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Hi everybody. So, today we are going to
talk about Schrodinger's equation, which

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is the basic equation of motion that
describes how quantum systems evolve.

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Before we get there, let's talk a little
bit about observables. So, what is an

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observable? It's a quantity like energy or
position or spin, something that you can

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measure. And, okay so what do we mean by
this? Well, there's some quantum state

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that we have, and we feed it into a
measuring apparatus. And the measuring

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apparatus has a, has a meter. If it's an
old fashioned measuring apparatus it has a

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meter which with, with, it has a needle
where, where the needle deflects. And so,

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the deflection of the needle is some real
number, it's labelled by real a number and

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so, so an observer is, is operator that
tells us, given this quantum state, what

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the real number is? Okay. So, more
precisely, suppose that we are working

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with a K level system. So that a quantum
state is a unit vector in a K dimensional

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complex vector space, Hilbert space. Then
the observable for the system is, is an

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operator A, which is a K by K emission
matrix. Alright. Remember emission matrix

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is, is a complex matrix such that, A = A
conjugate transpose. A = A dagger. So, for

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example, if K = two then we might have the
following emission matrix. So, so the

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diagonal must be real so it might be one -
two and then, and then this entry is the

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conjugate of that. Okay. But you might be
able a bit puzzled down because when we

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already have the notion of measurement?
So, if you wanted to measure a quantum

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state, we just picked an orthonormal
basis, 5N through 5K and then the

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corresponding outcomes would be one
through K, so, you, you can number them

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five, zero through 5K - one, and the
outcome zero through K - one. That's,

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it's, its sort of an arbitrary choice.
Okay, but that's what we said, said a

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measurement was, so what's this new notion
of an observable? Okay. So why, how can,

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how can it be that the measurement now
corresponds to this, this Hermitian

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matrix? Well, so, let's just think back.
What's special ab out a Hermitian matrix?

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A Hermitian matrix is special because
there's the spectral theorem. So is, if

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is, is Hermitian then it has a orthonormal
set of eigenvectors phi one through phi K

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with real eigenvalues lambda one through
lambda k. What this means is A phi I is

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lambda I phi. Okay, so, so now what does
it mean to use this observer to do a

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measurement? Well, all it means is that if
he write our state psi in the eigenbasis

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you know phi one through phi K. Then the
measurement outcome is going to be lambda

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I with probability. The square of the
magnitude of, of the I-th component of

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psi. And the new state is just phi. Okay,
so let's, let's step back and understand

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what this is saying. So what we are saying
is, when you specify a Hermitian operator

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A, Hermitian matrix A, you are really
specifiying an autonomal basis. Which

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autonomal basis? The autonomal basis of
the eigenvectors of A. But now, not only

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do you get this orthonormal basis but you
also get a, get corresponding real numbers

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which are the eigenvalues. So, what does
it mean when we, when we do a measurement

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according to E? Well, what they're saying
is. A specifies an orthonormal basis which

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is phi one, phi two through phi of K.
Okay, that's, that's once we write down A

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we've specified an orthonormal basis. Now
if quantum state was psi, what happens

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when we, when we do this measurement? Well
psi gets projected onto one of these, the

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measurement basis states. So if it happens
to be phi sub I. Well then the new state

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is phi sub I, but what do we actually read
out in our, in our, in our measurement?

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Well before we were saying we read out I,
but really what we read out is lambda sub

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I. So if, if, if the measurement outcome
was phi sub two. Then, then our meter

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would show a deflection of lambda sub two
which is a real number. Okay? So those are

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the, those are the rules of measurement.
So, for example, if, if our state was

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alpha zero + beta one, it's a qubit. And
suppose our observable is X, which is a

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bit flip. Now, remember the eigenvalues,
eigenvectors of x. So, the eig envectors.

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Let's call them phi one = to the + state
and phi two = to the - state, and the

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corresponding eigenvalues, lambda one =
+one lambda two = -one. Okay so now, we

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want to understand what happens when we do
a measurement. So, when we measure we, we

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need to write down this state psi in this
eigenbasis. So if you write it in the

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eigenbasis, you can sort of work this out.
It's, it's I believe it's alpha + beta /

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square root two + alpha - beta / square
root two - just do a change of basis. And

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so, so now the outcome of the measurement
is plus one with probability alpha plus

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beta over square root two magnitude
squared. And if so, the new state is just

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plus and it's -one with probability of a -
beta / square root two magnitude squared

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and then the new state is -. We could also
ask what is the expected value of the

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outcome? Well if it's one + one with this
probability, -one with this probability,

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the expected value of the, of the outcome
of the measurement is just one alpha +

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beta / square root two^2 + -one alpha -
beta / square root two magnitude square.

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Okay? Okay, Now there's something that I
actually what, what I said so far was not

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entirely correct and the reason its not
entirely correct is because we do have to

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worry about repeated eigenvalues. So lets
take this three dimensional example. So,

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so lets say that we have been observable
and it has, it has these eigenvectors phi

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one phi two, phi three but, but suppose
lambda one = lambda two = one, right? So,

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now you could ask what happens if the
measurement outcome is one? Do you get, is

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the new state phi one or phi two? Right,
well. Okay so here, here's the, here's the

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correct picture. You see the spectrum
theorem really tells us that. Okay. So in

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the case that there are repeated
eigenvalues, like lambda one = lambda two

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then in fact, you can find this
orthonormal basis phi one and phi two are

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definitely eigenvectors but actually, any
vector in this, in this, in this space

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spanned by phi one and phi two is also an
eigenvector with eigenvalue one. So, for

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example, if you took. One / square root
two phi one plus one / square root two phi

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two. You can check that this is also an
eigenvector and eigenvalue exactly one.

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Why? Because if you hit it with A. Well, A
phi one is phi one and A phi two is phi

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two. So you'll just get back one over
square root two phi one + one / square

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root two phi two. And of course, you can
see that this would hold for any linear

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combination of phi one and phi two. Okay,
so, so, you know the fact that we singled

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out phi one and phi two is completely
artificial. In fact, it's the whole

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eigenspace, the entire eigenspace with
which with eigenvalue one. So, so the

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outcome of this measurement would be,
well, it's either one or two. And, if the

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outcome is one, then the new status just
the state psi projected onto this

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eigenspace. I'd say projected into this
plane. Okay, and what's the probability

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that you will get this outcome? Well, you
can, you can, you know you can figure out

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the probability either by working it out
as the, you know alpha one is the

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amplitude in this direction phi one then
it's going to be alpha1 magnitude squared

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+ alpha two magnitude square or you could
use Pythagoras theorem and you could say

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directly the probability that you'll get
this outcome is just square of this

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projection so it's, it's this length
squared.