PHY120

Data Provided: Formula sheet and physical constants DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Answer question ONE (Compulsory) and TWO other questions, one each from section A and section B. All questions are marked out of ten. The breakdown on the right-hand side of the paper is meant as a guide to the marks that can be obtained from each part.

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PHY472

DEPARTMENT OF PHYSICS AND ASTRONOMY

Autumn 2016

Mathematics for Physicists and Astronomers 3 hours

Instructions:Answer ALL questions from BOTH sections.

There are 180 possible marks for this paper.

The breakdown on the right-hand side of the paper is meant as a guide to the marks that can beobtained from each part.

Please clearly indicate the question numbers on which you would like to be examined on the frontcover of your answer book. Cross through any work that you do not wish to be examined.

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SECTION A Mathematical Techniques

Unit 1 - Functions, Differentiation and Series

1.

Identify which of the functions drawn above correspond to each function below: [2.5]

y = x3 . . . . . .

y = 3px . . . . . .

y = 1/x . . . . . .

y = x2 . . . . . .

y = x . . . . . .

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2. Differentiate the following functions with respect to x:

(a) sinh 4x , [1]

(b) cosh x2, [1]

(c) sin x2, [1]

(d) sin2 x , [1]

(e) sin x cos x , [1]

(f) ln(sin x), [1.5]

(g)ln x

x. [1.5]

3. Find u when cosh u= 2sinh u− 1. [2]

4. Find the first 3 non-zero terms and the expression for the n-th term in a Taylor series atx = 0 for the functions:

(a) f (x) = xex , [2]

(b) g(x) = e−3x . [2]

5. A hyperbola can be described by the parametric equations

x = 2 secθ ,

y = 4 tanθ .

Evaluate the following when θ = π/4:

(a)dydx

, [2]

(b)d2 ydx2

. [2]

6. Given a cubic polynomial p(x) with conditions p(0) = 4, p′(0) = 3, p′′(0) = 4, andp′′′(0) = 6, find an expression for p(x). [2]

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7. (a) State two methods for evaluating limits of functions. [1]

(b) Use either of these methods to evaluate the following limits as x → 0:

i.sin x

x, [1]

ii.1− cos2 x

x2, [1]

iii.sin x − x

exp(−x)− 1+ x. [1]

8. (a) Expand [ln(1+ x)]2 in powers of x up to terms in x4. [2]

(b) Determine whether cos 2x + [ln(1 + x)]2 has a maximum, minimum or point ofinflection at x = 0. [2]

(c) Determine whether[ln(1+ x)]2

x(1− cos x)

has a finite limit as x → 0 and, if so, what value this takes. [3]

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Unit 2 - Introduction to Vectors

9. Given the three vectors a = i− j− 7k, b = 2i+ 2j− 2k, and c = −5i− j+ k evaluate thefollowing:

(a) 3b+ c [1]

(b) a+ b [1]

(c) a · c [1]

10. Given the three vectors p = 8i − j − 10k, q = i + 2k, and r = 3i + 5j − k evaluate thefollowing:

(a) p · q [1]

(b) p× q [2]

(c) |p× q| [1]

(d) p · (q× r) [3]

11. Given the point P = (1, 4,3) and the vectors n= i− j− 4k and v= 2i− 2j, find:

(a) the equation of the plane passing through P and perpendicular to n. [2]

(b) the projection of v along n. [2]

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Unit 3 - Complex Numbers

12. Let a = −2+ i; b = 3− 3i, and c = 2i. Calculate the real and imaginary part of :

(a) z1 = ab̄+ c [1]

(b) z2 = |a+ c| [1]

(c) z3 =ac2

b[2]

13. Let p = 4+ 4i; q = 5i, and r = −1+ i. Calculate the modulus and argument of :

(a) z1 = p+ q [2]

(b) z2 =pq

[2]

(c) z3 = q.pr [2]

14. Find all solutions of the following complex equation: [4]

z4 =6i(1− i)

�

�(p

5− 2i)�

� (p

2eiπ/4)�q

23[cos(π/8) + i sin(π/8)]

�4

15. Solve the equation by writing the answer as z = x + i y where x , y ∈ R: [3]

2z + 3i(z̄ + 2)1+ i

= 13+ 4i

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Unit 4 - Integration

16. Determine the following:

(a)

4∫

1

(x + 1)3 dx [1]

(b)

π/2∫

0

cos4x dx [1.5]

(c)

∫

ln 3x2x

dx [1.5]

(d)

∞∫

1

dxx2

[2]

(e)

∫

sin x cos2 x dx [2]

(f)

2∫

0

1

(3x + 1)3dx [2]

(g)

∫

x4 − 5x3 + 6x2 − 18x3 − 3x2

dx [3]

17. FinddIdx

where I(x) is given byx∫

0

t3 dt [2]

18. Using the method of integration by parts, find [5]∫

e3x sin 2x dx

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19. Determine the length of the curve y = 2x3 between x = 0 and x = 4. [3]

20. The area under y =2

3xbetween x = 2 and x = 4 is rotated completely about the x-axis.

Find the volume of revolution. [3]

21. Given the matrices

A=�

1 x1 0

�

B =�

2 1ex 1

�

calculate: [3.5 ]∫

2�

�AB�

�dx

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Unit 5 - Differential Equations

22. For each of the following differential equations,

(a) Give the dependent and independent variables, [2]

(b) State the order of the equation, [1.5]

(c) State whether it is linear or non-linear. [1.5]

i.d yd x+ x2 = 0

ii.d2 xd t2+ x2 = 0

iii.∂ 2φ

∂ x∂ y+φ =

∂ φ

∂ t

23. Find the solution to the equation [8]

dydx= (y + 1) sin x ,

given the boundary condition y = 1 at x = 0. Give an explicit expression for y in termsof x .

24. Find the general solution to the equation [10]

d2 yd x2− 4

d yd x+ 4y = 4x2.

25. What must be the relationship between the constants α and ω if the function [8]

φ(x , t) = Aeαx sin (ωt +αx) + Be−αx sin (ωt −αx)

is a solution of

∂ 2φ

∂ x2= 2

∂ φ

∂ t,

where A and B are arbitrary constants.

Find the specific solution if φ = sin2t for x = 0, given α > 0 and φ→ 0 as x →∞. [4]

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Unit 6 - Matrices

26. Given the matrices

α=�

1 25 1

�

β =�

2 10 −1

�

γ=

1 2 −10 1 13 −1 4

calculate:

(a) α+ β [1]

(b) βα [1]

(c) |γ| [2]

27. Consider two matrices A and B of size 2× 2:

A=�

a11 a12

a21 a22

�

B =�

b11 b12

b21 b22

�

(a) Show that the trace of their product is independent of the order, i.e. [1]

Tr(AB) = Tr(BA)

.(b) For matrices A and B of size n× n write an expression for the diagonal elements oftheir products (AB)ii and (BA)ii [1]

(c) Show that the relation Tr(AB) =Tr(BA) is valid for any size matrix pair [2]

(d) Generalise the result above for the order of the product of three matrices A, B, C [2]

28. Test the following set of linear equations to see if it possesses a nontrivial solution: [2]

x + 3y + 3z = 0,

x − y + z = 0,

2x + y + 3z = 0.

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29. Find the inverse of [4]

A=

3 2 12 2 11 1 4

30. Find eigenvalues and eigenvectors of the matrix: [5]

A=

p3

2 −12

12

p3

2

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SECTION B Applied Mathematics

31. Two masses m are resting on a frictionless table between two walls and are connectedto each other and the walls by identical springs with spring constants k, as shown in thetop-down diagram below.

We define the x-axis as running from wall-to-wall through the springs and masses, andthat all motion is confined to this axis.

(a) write two differential equations, one for the motion of each of the masses [7]

(b) derive an expression for the displacement of each mass as a function of time, as-suming x1(0) = 0 and x2(0) = a [9]

[Hint: You may wish to consider this as an eigenvector problem.]

32. The Leonard-Jones potential describes the van der Waals interaction energy V betweentwo neutral molecules separated by a distance r. This potential is given by

V (r) = 4ε�

�σ

r

�12−�σ

r

�6�

where σ is the hard-sphere radius of the molecule (assumed to be constant) and ε issome constant potential.

(a) Find the separation rmin at which the potential is minimised. [6]

(b) Use this expression for rmin to determine the minimum potential Vmin in terms of ε. [2]

(c) Consider a molecule of two identical atoms of mass m as a simple spring system . [6]

Derive an expression for the angular frequency ω0 =

√

√ km

of small oscillations

about the stable equilibrium position for an effective spring constant

k =d2V (r)

dr2

�

�

�

�

r=rmin

.

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END OF EXAMINATION PAPER

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