Quantum Mechanics II Homework: Perturbation Theory and Potential Energy, Assignments of Quantum Mechanics

Download Quantum Mechanics II Homework: Perturbation Theory and Potential Energy and more Assignments Quantum Mechanics in PDF only on Docsity! PHY662 - Quantum Mechanics II HWK #7, Due Tues., Mar. 2, at the start of class • Reading: Read Ch. 8 of Griths, Introduction to Quantum Mechanics. 1. Checking last week's answer to problem #2. Consider the potential V (x) =  ∞ x < −aF |x| −a < x < a∞ a < x , which you studied last week using the WKB approximation. Let the constant F be small in some sense, so that you can compute a perturbation series expansion for the ground and rst excited states for a particle of mass m in this potential. (a) What are the F = 0 energy eigenstates for this potential? (b) Compute the rst order (in F ) estimates for the energies of the ground state and the rst excited state, using time-independent per- turbation theory. (c) Write down the ground state energy E0 and the rst excited state E1, to 6 signicant gures (within this approximation), when F = κ h̄ 2π2 8ma3 , for the two cases κ = 0.2 and κ = 0.5, to rst order in F . That is, compute E0(κ = 0.2), E1(κ = 0.2), E0(κ = 0.5), and E1(κ = 0.5). (d) Compare your results from perturbation theory with the WKB re- sults from last week's problem. (e) Write out the expression for the sums you would need to compute to nd the second order correction to E0 and E1. You do not need to evaluate these sums, but simplify the terms in the sum as much as possible. 2. Extending last week's answer to problem #1: Airy functions and per- turbation theory. Let's consider the problem from last week, where the spectrum of mesons can be estimated using a linear potential for qq pairs. The potential was taken to be V 0(r) = V0 +Kr. In this problem, you will compute a new energy estimate, using second-order perturbation theory for a modied potential. Use the data from last week's problem and the key to that problem where necessary. (a) Eect of a δ(r) change to the potential. Fix up the energy esti- mates by trying a dierent model. At close distances the quarks attract more strongly. This is often described by a Coulombic at- traction. Here, try something simpler: add a δ-function attraction V 1(x) = −uδ(r) to the potential. That is, the total potential is V (x) = V 0(x) + V 1(x). To rst order in u, what are the corrections to the energies of the rst four radial (` = 0, n = 1, 2, 3, 4) states? 1 (b) By varying u, how well can you match the spectrum 9.460, 10.023, 10.355, 10.580 GeV/c2 for the n = 1, 2, 3, 4 S states of bb using your results to rst order in u? 3. Tunneling amplitudes for an electron. Quantum mechanics is central to understanding the behavior of small (mesoscopic or even nanoscopic) electronic devices. Fabrication techniques now allow one to play all sorts of games with potentials. Consider an electron moving in a narrow wire. Assume that the wire is narrow enough that its motion is only along one dimension. If the wire is metallic, the potential is nearly constant along the wire. Now interrupt the wire with a semiconducting segment of length a that has a potential −∆ relative to the metallic ends. Then, when the potential at both ends of the wire is zero, a plot of the potential would look like this: x V(x) −∆ a If the potential of the right end of the wire (the right lead) is lowered, the potential drop will be spread across the semiconducting region, like this: x V(x) −∆ a VR The potential drop between the left metallic end (where x < 0, V = 0) 2