Create a "molecular state" for one electron in a 1D potential consisting of 2 attractive delta functions of equal strength separated by a distance d. (This is a very simple model molecule; sort of H+ in 1D.)
Suppose your state is a linear combination of atomic states. Imagine fixed alpha, and that the value of the atomic wavefunction parameter corresponding to alpha is a. For specificity, let's consider two cases: i) where d=a and ii) d=2a.
Suppose your state is expressed as as ...exp{-(x-d)/(c a)} + exp{-(x+d)/(ca)}, where c is a confinement related parameter which can be varied. (While keeping d and a fixed.)
Here is the question for you: There are different possibilities regarding what value of c will minimize the total energy of the molecular state. Define the question by delineating the possibilities, and discuss what value of c you think will minimize the energy. You do not have to be right. Just try to argue/discuss energy in a meaningful way. In fact, you can, and are encouraged to, argue for different viewpoints (in different paragraphs).
(You can hand this in on Thursday, if you like.)
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3 comments:
For tomorrows class, we can at first discuss the issues around this problem for a few minutes. Then I would like to have two (~3 person) teams to debate the question:
A value of c greater than one will optimize the energy.
You can organize a team to take one side or the other if you like. Volunteers are welcome.
You don't have to be right. Just to be able to make arguments involving T and V (and T+V=E) makes you great*!
Remember, the focus will be on T and V, and how they are influenced by, and influence, "ca", the varialble confinement length parameter.
The first things to think about might be: how does ca effect T? How does ca effect V? What causes molecular binding? (Binding suggests a lowered energy.) Does that lowered energy come from a lowered T or V....?
Please feel free to get the ball rolling by commenting here...
* As Rilke says: " We should allow one another to grow great, for diminishing comes easily and does not need to be learned."
In the case that we have only one delta function potential, we found the optimal value of c to be 1. For our gaussian wavefunction, c=1 gives the full-width, half-max value as a or alpha. In our scenario, we will want a wavefunction that spreads out more and tries to occupy both potentials at once. Thus c must be greater than 1.
This will still correspond to a lower total energy. Our potential energy contribution will go down since the particle will now be more in regions with 0 potential (between the deltas). The kinetic energy will also go down because our relation of T ~= 1/c^2 as found from a single delta potential should still hold as a good approximation. Additionally, a wavefunction that is more spread out has a lower kinetic energy (look at the infinite square well energy as the length is increased).
The strongest argument I can come up with for the a>1 case is that the potentials now have to fight each other for ownership of the electron so it will force our gaussian wavefunction to spread out.
nice start. Keep those comments coming! One note: remember The delta functions are attractive hence the exp value of V is negative so spending time in between, where V(x)=0, might tend to make exp of V less negative (higher in a sense).
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