Question 1 (8 marks)

The results from the Sudbury Neutrino Observatory are consistent with a two-neutrino model that has the following parameters: a mixing angle θ s 69。and mass difference ∆m2  = 1me(2)  - mν(2)1 ~ 8.5 x 10-5 eV2 . (Reasonable value for the masses are mνe   = 0.002 eV- 1 and mνu   = 0.009 eV- 1 )

a) The flavour eigenvectors 1νe〉and 1νu〉for the electron and muon neutrinos and the Hamiltonian (H) are given by:

╱． νe   、．   ╱． cos θ

．        ．=．

sin θ  、．╱． ν 1  、．

．．       ．

cos θ ．．．( ν2  ．

╱  m1(2) H =．   2p

．

．(  0

0  、．

．

．

m2(2)       ．

2p    ．

(1)

where p s 8 x 106 eV.

Write a MATLAB code that calculates the probability that an electron neutrino will appear as a muon neutrino at t = 3.0 x 1015 eV- 1 .

b) The probability that an electron neutrino changes into a muon neutrino can be expressed as a function of the distance L from the source. The equation is

Pνe ->νu   = sin2 θ sin2 布(L)c3 \                                                                                               (2)

where 布 = c = 1, θ = 69。is the mixing angle and ∆m2  = 8.5 x 10-5  eV2 . Consider an electron neutrino with an energy of 8 MeV and write a MATLAB code that plots (use semilogx()) the probability Pνe ->νu

as a function of L, with L between 4 x 109 eV- 1 and 7 x 1012 eV- 1 .

Copy your plot and code in the box below.

(Axes should be labelled)

From your plot in part b), determine the shortest distance that the 8 MeV electron neutrino must travel before it oscillates to become a muon neutrino. Write the distance in eV- 1 and in m, in the box below.       Remember that 1m = 5.068 x 106 eV- 1. 

Question 2 (8 marks)

A beam of spin- 1 particles is prepared in the state

1w〉=  11〉+ i  10〉-  1-1〉

(3)

where

╱  1  、

╱ 0  、

．     ．

10〉=． 1 ．

．      ．

．      ．

．     ．

╱ 0  、

．     ．

1- 1〉=． 0 ．

．      ．

．      ．

．     ．

(4)

The matrix representation of the S z  operator for a spin- 1 system is

where a)

b)

c)

0   、

0    ．

．

．

．

Use MATLAB to find the possible results of a measurement (eigenvalues) and the corresponding quantum states (eigenvectors) of the operator S z . Write your code below.

Using the state in equation 3 and your result in (a), use MATLAB to find the probability of each of the outcomes of a measurement of S z . Write your code and the probabilities in the box below. Be careful to attribute the probabilities to the correct eigenstates.

Use MATLAB to calculate the expectation value for Sz by solving〈Sz〉=〈w1Sz 1w〉for the state in equation

3.

Write your code and answer below.

Question 3 (8 marks)

Consider a scenario in which particles travel in a circular quandrant section of a shield material, as in the figure below. The section has a radius (R) equal to 3 m.

All particles have the same initial position (x, y) = (0, 0) and velocity

11V1 1  =  0.4 ms- 1  .  At each step, the particles move by a random

distance d = 11V1 1τ, where τ = - log u with u being a random number

uniform between 0 and 1. In MATLAB, the distance can be written

as

d = - norm(v)*log(rand());

While travelling through the material, particles will be scattered by

an angle θ with respect to the downward y-direction. The angle θ is

uniformly distributed between 0 < θ <  .

In addition, there is a 10% probability that the particles can be absorbed.

In this question, you will simulate the motion of particles, developing the MATLAB code in stages.

a) We are interested in the motion of particles through the section, so if a particle exits, the simulation should stop for that particle. A particle is in the section if x > 0 and y < 0 or if its distance from the origin is less than R = 3 m.

Write a MATLAB function called inside .m that takes the current position of a particle and the section radius as input parameters. Your function should return 1 if the particle is inside the section, for example

1       >>R=3

2       >>pos=[2 .0,-1 .5]

3       >>inside(pos,R)

4       ans=  5                      1

and 0 otherwise, like:

1       >>pos= [-1 .0,2 .0]

2       >>inside(pos,R)

3       ans=  4                      0

Write your MATLAB function in the space below.

b) Write a MATLAB code that calculates the new position of a particle that moves by a distance d and it is scattered by a random angle θ (0 s θ s ) with respect to the downward y-direction.

c) Write a program that calculates and plots the paths of 5 particles inside the section, until they exit or get absorbed.

Write your code and sketch the plot in the box below.

To plot the circular quadrant section:

1       R=3

2       x=R*cos(0:0 .01:(pi/2));

3       y=-R*sin(0:0 .01:(pi/2));

4

5       plot(x,y, 'k ', 'LineWidth',2)

6       line(x,zeros(length(x),1),'Color',[0,0,0],'LineWidth',2)

7       line(zeros(length(y),1),y, 'Color',[0,0,0],'LineWidth',2)

d) Run your code with 1000 particles and count how many particles will finally exit the section. Write your result in terms of percentage.