Hi everybody. So, today we are going to talk about Schrodinger's equation, which
is the basic equation of motion that describes how quantum systems evolve.
Before we get there, let's talk a little bit about observables. So, what is an
observable? It's a quantity like energy or position or spin, something that you can
measure. And, okay so what do we mean by this? Well, there's some quantum state
that we have, and we feed it into a measuring apparatus. And the measuring
apparatus has a, has a meter. If it's an old fashioned measuring apparatus it has a
meter which with, with, it has a needle where, where the needle deflects. And so,
the deflection of the needle is some real number, it's labelled by real a number and
so, so an observer is, is operator that tells us, given this quantum state, what
the real number is? Okay. So, more precisely, suppose that we are working
with a K level system. So that a quantum state is a unit vector in a K dimensional
complex vector space, Hilbert space. Then the observable for the system is, is an
operator A, which is a K by K emission matrix. Alright. Remember emission matrix
is, is a complex matrix such that, A = A conjugate transpose. A = A dagger. So, for
example, if K = two then we might have the following emission matrix. So, so the
diagonal must be real so it might be one - two and then, and then this entry is the
conjugate of that. Okay. But you might be able a bit puzzled down because when we
already have the notion of measurement? So, if you wanted to measure a quantum
state, we just picked an orthonormal basis, 5N through 5K and then the
corresponding outcomes would be one through K, so, you, you can number them
five, zero through 5K - one, and the outcome zero through K - one. That's,
it's, its sort of an arbitrary choice. Okay, but that's what we said, said a
measurement was, so what's this new notion of an observable? Okay. So why, how can,
how can it be that the measurement now corresponds to this, this Hermitian
matrix? Well, so, let's just think back. What's special ab out a Hermitian matrix?
A Hermitian matrix is special because there's the spectral theorem. So is, if
is, is Hermitian then it has a orthonormal set of eigenvectors phi one through phi K
with real eigenvalues lambda one through lambda k. What this means is A phi I is
lambda I phi. Okay, so, so now what does it mean to use this observer to do a
measurement? Well, all it means is that if he write our state psi in the eigenbasis
you know phi one through phi K. Then the measurement outcome is going to be lambda
I with probability. The square of the magnitude of, of the I-th component of
psi. And the new state is just phi. Okay, so let's, let's step back and understand
what this is saying. So what we are saying is, when you specify a Hermitian operator
A, Hermitian matrix A, you are really specifiying an autonomal basis. Which
autonomal basis? The autonomal basis of the eigenvectors of A. But now, not only
do you get this orthonormal basis but you also get a, get corresponding real numbers
which are the eigenvalues. So, what does it mean when we, when we do a measurement
according to E? Well, what they're saying is. A specifies an orthonormal basis which
is phi one, phi two through phi of K. Okay, that's, that's once we write down A
we've specified an orthonormal basis. Now if quantum state was psi, what happens
when we, when we do this measurement? Well psi gets projected onto one of these, the
measurement basis states. So if it happens to be phi sub I. Well then the new state
is phi sub I, but what do we actually read out in our, in our, in our measurement?
Well before we were saying we read out I, but really what we read out is lambda sub
I. So if, if, if the measurement outcome was phi sub two. Then, then our meter
would show a deflection of lambda sub two which is a real number. Okay? So those are
the, those are the rules of measurement. So, for example, if, if our state was
alpha zero + beta one, it's a qubit. And suppose our observable is X, which is a
bit flip. Now, remember the eigenvalues, eigenvectors of x. So, the eig envectors.
Let's call them phi one = to the + state and phi two = to the - state, and the
corresponding eigenvalues, lambda one = +one lambda two = -one. Okay so now, we
want to understand what happens when we do a measurement. So, when we measure we, we
need to write down this state psi in this eigenbasis. So if you write it in the
eigenbasis, you can sort of work this out. It's, it's I believe it's alpha + beta /
square root two + alpha - beta / square root two - just do a change of basis. And
so, so now the outcome of the measurement is plus one with probability alpha plus
beta over square root two magnitude squared. And if so, the new state is just
plus and it's -one with probability of a - beta / square root two magnitude squared
and then the new state is -. We could also ask what is the expected value of the
outcome? Well if it's one + one with this probability, -one with this probability,
the expected value of the, of the outcome of the measurement is just one alpha +
beta / square root two^2 + -one alpha - beta / square root two magnitude square.
Okay? Okay, Now there's something that I actually what, what I said so far was not
entirely correct and the reason its not entirely correct is because we do have to
worry about repeated eigenvalues. So lets take this three dimensional example. So,
so lets say that we have been observable and it has, it has these eigenvectors phi
one phi two, phi three but, but suppose lambda one = lambda two = one, right? So,
now you could ask what happens if the measurement outcome is one? Do you get, is
the new state phi one or phi two? Right, well. Okay so here, here's the, here's the
correct picture. You see the spectrum theorem really tells us that. Okay. So in
the case that there are repeated eigenvalues, like lambda one = lambda two
then in fact, you can find this orthonormal basis phi one and phi two are
definitely eigenvectors but actually, any vector in this, in this, in this space
spanned by phi one and phi two is also an eigenvector with eigenvalue one. So, for
example, if you took. One / square root two phi one plus one / square root two phi
two. You can check that this is also an eigenvector and eigenvalue exactly one.
Why? Because if you hit it with A. Well, A phi one is phi one and A phi two is phi
two. So you'll just get back one over square root two phi one + one / square
root two phi two. And of course, you can see that this would hold for any linear
combination of phi one and phi two. Okay, so, so, you know the fact that we singled
out phi one and phi two is completely artificial. In fact, it's the whole
eigenspace, the entire eigenspace with which with eigenvalue one. So, so the
outcome of this measurement would be, well, it's either one or two. And, if the
outcome is one, then the new status just the state psi projected onto this
eigenspace. I'd say projected into this plane. Okay, and what's the probability
that you will get this outcome? Well, you can, you can, you know you can figure out
the probability either by working it out as the, you know alpha one is the
amplitude in this direction phi one then it's going to be alpha1 magnitude squared
+ alpha two magnitude square or you could use Pythagoras theorem and you could say
directly the probability that you'll get this outcome is just square of this
projection so it's, it's this length squared.