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Hello everyone so today I am going to
introduce some of the notation some of

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which have been sweeping under the rug, in
the context of topics, we've already

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covered and some that's going to be,
important for upcoming lectures. So, I'm

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sure some of you are, have been clamoring
for this, you know for this level of

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precision. And others of you are dreading
it but today is the day and so let's get

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started. Okay, so what we'll, what I'll
talk about today is well, first I'll start

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by talking about the Bra-ket notation. So
this is just completing this ket notation

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that, that I already introduced. Then I'll
talk about commission operators and the

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admission matrices and eigenvectors,
eigenvalues, and finally I'll talk about

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Tensa products. Tensa products of Hilbert
spaces. Tensor products of matrices, etc.

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Okay, so, so, let's get started. Okay so
just to make sure we're all on the same

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space, page. So far we've been talking
about quantum states which let's say we

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are working in a K dimensional Hilbert
space. K dimensional complex vector space.

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So our quantum state is a K dimensional
vector. K complex entries. And in our ket

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notation, we write it as alpha naught
(zero) + alpha1 (one) + so on. Alpha k -

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one / k - one. So, just to remind you, ket
notation was introduced by Paul Dirac, the

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great physicist and the useful thing
about, the ket notation for our purposes

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is that at the same time, represents the
fact that the state is a vector, and that

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it conveys information. So, for example,
if you had a qubit, you could write its

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state as alpha0(0) + alpha1(1). But we
could also write it as, you know maybe,

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maybe we don't want to write the states as
zero and one. Maybe, maybe it's a spin

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state, and we want to write it as alpha0
up + alpha1 down or maybe, maybe we want

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to write it instead as alpha0(blue) +
alpha1(red). Okay, so the, the great thing

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about a ket notation is that it makes it
very convenient to talk about a label for

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each vector in the bases. And at the same
time it, it allows you to express what the

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amplitudes are, o r the entries of your
vector. Okay, so now corresponding to this

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vector, which we'll call ket psi, there's
a corresponding vector which we will call.

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Bra psi and this is the conjugate
transpose. It's, it's a row vector alpha

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naught alpha1 alphak - one By star I mean
just of course, complex conjugate and

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we'll denote this, this vector, by this
symbol. So, where, where the, where the

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pointy end is to the left. And so,
together with the ket. This forms Dirac's

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Bra-ket notation. Okay so, so, now suppose
we have, we have another vector or state

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phi which has beta naught(0) + beta1(1) +
so on beta k - one, k - one. Okay, so then

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what's the inner product of, of psi and
phi?. Okay. So if you want to write out

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the inner product of psi and phi, the way
you would do it is you would, you would

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look, you would take alpha naught alpha1
alpha k - one (beta naught, beta1, beta k

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- one which is just. Summation of a J of
alpha j beta sub J. Right, that's your

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inner product. Now, how to you write it in
a ket notation? So the way we write it in

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the ket notation is we look, we, this is
Bra sign, and then we write a ket phi. And

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we, we don't duplicate the, the vertical
line in the middle and so that's our

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notation for inner product. Okay, so now
we can also of course once we have an

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inner product on our space, we can define
the length of a vector. So, so the norm of

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psi or it's length, is just the inner
product of psi with itself and then the

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square root of that. Okay. But what, what
would that end up being? Well, that,

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that'll just end up being square root of
the summation of alpha j alpha j. Which is

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of course the square root of the summation
of alpha j magnitude squared. And that's

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our usual notion of, of, of length of the
vector. Now, the other thing you can, you

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know the other thing to. If you, if you
wish to think about it, intuitively. Then

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the inner product gives you a natural
notion of the angle between two vectors,

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so if this was phi and that's psi then,
somehow we want to think of the, the inner

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product as telling us what this a ngle
theta is between these two vectors, and so

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it tells us what cosine theta is. Well but
of course you, you have to remember that

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in general this, this inner product is
going to be a complex number. And so, if

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you want to makes, make, make sense of
what angle means, we could, we could sort

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of say the angle between these two vectors
could be defined like this, the theta

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could be defined like this but cosine
theta is just a unit product between psi

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and phi. It's absolute. It's magnitude,
divided by the length of psi times length

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of phi. Okay. That's just an intuitive
notion of, of what the, you know it's,

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it's interpreting what, what, what the
angle between two vectors is. And here

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we'd, of course, want theta to be between
zero and phi or two. Okay. So for example,

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you see the, the reason we want to do this
is because if we were working with just

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real vectors then of course we might have
this situation or it could be that we had

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-psi, and if you had -psi then the inner
product would be negative of what it was

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here and of course the angle becomes phi
by two plus theta. And so we could do

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something similar in, you know in, in
general of course if we are working in the

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complex plane we don't only have to
consider, consider for corresponding to

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this, this state's phi + or - phi but we
also have to consider the E to the IX phi.

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So we could have an arbitrary face like
this. And okay, so that's what this

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definition takes care of.