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Okay. So in this video, let's continue to
talk about observables. So remember what

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an observable is. It, we, we said if you
have a k-level system then an observer for

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the system is a key-by-key Hermitian
matrix. Remember what a Hermitian matrix

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is? It's, it's a matrix where A equal to A
conjugate transpose. A equal to A dagger.

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Okay. And also you remember what, what is
special about this so when we are

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measuring using such, such an observable,
what we are really saying is we are going

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to measure in the basis of Eigenvectors of
this, of this observable. And, and so its

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particularly useful to us that Hermitian
matrix happens to have an orthonormal set

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of Eigenvectors with three Eigenvalues.
So, so the, the measurement rules are now

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that we are measuring on the basis of
Eigenvectors and if, the outcome is the

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j-th Eigenvector, the outcome of the
measurement is the j-th, Eigenvalue. So

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for example, you know, if we, if we go
back to our original example where we,

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where we have an atom and the electron in
an atom, let's say a hydrogen atom, and

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we, we think of you know the, the ground
state as, as the zero state, the first

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excited state, the one state, and so on up
to, you know, all the way up to k minus

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one. So that's our k-level system. So now
the, the observable, if you wanted to

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measure the energy of the system, would be
the observable that is you know, whose,

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whose Eigenstates are zero through k minus
one. And what would be the Eigenvalues?

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Well, the Eigenvalues would be the
energies of these corresponding states.

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So, what, what, what would this look like?
Well in this case, if you wanted to write

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out this observable, whether Eigenvectors
actually zero through k minus one, what

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does this look like? Well, it looks like a
diagonal matrix, right? Because it's the

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basis vectors as, as a basis vectors and
the standard basis are the Eigenvectors.

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And then, what should we put on the
diagonal? We'll put the energies of these

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corresponding states. So, we have E
naught, E1, E sub k minus on E. So the

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energies of, of the ground state, the
first excited and so on and everything

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else would be zero. So that would
correspond to the energy of zero. This is

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also called the Hamiltonian of the system.
Okay. So, so now let's ask the other

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question. How general is this notion of an
observer, right? We saw that its

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completely compatible with our definition
of a measurement that we, that we had

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before where we just take an orthonormal
basis and we measure in it. But now

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suppose that we have an arbitrary
orthonormal basis in mind. An arbitrary

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real outcomes lambda one through lambda k.
Can we find an, a Hermitian matrix A which

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has these Eigenvectors and corresponding
Eigenvalues? Okay. So, so let's try to

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work with an example. So, suppose that we
want Eigenvectors to be plus and minus,

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this is for qubit. And we want the Eigen,
corresponding Eigenvalues to be 2n -

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three. So how do we construct a Hermitian
matrix A whose Eigenvectors are plus,

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minus, Eigenvalues two and three. So
let's, let's start with trying to make

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plus Eigenvector. So the idea is let's,
let's use the projection matrix which

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projects onto the plus state. So you
remember what that was? So, you look at

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this matrix. Right. Do you remember what
this was? So, get plus, that's one over

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square root two, one over square root two.
Bra plus, it's just a conjugate transpose,

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everything's real so we don't have to
worry about that, it's just the roll

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vector. And if you multiply it out, you
get half, half, half, half. Okay, so this

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matrix projects onto the plus state. Okay.
Why is that? Well, if you, if you, if you

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apply it to a general state psi, what do
you get? Well, you get, okay, so, let's do

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it slowly. You get this times psi which
you know, again, grouping these in a

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different order just by transitivity and
we get this is, this is inner product of

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plus and psi which is a number. This is
equal to plus times in a product of plus

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and psi so a, and this is a complex
number. This is a complex number, so we

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can move it out front and so it's just inn
er product of plus and psi times plus.

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Okay. So in particular, if psi equal to
plus then, then, this times psi is just,

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is just plus. Alright? So, so, plus is an
Eigenvector. The plus state is an

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Eigenvector of this matrix with Eigenvalue
equal to one. What about if psi equal to

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minus? Well, if psi equal to minus, you
seem to get a multiple of plus, but which

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multiple? Well, you get the inner product
of plus and minus which is zero, right? If

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psi equal to minus, then you get zero
times plus which is just zero. So, so

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minus is also an Eigenvector of this, of
this matrix with Eigenvalue equal to zero.

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Okay. But remember, we wanted, we wanted
plus to have Eigenvalue two. So we could

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look at the matrix which is two times
this, okay? What's, what's two times this?

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It would be the matrix which is one, one,
one, one. And so, look at this which is

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two times this inner product. And so now,
plus is an Eigenvector with Eigenvalue

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equal to two. Similarly, we could make up
the projection matrix which, which

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projects on to the minus state and what's
this, this matrix equal to? It's just one

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over square root two minus one over square
root two that's get, get minus one over

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square root two minus one over square root
two and Bra minus, which is what? It's,

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it's a half, half minus half minus half.
Okay. And now, this is a matrix where

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minus and plus are both Eigenvectors but
minus is an Eigenvector with Eigenvalue

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one plus is an Eigenvector with Eigenvalue
zero. Again, what we'd like is Eigenvalue

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minus three so we can multiply by minus
three. So now, if you want, if you want

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simultaneously Eigenvalue two for plus and
minus three for minus, the, the natural

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choice is to look at the matrix which is
two times the projection on to plus, plus

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minus three, times the projection on to
minus. And now, what happens if you, if

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you multiply this? If you take this matrix
and, and, and act on the plus state with

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it? Right? So, so you'll get, for the
first stem you'll get plus times two, and

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for the second part you will just get,
zero. So y ou get two times plus. So plus

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is an Eigenvector with Eigenvalue two.
Similarly, minus is an Eigenvector with

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Eigenvalue minus three. And what's the
actual matrix? It's, it's actually, let's

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see, it's one, one, one, one and then plus
minus three times this. So what would it,

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what would it actually end up being? If
you work it out it would be, the matrix

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would be, you get five halves here, five
halves here, and then you get minus three

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halves plus one, so minus a half, minus a
half. Okay. What about the general case?

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Well, the general case is, completely
analogous, right? So, if you want an

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operator, an observable with Eigenvector
is phi i, Eigenvalue is lambda i, then we

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just look, look at the linear combination
which is which is the project on to phi i

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multiplied by lambda. Okay. So, so what
would it take to show that this has

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Eigenvectors phi i with corresponding
Eigenvalues lambda i? Well, just show that

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A times phi sub j equal to lambda sub j,
okay? Completely analogously to what we

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did earlier. Okay. So, so what have we
shown? We've shown that this language of

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observables is completely analogous to our
previous notion of measurement. So what

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we're, what we are saying is, whether we
pick an observer A to Hermitian matrix,

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this is completely equivalent to pick an
orthonormal basis of phi sub i's and, and

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corresponding measurement outcomes. So
what we see on the meter, the deflection

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of the needle, lambda sub i's.