Time-Dependent Perturbation Theory: The Photoelectric Effect - Finite System Approach, Study notes of Quantum Mechanics

Download Time-Dependent Perturbation Theory: The Photoelectric Effect - Finite System Approach and more Study notes Quantum Mechanics in PDF only on Docsity! PHY 6646 K. Ingersent Time-Dependent Perturbation Theory: The Photoelectric Effect • This handout mirrors the treatment of the photoelectric effect on Shankar pp. 499–506, with two principal differences: (1) The perturbing Hamiltonian is written H1E = eE·R instead of H1A = (e/mc)A · P. (2) The system is assumed to occupy a cubic box of sides L, whereas Shankar treats an infinite system. We comment on the significance of these differences at the end. • The initial state is taken to belong to the innermost (or K) shell of a hydrogen-like atom of effective nuclear charge Ze, with wave function 〈r|i〉 = π−1/2(Z/a0)3/2 exp(−Z|r|/a0), where a0 = h̄ 2/me2 is the Bohr radius. This state has energy εi = −Z2e2/2a0 = −(Zα)2mc2/2, α = e2/h̄c being the fine-structure constant. The characteristic size of the orbital is r0 = a0/Z = h̄/(Zαmc). We consider a monochromatic electromagnetic plane wave, E(r, t) = E0 cos(k · r−ωt). The electric dipole approximation is valid provided that |k|r0  1, or equivalently, h̄ω  (Zα)mc2. We will consider frequencies in the window (Zα)2mc2  h̄ω  (Zα)mc2, where not only can we make the dipole approximation, but the final-state energy is sufficiently high that the final state should be well-described by a plane wave of the form 〈r|f〉 = L−3/2 exp(ipf ·r/h̄) having an energy εf = |pf |2/2m. (See Shankar p. 500 and the end of this handout for discussion of this plane-wave approximation.) • In the dipole approximation, we need to calculate the dipole matrix element rfi = 〈f |R|i〉 = A ∫ d3r e−ipf ·r/h̄ r e−Z|r|/a0 = A × ih̄ ∂ ∂pf ∫ d3r e−ipf ·r/h̄ e−Z|r|/a0 = ih̄ ∂ ∂pf 〈f |i〉, where A = π−1/2(Z/La0)3/2. The overlap integral is straightforward to evaluate (see Shankar p. 504): 〈f |i〉 = A ∫ d3r e−ipf ·r/h̄e−Z|r|/a0 = 8πAZ/a0 [(Z/a0)2 + (pf/h̄)2]2 . Therefore rfi = ih̄ 8πAZ/a0 [(Z/a0)2 + (pf/h̄)2]3 (−4pf h̄2 ) . Noting that (Z/a0) 2 + (pf/h̄) 2 = 2m(εf − εi)/h̄2, we find rfi = − 2ih̄ m(εf − εi) pf〈f |i〉 = −i 4πAZh̄5 a0m3(εf − εi)3 pf . (1) • Fermi’s Golden Rule gives the scattering rate from |i〉 to |f〉 as Ri→f = 2π h̄ ∣∣∣∣e2 E0 · rfi ∣∣∣∣ 2 δ(εf − εi − h̄ω). 1