this was very well said!, currently taking a computational chem class and the entire lecture my professor went over long deviations for no reason when all she really needed to do was explain in this method, more so the theoretical aspect of it.
Should E only be along the diagonal in the Matrix? I guess when I think of eigenvalue problems I am subtracted lambda*identity matrix, (in this case E*identity matrix)? Am I missing something? Thanks and I really appreciate the series!
When you were solving the polynomial for E, couldn't you have simply brought beta to the right side and isolated for E, instead of having to solve the quadratic?
Hi Jonathan. In this case, you could have isolated beta^2 and (alpha - E)^2 and taken the square root to arrive at the same answer given here. In most cases, you won't be so lucky and the equation will have terms of other polynomial orders in beta which disallow such an approach. The quadratic equation will work for all second order polynomials, which is why it was used here. In general the polynomial will be n-th order in beta, and if it can't be factored the roots can only be found by advanced mathematical software (Matlab, Maple, etc.) using numerical methods, and/or guess-and-check by hand.
this was very well said!, currently taking a computational chem class and the entire lecture my professor went over long deviations for no reason when all she really needed to do was explain in this method, more so the theoretical aspect of it.
Sometimes a good example is worth a thousand derivations.
You did a really good job, man! Cheers from Brazil!
Thanks from California.
Short and sweet!
Just the way it should be.
How to culculate for napthaline
Should E only be along the diagonal in the Matrix? I guess when I think of eigenvalue problems I am subtracted lambda*identity matrix, (in this case E*identity matrix)? Am I missing something? Thanks and I really appreciate the series!
When you were solving the polynomial for E, couldn't you have simply brought beta to the right side and isolated for E, instead of having to solve the quadratic?
Hi Jonathan. In this case, you could have isolated beta^2 and (alpha - E)^2 and taken the square root to arrive at the same answer given here. In most cases, you won't be so lucky and the equation will have terms of other polynomial orders in beta which disallow such an approach. The quadratic equation will work for all second order polynomials, which is why it was used here. In general the polynomial will be n-th order in beta, and if it can't be factored the roots can only be found by advanced mathematical software (Matlab, Maple, etc.) using numerical methods, and/or guess-and-check by hand.