Today’s lecture is the second in a three-lesson series on quantum mechanics, and the only one required to get through this course. But a light reading of the previous chapter on the
physics of quantum computing, might help if at any time you feel lost. First and foremost, what is quantum mechanics?  
Well I can tell you what it’s not.  It’s not quantum physics.
Physics involves experiments in the real word -- hardware, atoms, golf balls and photons.  Those are called physical systems. Quantum mechanics is the mathematical symbol-
manipulation game that we do with pen and paper and serves as an abstract model of those physical systems.
The game allows us to predict what will happen if we ask an engineer or physicist to do an actual experiment with real  hardware. Quantum physics is the real world system.
Quantum mechanics gives us a mathematical analog in which to study the real world  system.  That analog is called the “state space” of the system. The simplest physical
system for quantum computing turns out to be a spin-1/2 sub-atomic particle like an electron. The state space for doing  the quantum mechanics in that physical system is C-2, the
2-dimensional Hilbert space of ordered pairs of complex numbers. We will construct higher dimensional state spaces for multi-qubit systems, but they’ll all be built by combining
these simple 2-dimensional Hilbert spaces, C-2.
Quantum mechanics is based on a few assumptions called postulates.”  You cannot prove them – they’re the result of a century of testing and retesting.  So far, no experiment has
contradicted these assumptions. The first postulate is the great gift that physicists have given us computer scientists and mathematicians.  It tells us that we can do all of our work
with pen and paper in some Hilbert space. We don’t need a lab, we don’t need any apparatus – we don’t even need physicists … we just need to know what the state space is.
And the reason it’s called a state space is that every state that the physical system is in, corresponds to some vector in the state space.
Notwithstanding a detail that we’ll get into later, states are vectors, and vectors are states. And please forgive me for continuing to repeat it, but the
fundamental state space for quantum computing is C-2, the ordered pairs of complex numbers. As we develop C-2, we’ll establish a preferred basis, called the z-basis, and meet
some alternate bases. The 2nd postulate of quantum mechanics tells us that anything we can measure is called an observable. And observables correspond to special
operators, that is linear transformations, in our Hilbert space – those operators have the fancy-sounding property of being “Hermitian.”
But keep it simple:  The states are vectors, and the observables are linear transformations.
Minor note to help you avoid future embarrassment at a party:  taking a measurement does not involve applying the observable’s corresponding operator to a state. Taking a
measurement is an irreversible, non-unitary operation.  The Hermetian operator associated with the observable is used in a different way that you’ll learn about later today.
We’ll meet the most important observable,  S-z, the z-component of an electron’s spin.
But before you freak out, remember – even though I talk about electrons and spin, all you really need is the simple 2x2  matrix for S-z.  We’re blissfully insulated by our warm, safe
Hilbert space. The third postulate of quantum mechanics says that if we try to look at a – or measure – the state of a system, we can only get a very limited number of possible
results that go by the unnecessarily scary sounding name “eigenvalues.” It gets even more interesting.  Before we measure anything, the state could be any one of the billions
of possible Hilbert space state vectors. But if we so much as look at it, it always collapses to a basis vector.  More vocabulary:  basis vectors in physics are called
“eigenvectors.” Stated a little simplistically, the measurement gives us one of the special eigenvalues, and the state collapses to that eigenvalue’s corresponding eigenvector.
In order to earn bragging rights about knowing quantum mechanics, you’ll have to learn how to compute eigenvectors and eigenvalues, and I’m going to teach you that in the next
few hours. But it’s nothing more than using determinants of small matrices, and then solving some simple linear equations.  I’ve got you covered. We can almost never know
what we get when we look at a qubit’s value, but the fourth postulate of quantum mechanics comes to our rescue. It tells us the exact probability of getting each of the possible
eigenvalue measurements when we do read that qubit’s location.
Remember, when we look at a qubit, it'll collapse and we’ll get one of the eigenvalues. For C-2 -- with the usual measurement operator, S-z -- the eigenvalues are just 0 or 1.
So this amounts to knowing the probability of detecting a 0 … or a 1. The symbol manipulation game I promised you is called the "Bracket” or bra – ket, notation.  It’s fun, it’s easy
and it’s good for you. So today, I’m going to give you examples and drill you with exercises to make sure you have that skill before releasing back into the wild. The final topic
today is “expectation values.”   These are the average measurements we get if we repeat a quantum experiment over and over and keep track of the results.
We’d like to be able to predict these expectation values, and the math to do so is something you already learned, but we’ll review in this chapter. And if you’re interested, you might read
the next optional chapter, because in it, you’ll learn that these expectation values are the connection between the new quantum physics and the older classical physics. In it,
there’s a section called “Larmor Precession”, which shows us why our classical idea of spin pointing in any spatial direction actually has a physical meaning that can be tested.  It’s
connected with the expectation value of our three observables, Sx, Sy and Sz. But that’s all optional.  We only need today’s chapter for the entire course, so begin the
reading and we’ll exchange questions and answers in the class forums.  See you there.
