Hello everyone so today I am going to introduce some of the notation some of
which have been sweeping under the rug, in the context of topics, we've already
covered and some that's going to be, important for upcoming lectures. So, I'm
sure some of you are, have been clamoring for this, you know for this level of
precision. And others of you are dreading it but today is the day and so let's get
started. Okay, so what we'll, what I'll talk about today is well, first I'll start
by talking about the Bra-ket notation. So this is just completing this ket notation
that, that I already introduced. Then I'll talk about commission operators and the
admission matrices and eigenvectors, eigenvalues, and finally I'll talk about
Tensa products. Tensa products of Hilbert spaces. Tensor products of matrices, etc.
Okay, so, so, let's get started. Okay so just to make sure we're all on the same
space, page. So far we've been talking about quantum states which let's say we
are working in a K dimensional Hilbert space. K dimensional complex vector space.
So our quantum state is a K dimensional vector. K complex entries. And in our ket
notation, we write it as alpha naught (zero) + alpha1 (one) + so on. Alpha k -
one / k - one. So, just to remind you, ket notation was introduced by Paul Dirac, the
great physicist and the useful thing about, the ket notation for our purposes
is that at the same time, represents the fact that the state is a vector, and that
it conveys information. So, for example, if you had a qubit, you could write its
state as alpha0(0) + alpha1(1). But we could also write it as, you know maybe,
maybe we don't want to write the states as zero and one. Maybe, maybe it's a spin
state, and we want to write it as alpha0 up + alpha1 down or maybe, maybe we want
to write it instead as alpha0(blue) + alpha1(red). Okay, so the, the great thing
about a ket notation is that it makes it very convenient to talk about a label for
each vector in the bases. And at the same time it, it allows you to express what the
amplitudes are, o r the entries of your vector. Okay, so now corresponding to this
vector, which we'll call ket psi, there's a corresponding vector which we will call.
Bra psi and this is the conjugate transpose. It's, it's a row vector alpha
naught alpha1 alphak - one By star I mean just of course, complex conjugate and
we'll denote this, this vector, by this symbol. So, where, where the, where the
pointy end is to the left. And so, together with the ket. This forms Dirac's
Bra-ket notation. Okay so, so, now suppose we have, we have another vector or state
phi which has beta naught(0) + beta1(1) + so on beta k - one, k - one. Okay, so then
what's the inner product of, of psi and phi?. Okay. So if you want to write out
the inner product of psi and phi, the way you would do it is you would, you would
look, you would take alpha naught alpha1 alpha k - one (beta naught, beta1, beta k
- one which is just. Summation of a J of alpha j beta sub J. Right, that's your
inner product. Now, how to you write it in a ket notation? So the way we write it in
the ket notation is we look, we, this is Bra sign, and then we write a ket phi. And
we, we don't duplicate the, the vertical line in the middle and so that's our
notation for inner product. Okay, so now we can also of course once we have an
inner product on our space, we can define the length of a vector. So, so the norm of
psi or it's length, is just the inner product of psi with itself and then the
square root of that. Okay. But what, what would that end up being? Well, that,
that'll just end up being square root of the summation of alpha j alpha j. Which is
of course the square root of the summation of alpha j magnitude squared. And that's
our usual notion of, of, of length of the vector. Now, the other thing you can, you
know the other thing to. If you, if you wish to think about it, intuitively. Then
the inner product gives you a natural notion of the angle between two vectors,
so if this was phi and that's psi then, somehow we want to think of the, the inner
product as telling us what this a ngle theta is between these two vectors, and so
it tells us what cosine theta is. Well but of course you, you have to remember that
in general this, this inner product is going to be a complex number. And so, if
you want to makes, make, make sense of what angle means, we could, we could sort
of say the angle between these two vectors could be defined like this, the theta
could be defined like this but cosine theta is just a unit product between psi
and phi. It's absolute. It's magnitude, divided by the length of psi times length
of phi. Okay. That's just an intuitive notion of, of what the, you know it's,
it's interpreting what, what, what the angle between two vectors is. And here
we'd, of course, want theta to be between zero and phi or two. Okay. So for example,
you see the, the reason we want to do this is because if we were working with just
real vectors then of course we might have this situation or it could be that we had
-psi, and if you had -psi then the inner product would be negative of what it was
here and of course the angle becomes phi by two plus theta. And so we could do
something similar in, you know in, in general of course if we are working in the
complex plane we don't only have to consider, consider for corresponding to
this, this state's phi + or - phi but we also have to consider the E to the IX phi.
So we could have an arbitrary face like this. And okay, so that's what this
definition takes care of.