Congratulations, part 1:  We did it.  We learned or reviewed the essential math necessary to tackle basic quantum algorithms.
Now, it’s time to move on to the other prerequisite: quantum mechanics.
This week, I’ll give you a full introduction to that subject, including enough different paths so that you can choose how deeply you want to go.
There are three chapters for our three class days.  For the most part, you only need to study and master the second one.
You can read this first chapter lightly without worrying about the details and if you want, skip the third entirely.
So what’s in today’s lecture optional lecture?  Well, it’s a story about how the classical physics stops working when we begin making more precise measurements of microscopic
systems. But instead of giving you history lesson, I made up my own arc the fits the needs of quantum computing yet still preserves the essence of the subject. We begin by describing
a very special physical system. An electron with one particular property that the electron possesses called “spin.”  We know electrons have spin because when we shoot them
through electromagnetic fields, they bend in ways that can only be explained by having this spin quality.  But the details of their behavior don’t match classical physics.
So we’ll start out imagining, naively, that electrons are tiny charged balls, rotating about some axis, an incorrect but easy-to-visualize picture.
We do few experiments and see how we’re forced to change our model.
In fact, when we’re done, we will have had to abandon our original picture entirely and replace it with some unintuitive  math, and that math is quantum mechanics.
It’s called spin-1/2 quantum mechanics to be precise.
I’ll give you the highlights, and you can decide whether or not this chapter’s for you.
If elecetrons were, in fact, spinning charged particls, their axis of rotation would have an orientation,
expressed by a unit vector called "n-hat," pointing in the direction of that axis of rotation.
It would also possess the quantity or magnitude of spin --  essentially how fast it’s rotating – expressed by some real number, let's call it S.
If that were true, billions of electrons would have billions of different spin orientations and magnitudes, and if we tested them, we’d get wide range of results.
Instead we only get two answers – up electrons and down electrons (details to be supplied in the chapter).
Well, we do a few follow-ups throwing away the down electrons and subjecting the up electrons to even further testing, and every time we do this we stranger and stranger
results. One by one, we’re forced to throw away our initial classical assumptions until finally… we have to toss the old model out the window, and adopt a strange new mathematics
to accurately predict the results of these experiments. The mathematics says that electrons have spin magnitude which I’ll call it ½ to avoid scaring you.
And although they do have  a kind of orientation that can have infinitely many values,
the secret of that orientation is destroyed every time we try to measure it. It snaps --  or as physicists like to say, collapses
to a state that's either an up angle of 55 degrees, or a down angle, of 55 degrees.
And what happened to our original classical model, where we thought that an electron’s spin could point in any 3-dimensional direction?
Did that have any use at all?
We’ll come back to that question in an optional section later this week, but the brief answer is yes. Surprisingly the full 3-D vector that we naively assigned to the electron’s spin, is
deeply present, but we can only see it if we do millions of experiments and analyze the data. But it has a different meaning than we thought;  it represents something called the
expectation value of three observables named Sx, Sy and Sz. I guess you’ll have do a little of the optional reading later this week if you want to hear the surprise ending of that sub-plot.
By the way, all these fancy symbols you’re seeing …  if you want to know what they mean, you do not have to look anywhere else. They’re all explained right here in these three
chapters. Some of them may appear in optional reading, but if you go for the options, I promise all is revealed without the need for any other reference.
