Quantum Mechanics: Time Variation & Scattering in 1D Harmonic Oscillator & Phonons, Assignments of Quantum Mechanics

Download Quantum Mechanics: Time Variation & Scattering in 1D Harmonic Oscillator & Phonons and more Assignments Quantum Mechanics in PDF only on Docsity! EEE434/591: QUANTUM MECHANICS 1. Starting from the time-dependent SWE: ψ= ∂ ψ∂ H t iℏ , that has a formal solution of the form: )0(exp)( ψ     −=ψ ℏ iHt t , calculate the time variation of the creation and annihilation operators used to describe the 1D harmonic oscillator problem. In your calculations, use: ℏℏ //)( iHtS iHt H eAetA −= , where AS is some arbitrary operator in the Schrödinger picture, and AH(t) is the representation of this operator in the Heisenberg picture. 2. Using Fermi's Golden rule described in class, calculate the total scattering rate out of state k for the case of a delta-function perturbing potential, i.e. )()( 0 rr δ=VV . In your calculations consider a three-dimensional crystal and plane-waves as basis functions. 3. The matrix element for scattering between states k and k' for polar optical phonon scattering, which is dominant scattering mechanism for GaAs bulk material at low electric fields, equals to ( )qkkkk ±−δ    +       ε − ε ω = ∞ ' 1)0( 11 2 )',( 0 0 2 2 2 N N Vq e M LO ℏ where: k = wavevector corresponding to the initial state, k' = wavevector corresponding to the final state, q = momentum transfer in the scattering process, N0 = phonons occupancy factor given by the Bose-Einstein statistics (top term in the square brackets corresponds to the phonon absorption and the bottom to the phonon emission process),