**1. Status update:**

The monsoon is *officially* here, in the Pune city, and somehow, my spring-break [^] too gets over—finally!

(…There were no rains on the day that the Met. department officially announced the arrival of monsoon in the Pune city. The skies were, in fact, fairly clear on that day! … However, this year, everything is different. It was raining on almost every day in the month of *May!*)

Anyway, to come back to the reason for the *permanent* break in the spring-break which I had taken…

Looks like I have found a minimum working clarity regarding the phenomenon of the quantum mechanical spin. … I guess the level of clarity which I have now got is, perhaps, as good as what might be possible within the confines of a strictly non-relativistic analysis.

So… I can now with some placid satisfaction proceed to watching the remaining lectures of the MIT courses.

I will begin writing the document on my new approach a little later. I now expect to be able to put it out by 15th July 2021, perhaps earlier. [Err… Any suggestions for the (Hindi) “muhurat”s for either / both?]

…But yes, the quantum spin turned out to be a tricky topic to get right. Very tricky.

…But then, that way, *all* of QM is tricky. … Here, let me highlight just one aspect that’s especially fun to think through…

**2. One fun aspect of QM:**

The Schrodinger equation for a one-particle system is given as:

,

where the notation is standard; in particular, is the imaginary unit, and is the system Hamiltonian operator.

The observation I have in mind is the following:

Express the complex-valued function explicitly as the sum of its real and imaginary parts:

,

where , and do note, also , that is, both are *real*-valued functions. (In contrast, the original ; it’s a *complex-*valued function.)

Substitute the preceding expression into the Schrodinger equation, collect the real- and imaginary- terms, and obtain a *system* of two *coupled* equations:

and

.

The preceding system of two equations, when taken together, is *fully equivalent* to the single complex-valued Schrodinger’s equation noted in the beginning. The emphasis is on the phrase: “fully equivalent”. Yes. The equivalence is *mathematically* valid—*fully!*

Now, notice that this latter system of equations has *no imaginary unit appearing in it*. In other words, *we are dealing with pure real numbers here*.

Magic?

… Ummm, not really. Did you notice the negative sign stuck on the left hand-side of the second equation? That negative sign, together with the fact that the in the first equation, you have the real-part on the left hand-side but the imaginary part on the right hand-side, and vice versa for the second equation, pulls the necessary trick.

This way of looking at the Schrodinger equation is sometimes helpful in the computational modeling work, in particular, while simulating the time evolution, i.e., the *transients*. (However, it’s not so directly useful when it comes to modeling the stationary states.) For a good explanation of this viewpoint, see, e.g., James Nagel’s SciPy Cookbook write-up and Python code here [^]. The link to his accompanying PDF document (containing the explanation) is given right in the write-up. He also has an easy-to-follow peer-reviewed paper on the topic; see here [^]. I had run into this work many years ago. BTW, there is another paper dealing with the same idea, here [^]. I have known it for a long time too. …Some time recently, I recalled them both.

Now, what is so tricky about it, you ask? Well, here is a *homework* for you:

** Homework:** Compare and contrast the aforementioned, purely real-valued, formulation of quantum mechanics with the viewpoint expressed in a recent Quanta Mag article, here [^]. Also include this StackExchange thread [^] in your analysis.

Happy thinking!

OK, take care, and bye for now…

**A song I like:**

[TBD]