## Monday, February 26, 2018

I think one of the greatest moments of being a teacher or student is when the student understands or learns something that they find exciting, satisfying, or stimulating. In this "Ah hah!" or Wow! moment they will say "That is really cool!" or "That is beautiful!" or something similar.
These moments can be so significant that the student can years later even remember the exact time, location, or circumstance in which the event happened.

Did you have any such experiences when you were an undergraduate?

I reflected on my own experience. Even though it is almost 40 years ago I can remember what I learnt and sometimes the place, the book, the person, ...
Here is some of the things that immediately came to mind. They are listed in random order. It is interesting that many involve learning how one result follows from a more fundamental result with a simple mathematical proof. Often it meant there was a deeper reason for something we had previously been told was "just the way it is".
Most of these beautiful moments were in theoretical physics and pure mathematics. None were in chemistry. I think this was partly because of my own interests and orientation and partly because of the quality (or lack thereof) or approach to teaching of different subjects.

Ehrenfest's theorem
The equations of motion of classical mechanics are the average of the equations of motion for position and momentum operators.

Heisenberg's uncertainty relation follows from commutation relations.

The energy eigenvalues for the harmonic oscillator can be derived from the commutation relations of creation and annihilation operators.
No differential equations or Hermite polynomials were required!

Experimental test of time dilation from measurement of the lifetime of cosmic-ray mesons

van der Waals interaction from the Schrodinger equation
I learnt this derivation from reading my father's copy of Quantum Chemistry (1957) by Walter Kauzmann.

Electromagnetic radiation and the speed of light from Maxwell's equations

The ideal gas equation of state from the partition function

The logical structure of the laws of thermodynamics
I learnt this axiomatic approach from Hans Buchdahl, both from his book and his lectures.

Functional analysis and the equivalence of matrix and wave mechanics
This was in a pure mathematics class. It is really just an isomorphism of Hilbert spaces.

Evaluation of infinite series from residues in complex analysis
Cauchy's residue theorem can be used.

Dimensional analysis in fluid mechanics
It was amazing the physical insights one could gain simply from dimensional analysis.

Newtonian gravity from Einstein's gravitational field equations

How do we create such moments for students?

1. Learning how assembler instructions were carried out by CMOS transistors. Bridging the device physics and programming gap was an 'aha!' moment for me.

2. I agree with Ross' listing of seeing the ideal gas equation "drop out" from the classical partition function. Although in my case I didn't see this until I was in graduate school - perhaps this says something about how closely I was paying attention as an undergraduate?

3. I distinctly remember being wowed by the relationship between symmetries and conserved quantities in classical mechanics and how this is encoded in Poisson brackets. Furthermore, these brackets turn into commutators in quantum mechanics with the addition of an i*h. I thought that that was amazing.

4. The trick for evaluating the integral of a gaussian function, squaring it and using polar coordinates.

5. Well, in undergrad, all of the above. But they never felt like unique "aha" moments.

1) realizing that Julian Schwinger's 1 grad year quantum class lectures were the Chinese Lunch of lectures ... none of his beautiful arguments, explaining the above, held water upon close thought. Always a catch! See below.

2)learning from our experiments that chemical reactions (in a dilute gas) really DO obey the laws of physics and not some sort of organic chemist's magic. You CAN just calculate, more or less, where the atoms go using classical mechanics.

2b) corollary: this really should win a Nobel Prize, which it did. And the post-doc later showed quantum effects (not tunneling) on the acattering.

As a professor: Some chemical reactions are dominated by relativistic effects!
Example: the collisions of Br + I2, which require relativistic electronic structure calculations to get the spin-orbit surfaces split by up to an eV .. they dominate the chemistry!

After retiring I decided to learn relativistic quantum field theory. FINALLY, now, I had the real AHA! moment. It all fits, and while one still has to ask Mother Nature for advice, things like the Exclusion Principle are not magic, which they always seemed.

6. Now you have written in in your blog.
"These moments can be so significant that the student can years later even remember the exact time, location, or circumstance in which the event happened"

The reason for your premise ( the above lines) lies in the famous quote below, centuries ago by the greatest polymath Leanardo da Vinci

"The noblest pleasure is the joy of understanding"

7. " none were in chemistry" Though chemistry has great application value for example pharmaceuticals etc. the subject is not taught well either in school or even at tertiary level. They try to teach too much in a short period, unlike physics and maths. This overloading in chemistry syllabus leads to rote learning.

8. Dear Feb. 27 anonymous, I'm the very long anonymous. The reason chemists seldom experience aha moments is that very few are very physics-y ones, and hate P-chem classes with a passion.

And outside of that ... it really IS too gloppy.