PC 242 Assignment 4: Due in class on Wednesday March 18, 2015 1. A wave function can be expressed as a sum over different harmonics as ψ (x) = 0.7 cos(x) + 0.1cos(3x) +0.2 cos(5x) (a) What are the constituent wavenumbers and wavelengths ? (b) What wavelength has the largest amplitude a(k) ? 2. A particle is described by the probability density function −L # % AL + 2Ax, for 2 ≤ x ≤ 0 % L P(x) = $ AL − 2Ax, for0 ≤ x ≤ % 2 % 0, otherwise & a) Make a sketch of the probability density function, P(x) as a function of x. b) If the wave function is real, write down a possible expression for the wave function ψ(x). c) Normalize the wave function and determine the constant A in terms of L. d) What is the probability that the particle lies in the region x < L/6? 3. An electron in the n=4 state of a 5nm wide infinite well makes a transition to the ground state, emitting a photon. What is the wavelength of the photon? 4. An electron is trapped in an infinite well. If the lowest energy transition possible in the well produces a photon of 450 nm wavelength, what is the width of the well? 5. What is the probability that a particle in the n=2 state of an infinite well will be found in the middle third of the well? 6. Show that the uncertainty in the momentum of a particle in level n of an infinite well is given by nπ Δp = L 7. An electron is trapped in a finite square well. How far in energy (eV) is it from being free if the penetration depth into the walls of the well is 1nm? 8. A 2kg block oscillates with an amplitude of 10cm on a spring with spring constant 120N/m. What energy level is the block in? What is the energy spacing between successive energy levels?
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