There is a powerful belief amongst us that teaching and learning are coupled in some magical way.
Recently, while searching for an article I'd written and had accepted, I came across a letter to the editor referring to earlier work of mine. How wonderful; an indication that someone had actually read something I'd written.
But, in thinking about it, I was apalled that the editor had not told me about this letter. I refer to J. Chem. Ed. 61,268(1984), by F. O. Ellison. Ellison objected to my using Cartesian coördinates to show that an LCAO-MO for the hydrogen molecule cation was not an eigenfunction of the appropriate Hamilonian. Instead, Ellison showed symbolically that this was the case. Of course, we were both "correct", but the question in my mind then, and now, is- what pedagogically works for students of physical chemistry.
In teaching valence bond or molecular orbital theory, one needs to deal explicitly with coördinate systems, so that one can locate the nuclei. The figure shows three coördinate systems applicable to the problem at hand. Which should one use? How should it be used? What does using it do for the student?
The typical chemistry student has ended his/her mathematical formal instruction with differential equations. Thus partial differential equations in two or more independent variables is not usually included in the curriculum. So a Hamiltonian of the form
$$
-\frac{\hbar^2}{2m_e}
\left (
\frac{\partial^2}{\partial x^2}+
\frac{\partial^2}{\partial y^2}+
\frac{\partial^2}{\partial z^2}
\right )
-\frac{Ze^2}{r}
$$
can not be referring to $r_A$ or $r_B$, but to $r = \sqrt{x^2+y^2+z^2}$. That is
the radius from the origin in spherical polar coördinates, viz., $r,\vartheta,\varphi$.
So, when Ellison writes
$$
H\psi_{trial}=\left [
-\frac{\hbar^2}{2m}\nabla^2 -\frac{ Ze^2}{r_A} - \frac{ Ze^2}{r_B}
\right ](1s_A+1s_B)
$$one
has the right to ask, does the student understand the difference between $r$ and $r_A$?
As one traces backwards the accumulated learning that was required to get to these points the student/reader has the right to ask 'what does this symbolism mean?'. After all, $r_A = \sqrt{x^2 + y^2 + (z- R/2)^2}$ and $r_B= \sqrt{x^2 + y^2 + (z+R/2)^2}$ so the LCAO proposed is a little complicated.
So, we return to the idea of third year chemistry majors studying physical chemistry for the first time
and dealing with mathematics based on two years of standard calculus instruction.
Having been out of classrooms for more that 12 years, I can no longer attest to what students have been exposed
to, but my guess is that $\nabla$ is not one of symbols they've seen.
But I'm sure that they've never seen an equation of the form $Operator$ operating on $function$ yields a $number$ times $the\ same\ function$, i.e., $H\psi = E \psi$. Actually, most likely they've "seen" it, but never actually used it.
Following backwards the accumulated knowledge that these students have
acquired is an amazing journey.
Consider the gradient
$$
\vec{\nabla} =
\hat{i}\frac{\partial}{\partial x}+
\hat{j}\frac{\partial}{\partial y}+
\hat{k}\frac{\partial}{\partial z}
$$
Does the student know this operator? Is it related to $\nabla^2$?
Where did we learn this?
And, relative to the diagram, what is $\nabla$ and $ \nabla^2$
in elliptical coördinates?
And what is an LCAO-MO in these elliptical coördinates?
As we trace backwards from the current state of the art, looking at the assumed knowledge
required for understanding, we see the reverse process of educational curriculum development
through the lens of practitioners.
This enormous pyramid of learned techniques and facts has to be reverse engineered so that children can absorb this material and incorporate this knowledge into their armamentarium.
How we do this is the task of educators who do not know what their charges will need; the material they are presenting or the material they are omitting. The future will unfold, and leaving out a topic could easily warp understanding many years hence.
On the other hand, most of their charges will never use any of the mathematics they are taught!
Consider the backwards path from partial differential equations, through calculus, through advanced algebra, solid geometry, intermediate algebra, trigonometry, plane geometry, elementary algebra, arithmetic, and finally numbers, and more than finally, fingers and toes.
Can we really truncate this learning path?
It seems doubtful we can; but instead, we leave it to the child to drop off of the heirarchical tree. That seems to
work in most cases, but the doors that are closed for these children are rarely re-opened.
Perhaps this is the best a civilization can do.
