Download 2 Questions on Griffiths and Perturbation Theory - Homework 5 | PHYS 487 and more Assignments Quantum Physics in PDF only on Docsity! Physics 487 Homework 4: Due Oct. 8, 2008 1. ) Griffiths: 6.19, 6.21. 2.) 1) Perturbation theory Hydrogen Atom. This problem is designed to provide a refresher for the perturbative analyses of the hydrogen atom of the last weeks. Below we will make order of magnitude estimates of the various effects and we will derive the hierarchy given in Griffiths, Table 6.1 (you will be spared of involved calculations on this topic!). Since the order of magnitude is already given in the table, it is important that you show all the steps of your derivation. Oftentimes, it is these types of estimates, and not the involved calculations, that can give you the required insight, and even go far in solving problems. (a) First let us consider the (unperturbed) hydrogen atom discussed in Grif- fiths, Chapter 4. What is the Hamiltonian describing the atom? Consider the ground state energy of the hydrogen atom given by Griffiths, Eq. 4.70. Express the energy in terms of the fine structure constant α defined in Griffiths Eq. 6.43 and other constants such as the mass of the electron. Derive the value of α by substituting the values of the fundamental constants in the formula. (b) The first order corrections to the Hamiltonian of part (a), known as the fine structure, arise due to relativistic effects and spin-orbit coupling. What is the physical origin of each of these effects? The Hamiltonians appropriate for each of these effects are given in Griffiths Eq. 6.50 and Eq. 6.61, respectively. For each of these forms, make an order of magnitude estimate of the effects by replacing operators by appropriate scales (lengths by the Bohr radius; spin and angular momentum by h̄, etc; remember that momentum can be equated to an inverse length). Rewrite these estimates in terms of α and other constants. Compare your result to the result obtained in Eq. 6.66 show that it is of the same order of magnitude. (c) Consider the Zeeman coupling of the electron to an external magnetic field given in Griffiths Eq. 6.71. For what fields B does the magnitude of the Zeeman effect compare with the fine structure correction of part (b)? (Making sure you understand the reasoning of Section 6.3.2, you can use Eq. 6.59 for comparison.) (d) A still smaller correction to the hydrogen spectrum comes from the hyperfine interaction discussed in Griffiths Sec. 6.5. What is the physical origin of this effect? As in part (b), make an estimate of this effect in terms of α , etc, using the form HHF µ0gpe 2Sp · Se/(8πmpmer 3) where gp is about 6, and recall, µ0ǫ0 = c 2. (e) Given the hierarchy of intrinsic effects in parts (a), (b) and (d), (no applied fields) how would one do perturbation theory? Which would be the unperturbed basis states for handling each of the perturbative effects? 1
