PS 9 Sol - CHEM 120A Problem Set 9 Solutions David Hoffman...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CHEM 120A Problem Set 9 Solutions David Hoffman , Doran Bennett , Tara Yacovitch 4/26/2010 1. (a) We’re asked to verify that | Φ k i = | ψ- k i-| ψ k i is also and eigen solution of our Hamiltonian. Let’s see we know that | ψ- k i is degenerate with | ψ k i which means they have the same energy eigenvalue associated with them (you can prove this to yourself by finding the action of H on both eigenstates keeping in mind that cosine is an even function). If they both have the same energy eigenvalues then we have, H | Φ k i = H | ψ- k i - H | ψ k i = E | ψ- k i - E | ψ k i = E | Φ k i (1a) The second part of this problem involves evaluating Φ k (0) which I’m confident you are capable of doing. (b) We want to show that the boundary condition is satisfied for certain values of k . Before proceeding note that, Φ k ( x ) = sin( kx ) (1b) Continuing, Φ k ( b + Mb ) = sin( kx ) = sin( πs ) = 0 (1c) The final equality holds because s is an integer. (c) Here we have a long, poly-unsaturated alkyl chain and we want to estimate the energies of the ions relative to the neutral species. We know that we have n possible k values and that there are likewise n different wave functions (you might say “orbitals”) corresponding to these k values. Each orbital can be occupied by two electrons, so the energy of the neutral system is, E = 2 n/ 2 X s =1 α + 2 β cos πs ( n + 1) (1d) Where n is the number of π electrons for n even and, E = α + 2 β cos π (( n- 1) / 2 + 1) n + 1 + 2 ( n- 1) / 2 X s =1 α + 2 β cos πs n + 1 (1e) 1 For j odd. For l = 0 that means that j = n . Assuming that we have an even number of electrons and setting the zero of energy to be equal to α the difference in energy between our neutral molecule and the- 1 ion is, Δ E = 2 β cos π ( n/ 2 + 1) n + 1 = 2 β cos π 2 1 + 1 n + 1 (1f) While the difference in energy between our neutral molecule and the +1 ion is,...
View Full Document

{[ snackBarMessage ]}

Page1 / 6

PS 9 Sol - CHEM 120A Problem Set 9 Solutions David Hoffman...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online