Download Physics Homework: Neutrons, Wave Functions, and Quantum Uncertainty and more Assignments Advanced Physics in PDF only on Docsity! Physics 261, Written Homework #2 Due Fri 4/21 at class time (or beforehand, in Has404) 1. Neutrons traveling at a speed of 0.40 m/s are directed through a pair of slits having a 100-μm (micron) separation. An array of detectors is placed 50.0 m downstream of the slits. The mass of a neutron is 1.67×10-27 kg. (a) What is the de Broglie wavelength of the neutrons? (b) How far off axis is the first zero-intensity point on the detector array? (a distance with units) (c) When a neutron reaches a detector, can we say which of the two slits the neutron passed through? If ‘yes,’ please say how. If ‘no,’ please explain why not. 2. An electron is confined to move in one dimension in the range 0 < x < L. It has a wavefunction given by… ψ(x) = A (Lx-x2) for 0 ≤ x ≤ L, where L has a positive value. -and- ψ(x) = 0 for x < 0 or x > L. (a) Find the value of A needed to normalize the wave function. (b) Find an expression for 〈x〉 (which should only involve L and some numbers). (c) Find an expression for 〈x2〉 ( “ “ “ ). (d) Find an expression for Δx = √[〈x2〉 - 〈x〉2]. 3. (A continuation of the above problem). Using the ψ(x) for the electron as in problem 3, (a) Find an expression for 〈px〉 (which should only involve L and some numbers) (Recall that pψ = -i(h/2π) dψ/dx ). (b) Find an expression for 〈px2〉 (which should only involve L and some numbers). (c) Find an expression for Δp = √[〈p2〉 - 〈p〉2]. (d) Explicitly test the Heisenberg Uncertainty Principle. Is it obeyed? 4. An electron is localized in a one-dimensional region that is 2.00 nm wide. (Assume that the electron moves in one dimension.) (a) What are the quantum numbers of the ‘ground’ and ‘first excited’ states (the states that contain one electron and that have lowest energy, and second-lowest energy, respectively)? (b) Construct an energy-level diagram for this electron, with the energies of each state clearly labeled with numbers and units, not just formulas. Include at least 4 states, including the ground state. 5. Now consider the three-dimensional version of the above: the electron is confined to a cube of width 2.00 nm. You may treat the x, y, and z directions independently and find that the total kinetic energy is p2/2m = (px2 +py2+ pz2)/2m. Now you must specify three quantum numbers for each state. (a) What are the quantum numbers for the ground state? (b) List all possible combinations of quantum numbers for the state of next highest energy (the first excited state). (c) What is the difference in energy between the first excited state to the ground state? Please give the answer as an expression and as a numerical value with units. (this question was revised 4/13). (This is a crude model of the energy of a free electron in a nanoparticle e.g., of the semiconductor CdSe. Among other things, it does not account for interactions among the many electrons in the particle, so the numerical answer is not quite correct). – end –
