Homework Set #5 - Introduction to Quantum Mechanics 1 | PHYS 5260, Assignments of Quantum Mechanics

Download Homework Set #5 - Introduction to Quantum Mechanics 1 | PHYS 5260 and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Spring 2008 PHYS 5260: Quantum Mechanics - II Homework Set 5 Issued March 14, 2008 Due April 4, 2008 Reading Assignment: Shankar, Ch.19; Sakurai, Ch 7. 1. Compute the probability current density j for a state ψ(r) = eik·r + f(θ) e ikr r . Argue that far away from a target (of interest in a scattering problem) and θ 6= 0, many of the terms oscillate fast with θ and therefore average to zero, when integrated even over a narrow range of θ (physically corresponding to acceptance/resolution window of the detector). Thereby show that j reduces to j = h̄k m + h̄k m |f(θ)|2 r2 r̂, with the second contribution corresponding to the scattered current density. 2. Diffraction from a crystal: Bragg scattering Within Born approximation, compute a scattering amplitude f(θ, φ) and the corre- sponding differential scattering cross section dσ dΩ for scattering from a large cubic crys- tal, i.e., from a periodic array of identical “blobs” (e.g., atoms) located at lattice sites Rn = an1x̂ + an2ŷ + an3ẑ, with −N ≤ ni ≤ N ; (2N)3 is the number of lattice sites. The scattering potential of the crystal is given by V (r) = ∑ Rn v(|r−Rn|), where v(r) is an isotropic potential characterizing an atom. (a) Work out above scattering for i. v(r) = v0a 3δ3(r), ii. v(r) = v0e −r2/a2 , iii. v(r) = v0θ(a − r), (where the theta-function θ(r) is of course not to be confused with the polar angle θ), where a models a finite radius an atom. (b) What is the condition on q that determines the location of Bragg peaks that characterize the scattering amplitude (as you should discover)? (c) Show that the three cases of v(r) above only differ in the form factors, that provide an envelope for the array of Bragg peaks. (d) Show that the form factors for cases (ii) and (iii) become isotropic in the limit of low angle θ scattering, such that ka → 0 and/or long wavelength and reduce in form to that of (i). Hints: (a) You should find our friend, Poisson summation formula N∑ n=−N eiqn = sin q(N + 1/2) sin q/2 , (1) N1 = ∑ p 2Nδq,2πp (2) = ∑ p 2πδ(q − 2πp) (3) extremely useful. (b) Your calculation of the scattering for the crystal should break up into computation of a scattering amplitude for an atom (giving you a so-called “form factor”), and a computation of coherent superposition from all atoms, with this second part independent of v(r) and only determined by crystal structure (cubic here). 3. Scattering amplitude properties (a) Using the relation between scattering amplitude f`(k) for partial wave ` (angular momentum `) and the scattering matrix S`, show that unitarity of S` (particle conservation) implies that f`(k) = 1/(F̃`(k) − ik), where F̃`(k) is a function of k (or equivalently energy E = h̄2k2/2m), whose details are determined by the potential V (r). Show that F̃`(k) = k/ tan δ`, with δ` scattering phase shift in `th partial wave. (b) By analyzing general properties of the Schrodinger’s equation (that determine F̃`(k)), it can be shown that F̃`(k) = F`(k 2)/k2`, where F`(k 2) is another function that is analytic in k2; you do not need to show this for a general case, but we will see examples of this for specific V (r) below. Use this form of f`(k) to argue that generically low energy scattering is dominated by s-wave (` = 0) scattering, with other partial waves (channels) subdominant. (c) At low energy F`(k 2) can be well approximated by its two lowest Taylor expansion terms, F`(k 2) ≈ c0 + 12c1k 2. For the dominant s-wave scattering c0 ≡ −1/a, c1 ≡ r∗, with a the scattering length and r∗ the effective range of the potential, giving f0(k) = 1 −a−1 + 1 2 r∗k2 − ik . (4) Hence show that at low energies the total cross section is generically given by σ = 4πa2.