PHY4604 Fall 2004 - Practice Test 2: Introduction to Quantum Mechanics, Exams of Physics

Download PHY4604 Fall 2004 - Practice Test 2: Introduction to Quantum Mechanics and more Exams Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Practice Test 2 October 31, 2004 These problems are similar but not identical to the actual test. One or two parts will actually show up. 1. Short answer. • You are told that a potential can be written V (x) = 10∑ n=1 Vnx 2n. All Vn ≥ 0. What can you say about the solutions ψ to Hψ = Eψ? • What is tunnelling? Draw a picture describing the process, and describe explicitly a situation which distinguishes classical physics from quantum mechanics. • Write the Dirac expression 〈α|H|α〉 as an explicit integral in three dimen- sions, assuming that 〈r|α〉 represents a wave function ψα(r). If |α〉 is an eigenvector of H with eigenvalue Eα, evaluate the integral. • Identify: Ehrenfest’s theorem. • If A and B are self-adjoint, is the combination A + iB self-adjoint, anti- self-adjoint, unitary, antiunitary, none of the above, or more than one of the above? Defend your answer. • Prove that if (ψ,Oψ) = (Oψ, ψ) for all ψ, then O is self-adjoint. • Prove that eigenvectors corresponding to distinct eigenvalues are orthogo- nal. • Find the expansion coefficient of the first excited SHO state in the function (x2 + x20) −2 • Draw a picture of a finite attractive square well in 1D, V = −V0 inside and V = 0 outside, and sketch the form of the ground and 1st excited states of negative energy, assuming such states exist. Be sure to indicate how these functions differ from the analogous eigenfunctions of the infinite square well problem. What happens to the number of bound states (E < 0) when you make the well in this problem deeper? 1