1 Phy 3101 Modern physics, Spring 2008 Homework set week #15. For final exam preparation For numerical answers give two significant digits. The first should be correct, the second can be approximate. 1. A particle moving in one dimension between two rigid walls separated by a distance L has the wave function Ψ(x) = A sin(πx/L). Since the particle must remain between the two walls, what must be the value of A? 2. 222 Rn decays by the emission of an α particle with a lifetime of 3.823 days. The kinetic energy of the α particle is measured to be 5.490 MeV. What is the uncertainty of this energy? Your answer must include an equation that shows how this answer is found. 3. If the uncertainty in the position of a wave packet representing the state of a quantumsystem particle is equal to its de Broglie wavelength (∆x ≈ λ = h/p), then how does the uncertainty in momentum compare with the value of the momentum of the particle? 4. In order to determine the location of an electron to within 5 × 10−12 m by use of electromagnetic waves (light), the wavelength of the light must be at least as small as 5 × 10−12 m. (a) Calculate the momentum of a photon with a wavelength λ = 5 × 10−12 m. (b) Calculate the energy of the same photon. (c) After the electron is “located,” up to an uncertainty ∆x = 5 × 10−12 m, what is the minimum uncertainty of its momentum? 5. The energy of a certain nuclear state can be measured with an uncertainty of 1 eV. What is the minimum lifetime of this nuclear state? 6. A particle of mass m is confined to move in only one dimension over a range from x = 0 to x = L. (a) Use the uncertainty relationship to estimate the smallest possible energy. (b) Assume that there is no force on the particle between the two ends of the region. Use the Schrödinger equation to find the ground state energy for the particle. (c) Are your answers to part (a) and part (b) close to each other? 7. This is too difficult for a test, but some of the ideas are not An electron is trapped in a one-dimensional infinite square well in the n = 5 state. (a) Find the probability of finding the electron between x = 0.2L and x = 0.4L. (b) Find the probability of finding the electron between x = L/2 − 0.005L and x = L/2 + 0.005L.