Quantum Mechanics Homework Set 7: Solutions and Explanations - Prof. Leo Radzihovsky, Assignments of Quantum Mechanics

Download Quantum Mechanics Homework Set 7: Solutions and Explanations - Prof. Leo Radzihovsky and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Fall 2007 PHYS 5250: Quantum Mechanics - I Homework Set 7 Issued November 17, 2007 Due December 14, 2007 Reading Assignment: Shankar, Ch. 13, 14, 15; Sakurai: 3.5-3.8, 3.10, Appendix A.4, A.5, A.6. 1. Three-dimensional spherical square well potential with V (r) = −V0, for r < d and 0, for r > d. (a) For the bound states (E < 0): i. Find the eigenfunctions with angular momentum ` and a matching condition determining the corresponding coefficients of the radial part of the wavefunc- tion. Express your answer in terms of special functions, but do not try to solve the transcendental equations for the matching coefficients. Argue based on this general structure of the solution that the spectrum must be discrete. Hint: note that at large r the spherical Henkel function of h (1) ` (kr) = j`(kr)+ in`(kr) ∼ eikr/(kr). ii. Specializing to ` = 0, use above general result to explicitly find eigenfunc- tions in terms of elementary functions and greatly simplified transcendental matching equation. Formulate the transcendental equation graphically. iii. What is the critical value V (n) 0c at which nth bound level appears? Use this result to find the minimum value V c0 of the potential depth V0 below which an ` = 0 bound state is impossible. Compare your answer with that of a 1d square well potential from an earlier homework assignment. Explain the connection and why here (in contrast to that earlier true 1d problem) a bound state only appears if the potential well is sufficiently deep, i.e., V0 > V c 0 . iv. Study the problem for a very deep well and arbitrary `. For states with energy |En,`| comparable to V0 (i.e., very deeply lying low energy bound states), determine the order in which these appear; use the |n, `〉 notation or the spectroscopic notation 1s, 1p, 2f, etc...to list the lowest 10 states. Hint: In this last case a considerable simplification takes place, but the eigen- values must still be computed numerically. (b) For the continuum states (E > 0): i. Write down the appropriate wavefunctions for inside and outside regions and the corresponding matching condition. Based on this make an argument that the eigenenergies form a continuum (rather than a discrete set). ii. Write down large r asymptotic form of these wavefunctions and use it to relate the coefficients B and C of the outer solution to the (so-called) scattering phase shift δ`(k), latter defined by the asymptotic form of the wavefunction R(r →∞) ∼ (kr)−1 sin(kr − π 2 `+ δ`). iii. Use above results to approximately compute δ0(k) for the special case of ` = 0, showing that at low energies (small k) k cot δ0(k) ≈ −a−1 + 12r0k 2, and giving explicit expressions for parameters a (scattering length) and r0 (effective range). iv. Show that in V0 →∞ limit, a reduces to the potential range d. v. Using your results above, demonstrate a general result that δ`(k) → 0 as k → 0, and derive how rapidly (with k) it approaches zero for a given `. 2. Hydrogen atom (a) An electron in the Coulomb field of a proton is in a state described by a wave function ψ(r) = Ae−r/r0 . What is the probability that (while in this state right before the measurement) it will be found in (i) a state with angular momentum ` 6= 0 and m 6= 0? (ii) its ground state; for the latter, compute and sketch P (r0/a0) and show that it is maximized at 1 for r0 = a0. Why? (a0 is Bohr radius). (b) Show that eigenfunctions of a spherically symmetric Hamiltonian are also eigen- states of parity with eigenvalue P = (−1)` determined only by the angular mo- mentum quantum number `. 3. Angular momentum (a) Derive Pauli matrices ~σ by explicitly computing matrix elements of angular mo- mentum components ~J for the j = 1/2 representation and using ~J = 1 2 h̄~σ. (b) Using their commutation and anticommutation relations derive Pauli matrices identities: i. Tr(σiσj) = 2δij, also verifying it by explicit matrices found above. ii. (~a · ~σ)(~b · ~σ) = ~a ·~bI + i(~a×~b) · ~σ. (I is an identify matrix.) iii. Use above identities to show that a square of the 2D Dirac Hamiltonian HD = c(~p−e ~A/c)·~σ+mc2σz (describing a spin 1/2 particle; ~p = (px, py)) gives the Klein-Gordon Hamiltonian HKG = H 2 D = (~p− e ~A/c)2c2 +m2c4−2ce ~B · ~S (describing a spin zero particle). Taylor expand the square-root of the latter to lowest order in (~p−e ~A/c)2/m2c2 to show that (upto a constant, that is the rest energy mc2) one gets the Hamiltonian for the nonrelativistic Schrodinger’s