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UNIVERSITY COLLEGE LONDON

EXAMINATION FOR INTERNAL STUDENTS

MODULE CODE

: PHAS1246

ASSESSMENT

:

PHAS1246A

PATTERN

MODULE NAME

:

Mathematical Methods II

DATE

:

17 May 12

TIME

:

14:30

TIME ALLOWED

:

2 Hours 30 Minutes

2011/12 PHAS1246A 001 EXAM 205

2011

University College London

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Answer ALL questions from Part A and THREE questions from

Part B

A formula sheet is attached after the end of the

paper

The numbers in square brackets in the right-hand margin indicate the provisional

allo-

cation of maximum marks per sub-section of a question.

[Part marks]

Part A

1. Given the matrix

[5]

/ 1

2

3

A =

4 1 5

6

0 2

calculate its trace and its determinant, and write down its transpose AT

.

2. Write the momentum four-vector of a particle that has

a rest mass of 80 GeV/c2,

energy 100 GeV and moves along the z-axis. What is its speed as a fraction of the

speed of light?

[5]

3. A muon (m

0.1 GeV) is produced with energy 10 GeV in

a cosmic ray cascade,

[6]

10km from the surface of the earth and travels vertically towards the surface.

Given that the muon lifetime is

2 as in its rest frame, use time dilation and

length contraction to explain how an observer riding along with the

muon and an

observer from the earth interpret the fact that most of the time such

a muon makes

it down to the surface of the earth before decaying.

4. Calculate the work done by the force F

= (x

y)i

x) j, when it is applied to

a body that moves on the x

y plane from point (2, 2) to point (4,8) along:

(a) the straight line connecting the two points; and

[5]

(b) the parabola y

= x2/2.

[3]

PHAS1246/2012

PLEASE TURN OVER

1

(y

a:

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[Part marks]

5. a) Show that a linear, first-order differential equation

[4]

dy

dx + P x)y = Q x)

has the general solution

x) S

x)L1 S x)Q x) dx +

where C is an arbitrary constant and

8 x) = exp [f P x) dx] .

b) Use the above method to solve the following differential equation

dy

x 2)

dx

y = x

2)3

and find the solution that satisfies y = 10 for x = 4.

[4]

6.

a) Given the differential equation

[4]

A x y) + B x , y)

dx

y

explain what the condition is for the above to be solved using the perfect

differential method.

b) Show that the LHS of the differential equation

y2 x)dx 2ydy = 0

is not a perfect differential, but if you multiply it by e it becomes one. Use

this to derive a general solution to the above equation.

PHAS1246/2012

CONTINUED

2

[4]

= 0 ,

+

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Part B

[Part marks]

7. A simple harmonic oscillator, with natural frequency wo, experiences an oscillating

driving force f(t) = cos wt .

Therefore, its equation of motion is

d2x

dt2

wo2x

= cos wt ,

where x is the position.

(a) Given that at t = 0 we have x

0 and dx/dt = 0

,

show that the trajectory

[16]

of the oscillator is given by

x(t) =

2

2

(cos Wt

cos CAJot) .

C4i0 (4)

(b) Describe the solution (i.e. the motion) if w is approximately, but not exactly,

[4]

equal to wo. (Hints: cos a

cos b = 2 sin[(b

a)/2] sin[(b

a)/2] and for small

0, sin 0

O.)

8. Using volume integrals and appropriate coordinate systems, calculate:

(a) the volume of a sphere of radius Rs, assuming that the sphere is centred at

[4]

the origin of the coordinates system; how would you change the limits of

integration to calculate that part of the volume of the sphere that lies in the

positive z hemisphere?

(b) the volume of a cylinder that has radius R, and height h;

[3]

(c) the volume of a cone of base radius R, and height h; and

[5]

(d) The fraction of a sphere s volume that is above the plane z

= a Rs, where

[8]

0 < a < 1, assuming again that the sphere is centred at the origin of the

coordinates system. Verify that your solution gives 0 for a

= 1 (when the

plane touches the sphere tangentially) and 1/2 for a = 0 (when the plane cuts

the sphere in two halves).

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[Part marks]

9. A particle H, with rest mass mH and energy E, is traveling along the x-axis in the

lab frame, in the positive direction.

(a) Express the momentum, p, of H, in terms of E and mH, and write down its [4]

momentum four-vector in its rest frame and in the lab frame.

(b) Prove that E = limHc2, where -y is the particle s boost factor.

[4]

(c) At some point, H decays to two photons. In the rest frame of H, the two [6]

photons are emitted on the x

y plane, in opposite directions along a line

that forms an angle a with the x-axis. Derive the momentum four-vectors of

the two photons in the lab frame.

(d) Show that the cosine of the angle between the trajectories of the two photons

[6]

in the lab frame, as a function of the angle a above, is

02

02 sing

cos 0 =

1

)32 cost

where 6 = v/c. If a = 7r/2 and mH = 125 GeV /c2, what should be the energy

of H so that the angular separation between the two photons is 60 degrees?

10.

(a) Evaluate the determinant

[5]

gc

ge

a+ ge

gb+ ge

0 b b

c e e b

e

b b+f

b+d

using as much as possible properties of determinants that can help simplify

your calculations.

(b) Express the following equations

[7]

x+2y+z

=

4

3x

4y

2z =

2

5x+ 3y+ 5z

=

1

in matrix form and solve them using the inverse matrix technique.

(c) Using Cramer s rule, evaluate x from the equations above.

[4]

(d) Show that the product of two orthogonal matrices is also an orthogonal matrix. [4]

What do orthogonal matrices represent?

PHAS1246/2012

CONTINUED

4

1

+

a

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[Part marks]

11.

(a) Write the cartesian coordinates (x, y, z) as functions of the spherical polar

coordinates (r, 0, 0).

(b) Derive the Jacobian of the transformation from spherical to cartesian coordi-

[6]

nates. What does the Jacobian represent?

(c) By making two successive, simple changes of variables, evaluate

[6]

[3]

I=1,11 x2dxdydz

inside the volume of the ellipsoid

y2 z2

a2

b2

c2 R2

where a, b, c and R are constants.

(d) Give the definitions of Hermitian and Unitary matrices and show that if A is

Hermitian and U is unitary then U AU is also Hermitian.

PHAS1246/2012

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[5]

END OF QUESTION P PER

TURN OVER FOR FORMUL

SHEET

x2

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Formula Sheet

The Lorentz transformation relating a four-vectoi defined in frame S to that defined

in the frame S which is moving along the x axis of S with speed v is

a \

0

0

\

/ xl \

b

0

1 0

0

X2

0 0

1

0 x3

\ d

0

0

y

I

\ x4 /

where c is the speed of light, and the boost factor -y is defined as:

1

7

ic

For the position four-vector, x1 = x, x2 = y, x3

z, x4 = ct, and for the energy momen-

tum four-vector, xi = px, x2 = Py, X3 = pz, x4 = El c.

Planck s constant h = 4.14 x 10-15 eV

Speed of light c = 3.00 x 108 m

Mass of the electron (and positron) me = 0.5 MeV/c

PHAS1246/2012

END OF FORMULA SHEET

6

.

.

.

/

/

7

c

/

71)-

A/1

=

s.

. .

.