacian operator, r is the position vector, t is the time and λ is a constant.
Suppose the lump of uranium is a sphere of radius a and the density of neutrons is spherically symmetric. Furthermore, suppose this equation can be solved by the method of separation of variables so that n = R(r)T (t)
where r is the distance from the centre of the sphere. Find two ordinary differential equations for R(r) and T (t). [4]
Suppose that the density of neutrons is never zero except at the surface of the sphere and finite everywhere inside. Find n(r, t). [8]
Show that the concentration of neutrons will grow as a function of time provided that
λ > π^2 a^2
[8]
[Hint: To find R(r), make the substitution R(r) = rpf (r) for some p.]
Natural Sciences IB & II, Paper 1
4C
Calculate the Fourier transform of the function
f (x) = e−λx 2 ,
where λ is a positive constant. [5]
Consider the partial differential equation for ψ(x, t)
∂^2 ψ ∂x^2
∂ψ ∂t
Find the ordinary differential equation that
ψ˜(k, t) =
−∞
ψ(x, t) e−ikx^ dx
obeys. [5]
Find ψ˜(k, t) given ψ(x, 0) = e−λx 2
. [5] Hence find ψ(x, t) for t > 0. [5] [Hint: You may use the following relation ∫ (^) ∞
−∞
e−λ(x+iα)
2 dx =
π λ
for α and λ real constants and λ > 0 .]
Natural Sciences IB & II, Paper 1
5A
(i) Show that an n × n matrix A is diagonalisable if it has n linearly independent eigenvectors. Show that if A is diagonalisable, then so is B = P−^1 AP, where P is an n × n matrix and P−^1 is its inverse. [6]
(ii) Let λi (with i = 1, 2 ,... , n) be the eigenvalues of an n × n Hermitian matrix A.
(a) Show that
Tr Ak^ =
∑^ n
i=
λki and det Ak^ =
∏^ n
i=
λki ,
for all positive integers k. [6] (b) Show that det(exp A) = exp(Tr A). (^) [4]
(c) If A^2 = A, prove that either (i) det A = 1 and Tr A = n or (ii) det A = 0 and Tr A = m < n, where m is an integer. [4]
Natural Sciences IB & II, Paper 1 [TURN OVER
7A
(i) Two-dimensional fluid flow can be described by a complex potential
f (z) = u(x, y) + iv(x, y) ,
where z ≡ x + iy. Let the fluid velocity be V = ∇u. If f (z) is analytic, show that
(a) ∇ · V = 0, [4] (b) df dz = Vx − iVy. [4]
[You may assume the Cauchy-Riemann equations without proof.]
(ii) Consider the Gamma function
Γ(z) =
0
tz−^1 e−tdt for Re(z) > 0.
(a) Using integration by parts, derive the recursion relation
Γ(z + 1) = z Γ(z). (⋆)
Also show that Γ(1) = 1. [4] (b) Assuming that (⋆) holds for all z ∈ C, show that Γ(z) has simple poles at all non-positive integers. Compute their residues. [8]
Natural Sciences IB & II, Paper 1 [TURN OVER
8A
Legendre’s equation
(1 − x^2 )y′′^ − 2 xy′^ + ℓ(ℓ + 1)y = 0 ,
admits series solutions of the form y =
∑^ ∞
n=
anxn.
(a) Derive the recurrence relation for an. [4]
(b) Show that for integer ℓ, one of the solutions, Pℓ, is a polynomial of order ℓ; while the other solution is an infinite series Qℓ. [2]
(c) Find the first four polynomials Pℓ(x) (i.e. ℓ = 0 , 1 , 2 , 3) given the normalisation Pℓ(1) = 1. [4]
(d) Show that the Wronskian of Pℓ and Qℓ is given by
PℓQ′ ℓ − P ℓ′ Qℓ =
Aℓ
1 − x^2
for Aℓ independent of x. [6]
(e) Derive Q 0 (x) in closed form, assuming Q 0 (0) = 0. [4]
Natural Sciences IB & II, Paper 1
10A
(i) Consider the Sturm-Liouville equation
d dx
p(x)
dψ dx
where p(x) > 0 and w(x) > 0 for α < x < β.
(a) Show that finding the eigenvalues λ is equivalent to finding the stationary values of the functional
Λ[ψ(x)] =
∫ (^) β
α
(pψ′^2 + qψ^2 )dx ,
subject to the constraint (^) ∫ β α
wψ^2 dx = 1.
You may assume that ψ(x) satisfies suitable boundary conditions at x = α and x = β (which should be stated). [6] (b) Explain briefly the Rayleigh-Ritz method for estimating the lowest eigenvalue λ 0. [4]
(ii) The wavefunction ψ(x) for a quantum harmonic oscillator satisfies [ − d^2 dx^2
]
ψ = λψ. (⋆)
(a) Use the trial function
ψ(x) =
15 16 a^5 (a
(^2) − x (^2) ) for |x| 6 a 0 for |x| > a
to estimate the lowest eigenvalue λ 0. [8] (b) The exact ground state wavefunction is
ψ 0 (x) =
2 π
e−^
1 2 x^2.
Find the corresponding eigenvalue and compare it to the previous estimate. [2]
END OF PAPER
Natural Sciences IB & II, Paper 1
