Okay. So in this video, let's continue to talk about observables. So remember what
an observable is. It, we, we said if you have a k-level system then an observer for
the system is a key-by-key Hermitian matrix. Remember what a Hermitian matrix
is? It's, it's a matrix where A equal to A conjugate transpose. A equal to A dagger.
Okay. And also you remember what, what is special about this so when we are
measuring using such, such an observable, what we are really saying is we are going
to measure in the basis of Eigenvectors of this, of this observable. And, and so its
particularly useful to us that Hermitian matrix happens to have an orthonormal set
of Eigenvectors with three Eigenvalues. So, so the, the measurement rules are now
that we are measuring on the basis of Eigenvectors and if, the outcome is the
j-th Eigenvector, the outcome of the measurement is the j-th, Eigenvalue. So
for example, you know, if we, if we go back to our original example where we,
where we have an atom and the electron in an atom, let's say a hydrogen atom, and
we, we think of you know the, the ground state as, as the zero state, the first
excited state, the one state, and so on up to, you know, all the way up to k minus
one. So that's our k-level system. So now the, the observable, if you wanted to
measure the energy of the system, would be the observable that is you know, whose,
whose Eigenstates are zero through k minus one. And what would be the Eigenvalues?
Well, the Eigenvalues would be the energies of these corresponding states.
So, what, what, what would this look like? Well in this case, if you wanted to write
out this observable, whether Eigenvectors actually zero through k minus one, what
does this look like? Well, it looks like a diagonal matrix, right? Because it's the
basis vectors as, as a basis vectors and the standard basis are the Eigenvectors.
And then, what should we put on the diagonal? We'll put the energies of these
corresponding states. So, we have E naught, E1, E sub k minus on E. So the
energies of, of the ground state, the first excited and so on and everything
else would be zero. So that would correspond to the energy of zero. This is
also called the Hamiltonian of the system. Okay. So, so now let's ask the other
question. How general is this notion of an observer, right? We saw that its
completely compatible with our definition of a measurement that we, that we had
before where we just take an orthonormal basis and we measure in it. But now
suppose that we have an arbitrary orthonormal basis in mind. An arbitrary
real outcomes lambda one through lambda k. Can we find an, a Hermitian matrix A which
has these Eigenvectors and corresponding Eigenvalues? Okay. So, so let's try to
work with an example. So, suppose that we want Eigenvectors to be plus and minus,
this is for qubit. And we want the Eigen, corresponding Eigenvalues to be 2n -
three. So how do we construct a Hermitian matrix A whose Eigenvectors are plus,
minus, Eigenvalues two and three. So let's, let's start with trying to make
plus Eigenvector. So the idea is let's, let's use the projection matrix which
projects onto the plus state. So you remember what that was? So, you look at
this matrix. Right. Do you remember what this was? So, get plus, that's one over
square root two, one over square root two. Bra plus, it's just a conjugate transpose,
everything's real so we don't have to worry about that, it's just the roll
vector. And if you multiply it out, you get half, half, half, half. Okay, so this
matrix projects onto the plus state. Okay. Why is that? Well, if you, if you, if you
apply it to a general state psi, what do you get? Well, you get, okay, so, let's do
it slowly. You get this times psi which you know, again, grouping these in a
different order just by transitivity and we get this is, this is inner product of
plus and psi which is a number. This is equal to plus times in a product of plus
and psi so a, and this is a complex number. This is a complex number, so we
can move it out front and so it's just inn er product of plus and psi times plus.
Okay. So in particular, if psi equal to plus then, then, this times psi is just,
is just plus. Alright? So, so, plus is an Eigenvector. The plus state is an
Eigenvector of this matrix with Eigenvalue equal to one. What about if psi equal to
minus? Well, if psi equal to minus, you seem to get a multiple of plus, but which
multiple? Well, you get the inner product of plus and minus which is zero, right? If
psi equal to minus, then you get zero times plus which is just zero. So, so
minus is also an Eigenvector of this, of this matrix with Eigenvalue equal to zero.
Okay. But remember, we wanted, we wanted plus to have Eigenvalue two. So we could
look at the matrix which is two times this, okay? What's, what's two times this?
It would be the matrix which is one, one, one, one. And so, look at this which is
two times this inner product. And so now, plus is an Eigenvector with Eigenvalue
equal to two. Similarly, we could make up the projection matrix which, which
projects on to the minus state and what's this, this matrix equal to? It's just one
over square root two minus one over square root two that's get, get minus one over
square root two minus one over square root two and Bra minus, which is what? It's,
it's a half, half minus half minus half. Okay. And now, this is a matrix where
minus and plus are both Eigenvectors but minus is an Eigenvector with Eigenvalue
one plus is an Eigenvector with Eigenvalue zero. Again, what we'd like is Eigenvalue
minus three so we can multiply by minus three. So now, if you want, if you want
simultaneously Eigenvalue two for plus and minus three for minus, the, the natural
choice is to look at the matrix which is two times the projection on to plus, plus
minus three, times the projection on to minus. And now, what happens if you, if
you multiply this? If you take this matrix and, and, and act on the plus state with
it? Right? So, so you'll get, for the first stem you'll get plus times two, and
for the second part you will just get, zero. So y ou get two times plus. So plus
is an Eigenvector with Eigenvalue two. Similarly, minus is an Eigenvector with
Eigenvalue minus three. And what's the actual matrix? It's, it's actually, let's
see, it's one, one, one, one and then plus minus three times this. So what would it,
what would it actually end up being? If you work it out it would be, the matrix
would be, you get five halves here, five halves here, and then you get minus three
halves plus one, so minus a half, minus a half. Okay. What about the general case?
Well, the general case is, completely analogous, right? So, if you want an
operator, an observable with Eigenvector is phi i, Eigenvalue is lambda i, then we
just look, look at the linear combination which is which is the project on to phi i
multiplied by lambda. Okay. So, so what would it take to show that this has
Eigenvectors phi i with corresponding Eigenvalues lambda i? Well, just show that
A times phi sub j equal to lambda sub j, okay? Completely analogously to what we
did earlier. Okay. So, so what have we shown? We've shown that this language of
observables is completely analogous to our previous notion of measurement. So what
we're, what we are saying is, whether we pick an observer A to Hermitian matrix,
this is completely equivalent to pick an orthonormal basis of phi sub i's and, and
corresponding measurement outcomes. So what we see on the meter, the deflection
of the needle, lambda sub i's.