Homework #9

All problems MUST be in a SINGLE *.m file and be in separate blocks using '%% Problem X'. Otherwise you will not receive credit. You may need additional files for class definitions or functions. See class webpage for naming convention.

Contents

Problem 1

Muons are constantly being generated in the upper atmosphere and, although they are an unstable particle, a significant fraction survives until they reach the ground. The muon lifetime can be found using a barrel of liquid scintillating material. When the muons interact with the scintillator, they can lose most of their kinetic energy and a flash of light is generated. When the muon decays, a high energy electron is generated, which immediately deposits its energy in the scintilator, causing a second flash of light. The time between the first flash and the second is exponentially distributed as:

$P(t|\tau)=\frac{1}{\tau} e^{-t/\tau}$

A large amount of data can be collected an sorted into finite sized time bins. This data is Poisson distributed. Make a maximum likelihood estimate of the muon lifetime from the data set by fitting the binned data to the two parameter model:

$y(t)=A\exp{-t/\tau}$

The data is found in 'HW10_Data.mat'. 'Counts' is the counts in each time bin. 'T' is the bin centers in microseconds.

Plot the data, the model calculated from your starting guess, and the model calculated from the found parameters.

Problem 2

Given the found value of 'A' and 'tau' in Problem 1, use Monte-Carlo simulations to find the standard error of your estimate of tau.

Problem 3

Demonstrate the use of 'subplot' by making 6 plots in the same figure. The left three should be the first three lowest energy wavefunctions for the quantum harmonic oscillator ($\Psi_0,\Psi_1,\Psi_2$). The right three should be the cooresponing probability density ($\Psi_0^2,\Psi_1^2,\Psi_2^2$)

Problem 4

Use 'meshgrid' and 'surf' to show a 2D Gaussian shape with peak value equal to 10 and $\sigma_x=2,\sigma_y=5$