If you chose to read this optional chapter, you obviously have the interest and time in going a step beyond the basic quantum mechanics needed for the course.  Congratulations.
If not, you’ll be fine as long as you study the material in the previous lesson.  This chapter only comes into play in our next Foothill College course, CS 83B. Today, we consider
what happens to a quantum system – for us a qubit – when it’s allowed to evolve over time. We’ll add a few more words to our vocabulary.  Terms like "Hamiltonian," "energy eigenkets"
and the famous "Schrodinger equation."  We've met a few of the operators corresponding to the key measurables in our spin-1/2 world.  The computer scientist’s ever present Sz,
as well as the very useful Sx and Sy. Imagine that instead of measuring the x, y or z-component of spin, we decide to read off the total energy – the sum of potential and kinetic energy
-- of our quantum system.  The observable for that quantity is called the Hamiltonian.
Developing the equation for this new measurable requires a little first year physics, so I’ll spoon feed you a few results. When I’m finished, you’ll see something that should look
familiar.  Like all operators in C-2, it’s a 2x2 matrix. This matrix is the measurable corresponding to the system’s total energy – the spin-1/2 Hamiltonian.   And we solve it just like
we solve any operator.  We compute its eigenvalues – the allowable energies that appear when we look at the system – and eigenvectors – also known as eigenkets --  that
correspond to those energies. When you follow along in the chapter, you’ll see that the energy eigenkets are the same two vectors that we got when we solved the S-z observable:
ket-plus and ket-minus.
Well, what about time evolution? Remember how any vector in our state space can be expanded along an observable’s basis?  Well that’s till true.  Take any state vector.  We can
express it as a superposition of the energy eigenkets. But now there’s something new.  Since we're allowing the states to evolve in time, the expansion coefficients must be
functions of time --  functions that, for the time being, are unknown. Now, a short footnote to the physicists:
I’m presenting the Schrodinger, not the Heisenberg, picture of evolution, which is why we can say that the observables remain fixed and the states, or kets, are the thing that
change over time.
In any event, this leaves us with one final challenge.  To find the actual formulas for the two unknown functions. I’d like you to say hello to the Schrodinger equation.  It provides the
framework that enables us to solve for the unknown time-dependent expansion coefficients. When the smoke clears, our original state vector  -- which can be any vector in the
space – will be expressed as a superposition of the energy eigenkets, and the coefficients of those eigenkets will be functions of time. Let’s pause for a second and notice what
happens if the state whose evolution we’re trying to compute happens to be one of the energy eigenkets. The time evolution doesn’t seem to change anything, since vectors that
differ by a mere scalar -- especially a length one scalar -- are considered identical states.
Because of this, energy eigenkets are often called "stationary states."  They don’t change – they’re stationary. Now, energy is nice, but we may want to study one of our other
observables, say Sy.  Do we have to do everything  all over again? No. We just solved a time-dependent expansion along the energy basis, so all we have to do is convert that solution
from the energy basis to the S-y basis. Now it’s time to say “thank you” to our old friend the dot-with-the-basis vector trick. We take our evolving ket, currently expressed in terms
of the energy basis, and dot it with the two S-y basis vectors to get the coefficients along the that  basis.
Now we know what we can expect to see if we measure this other observable, S-y, at any future time t.
That’s a lot to take in so I’ll summarize. ONE: Compute the energy eigenkets for the system.  But for quantum 
computing, they’re always the same, ket+ and ket-, so
there’s nothing really to do. TWO:  Expand our vector of which is the same as the Sz or preferred basis. THREE: Solve the Schordinger
equation to get that state’s energy at any future time t.  Again, I did this for you, so nothing to do. FOUR: For any other observable that we plan on measuring, say for example
S-y, we dot the time-dependent solution with each of S-y’s eigenkets. And FIVE: We declare victory.
In the final section of this chapter, I’ll pick some initial state, and we’ll do this for all three observables, Sx, Sy, and Sz,  and after we get the answers, we’re going to go a step further.
We’ll compute their evolving expectation values.  This is just as easy as computing ordinary expectation values, only we’ll have a time variable, t, in all of our answers. What we get
tells us how the expectation value of the three observables evolves in time. It’s a three-dimensional vector – which we can think of as a spin direction  that precesses about the
axis of our magnetic field. It’s called "Larmor precession," and the surprise here is that it will reproduce the original, classical, naïve and incorrect, model of our spin-1/2 electron
in full classical detail, but with a different interpretation than we initially expected two chapters ago. And, there you have it.  A summary and introduction to the final installment in your
lightning education in quantum mechanics.  Whatever path you plan to take – or took -- I hope you find this week’s material enlightening.
