Quantum Mechanics II: Electric Dipole Transitions & Selection Rules - Problem Set 8, Assignments of Physics

Download Quantum Mechanics II: Electric Dipole Transitions & Selection Rules - Problem Set 8 and more Assignments Physics in PDF only on Docsity! PHY4605–Introduction to Quantum Mechanics II Spring 2005 Problem Set 8 March 21, 2005 Due: Mar. 28, 2005 Reading: Griffiths Ch 9 1. Dipole transitions. The electric dipole approximation to a radiative transition between states of a system of charged particles assumes the velocities of particles are small, so the size of the radiating system is negligible compared to the wavelength of the radiation, and it neglects the interaction of the system with the magnetic field of the radiation (exactly the opposite of what we did when studying hyperfine transition–why?). The energy of a particle with charge q at position r in an electric field E is V = −qr · E. (1) In the electric dipole approximation, the variation of the electromagnetic field across the system is neglected, so the electric field at the position of the system is E(t) = E0~² cos ωt, (2) where E0 is the real amplitude of the wave, ω the angular frequency of the radiation, and ~² is a unit polarization vector. The term in the Hamiltonian describing the interaction of a single particle with the radiation is V = −qE0~² · r cos ωt. (3) In the first order electric dipole approx. to the photoionization of atomic hy- drogen, the initial electronic wavefunction is ψi ∝ e−r/a0 , i.e. an electron in the ground state of H, and the final state is approximated as a plane wave ψf ∝ eikf ·r, (4) i.e. an electron zipping away from the atom with momentum h̄k. ? Use first-order t-dependent perturbation theory to find the angular distribu- tion of the ejected electron. That is, find the probability of observing kf in a given direction relative to initial E-field polarization ~². Don’t bother to compute any factors (normalization constants, etc.) which are independent of direction. (Hint: the computation can be simplified by using polar coordinates with z axis along kf . A useful result from spherical trig. is that if r has spherical angles θ and φ in this coordinate system, and ~² has angles θ′, φ′, then the angle Θ between r and ~² satisfies cos Θ = cos θ′ cos θ + sin θ′ sin θ cos(φ− φ′). (5) If you find yourself doing long messy integrals, think again...) 1