1 Heisenberg Uncertainty Principle - Groups C & D In this class, we will use the Heisenberg uncertainty principle fairly approximately, which we will follow here. The strict definition is that Ar and Ap represent the standard deviation of the position and momentum, as you may have seen in statistics courses. In this problem, we will be more loose with the definitions. A common wave function is the Gaussian: V(x) = Ce do? (6) where a is a parameter that controls the width of the wave function and C = a(21/ is a normal- ization constant, which we will not need to solve for here. This is the ground state of a particle in a harmonic oscillator (quadratic) potential well. We will assume that the particle is an electron for this problem. 1.1 Probability density Write down a formula for the probability density in terms of C and a. 1.2 Spread Find the values of r such that the probability density is |CIe-2 ~ 0.135|C|2. We will estimate the spread Ar of the wave function as the distance between those two points. It turns out that for this wave function, there is about a 95% probability that the electron will be found between those two points.31. Consider the system of equations azcl + :62 = 1 and 351 + 112 = 2. (a) Show that the solution is $1: 1/(1 a) and 332 = 2 1/(1 a). (b) Set up and solve the system using the routine gauss that we wrote in class with a = 10\A study randomly assigned adult subjects to one of three exercise treatments (1) a single long exercise period five days per week, (2) several ten-minute exercise periods five days per week, and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment Does the type of exercise treatments influence weight loss significantly? Use a = 5% Treatment Mean Std. Dev. n long periods 11.2 2.2 Multiple short periods 8.3 3.2 9 Multiple short periods with treadmill 11.3 2.6 Define the population parameter or distribution about which hypotheses are to be tested. O O Normal (Gaussian) O Chi-Squared OF