Analysis of Physical Systems: Harmonic Oscillators and Spherical Harmonics, Exams of Chemistry

Download Analysis of Physical Systems: Harmonic Oscillators and Spherical Harmonics and more Exams Chemistry in PDF only on Docsity! 1.) Consider a system which is described by the state 1,10,11,1 8 1 8 3 −++= AYYYψ , where A is a real constant, and Y is a spherical harmonic (often noted as | ). m,l >ll m, (a) Show that A = 2 1 if ψ is normalized. (b) Calculate the expectation values of Lz and L2 in the state ψ . (c) Find the probability associated with a measurement that gives zero for the z- component of the angular momentum. 2.) a) Do the following matrices commute? ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − −= ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = 211 101 112 ~ 001 000 101 ~ BA b) Are any of the following matrices Hermitian? If so, which ones? ⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ + − =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +− + =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = 744 449~ 744 449~ 43 19~ i i C ii ii BA c) Find the eigenvalues and normalized eigenvectors of the following matrix: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = 34 43~A 2 docsity.com 3.) Consider the following Hamiltonian H~ for the first four levels of a perturbed harmonic oscillator: ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − + ⎟⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ =+= 0000 0 2 10 2 1 0000 0 2 101 2 7000 0 2 500 00 2 30 000 2 1 ~~~ 0 )1()0( ωεω hh oHHH where 1<<ε and oω is the characteristic frequency of the oscillator. Use time independent non degenerate perturbation theory to calculate the energy of the third level to second order, and the first order correction to the third wave function. 4.) Consider the following Hamiltonian matrix: ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ =+= 00 0 00 200 010 001 ~~~ )1()0( ε εε ε HHH a) Use degenerate perturbation theory to find the first order corrections to the energy for any degenerate levels in this 3 level system. b) Find the “correct” zeroth order eigenfunctions corresponding to the first order energies deduced in part a) c) Find the second order energy corrections for the degenerate levels. 5.) In class, the following expression was derived for the transition probability for absorption, Pq(t), using time-dependent perturbation theory for an oscillating light field interacting with the electric dipole moment of the system: ( ) ( )2 2 2 )1()1( 2 1sin4 )( ωω ωω − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅= qj qj qjjq q tHH tP h Briefly define all the symbols in this result, and explain how this expression incorporates features that are important to spectroscopy. 3 docsity.com