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1 Homework set #11 Phy 3031 Modern physics, Spring 2007 Due at the start of class on Wednesday, March 28, 2007. 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. Suppose that a particle of mass m is constrained to move in one dimension between two barriers that are a distance A apart. (a) (b) (c) (d) Use the uncertainty principle to estimate the minimum energy of the particle. With the previous result, find the minimum energy of an electron if A is 0.1 nm. Find the minimum energy of an electron if A is 1 cm. Calculate the minimum energy of a 100 mg = 0.1 g bead on a 1 cm long wire.