PCh10 Lec 3

(4/29/20, bes)

Well...we are making great progress in the material. Great job turing in your lab reports!

Let's take a step back and look at the big picture...we have come a long way...

The Big Picture

As stated on the Wikipedia site, the time-independent Schrodinger equation looks like the following:

YES...sorry this is considered the easier, time-independent Schrodinger equation...the time-dependent Schrodinger equation is used in spectroscopy since there is a time-dependence of when you excite the system...it apply a pulse of RF in FT-NMR...or any light (UV, vis, IR, etc). Time-dependent quantum or "Quantum: Spectroscopy" is a graduate level class that most pchem PhD students will take.

Parts of Schrodinger Equation

As we transition from the analytical solutions for the model QM systems and H-atom (Ch < 10) to the numerical solutions for the real-world chemical systems (Ch > 10), we can separate our discussions based on the 3 (interconnected) aspects of a the Schrodinger equation. I will outline these discussion below and then go into more details later.

Calculation of Energy

Sec 10.4: Using the Variational method to solve the Schrodinger equation.

As the name states the "variational method" means that you "vary" some parameter in the Schrodinger equation (mainly the wavefunction) and then calculate the energy. This method introduces some vocabulary...to start this method we select a trial wavefunction. Notice that we use the term wavefunction and not eigenfunction; the eigenfunction is the analytical solution and the wavefunction is the numerical solution...our hope of finding the true eigenfunction is long gone!

Variational method steps...

1) using the trial wavefunction in the Schrodinger equation we calculate an "energy_01"

2) we vary some aspect of the trial wavefunction and calculate energy_02.

3) We compare energy_01 with energy_02

if energy_01 > energy_02, then we accept the variation we made to the trial wavefunction,

if energy_01 < energy_02, then we decline the variation we made to the trial wavefunction,

the variational theorem says the best wavefunction is the one that generates to lowest energy.

4) return to step 1, then 2, accept/decline, until any changes in the trial wavefunction results in the ~same energy, energy_n ~ energy_n+1.

at this point your answers have become "consistent"...or "consistent with its self" or self-consistent...

Questions that arise...

How does one choose the best trial wavefunction to start?