Problems on Fourier Coefficients, Quantum Mechanics, and Spectra - Prof. David Ginger Jr, Assignments of Quantum Chemistry

Download Problems on Fourier Coefficients, Quantum Mechanics, and Spectra - Prof. David Ginger Jr and more Assignments Quantum Chemistry in PDF only on Docsity! Homework 5A Due Mon Nov 6 Part A. More problems will be assigned throughout next week. Levine 6.16, 6.17, 6.47, 7.27, 7.29 1) A Dirac notation equivalent of our integrals for finding Fourier coefficients is the “projection operator” Pn = |n> <n| (it may look strange but you know how to “use” it). Pn projects out the component of any arbitrary ket that lies along the basis ket |n> . (i.e. if ψ= a|1> + b|2> + c|3>) then Pnψ = |1>(<1| a|1> + <1|b|2> + <1|c|3>)= a|1>. Combine the definition of completeness in Levine 7.39 with Levine 7.71 to show that 1|| =><∑ n nn This is the more common statement of completeness for an orthonormal basis set. Note: this is really a notational problem—just translate Levine’s equations into Dirac notation, and using the definitions you are familiar with show that this new operator must return f when operating on any function f. 2) What is the most probable distance (in meters) of finding a 1s electron from a Uranium nucleus that has been ionized of all but 1 electron? What is the potential energy of an electron this far away from a uranium nucleus (in Joules)? The virial theorem (an important result from chapter 14 that we haven’t had time to cover) tells us that 2 * <KE> of this electron will equal -<U> (or that <KE>=-<E>), the average PE of this electron. Use a classical estimate to calculate how fast such an electron is moving. Comment. 3) Following Levine, use a computer to make linear combinations of Y(1,-1), Y(1,1), Y(1,0) to generate the familiar px, py and pz orbitals. Make 3D plots of the angular components to verify that you indeed get the angular factors from px and py. Why can we take linear combinations of these eigenfunctions and create new eigenfunctions of H, when we know that in general, linear combinations of eigenfunctions are not also eigenfunctions. 4) See next page: