Okay. So now it turns out this bra-ket notation is, is extremely useful for a
number of purposes. So, let's, let's look at how we write down the projection matrix
projection matrix, P which projects onto the phi. So, here's our picture, here's a
state phi and what we want is an operator, we want a Matrix P that given an arbitrary
state psi tells us how to project psi onto, onto, onto this vector so it, it
should give us, so if we, if we apply, if you look at P psi, the result of P psi
should be this vector. It's the projection of psi onto phi. Okay. So, what, what is,
what is this matrix P which does this projection for us? So, P is, okay, so
again, assume that, psi was the phi was given by this, this vector, beta not
through beta k - one. Then P is given to us by, by this matrix, which is beta not
through beta k - one times the corresponding row vector, beta north star
through beta k - one star. So, this is going to be a, k by k matrix and you can
check that this, this will actually project any given, given vector onto, onto
phi. So, how do we write P out in the ket notation? Well, it's just ket phi followed
by bra phi. Okay. Now if, if I were writing this quickly, if you were writing
it quickly though, maybe you'd write it as phi, and then we put an x here and phi,
right? And then you can read is as ket bra phi. Alright. So now, this is P. So,
what's P times psi. So, what, what happens if you apply P to psi? Well, we'll get
that followed by psi. Okay, so you'll notice, in this notation, we always omit
the, you know, if you have two parallel lines, we just omit one of them. And now,
we can use associativity of you know, of these operation, of these multiplication
operations. And we can write this, instead parenthesize this, like this. So, before
we had, we were given this times phi. And now, we can write it as this times that
and what's this, what's this quantity? Well, this is just an inner product. So,
this is just a inner product between phi and psi, which is this belongs to the comp
lex numbers. The inner product is a complex number times five. So, what, what
this tells us is when you project psi, when we apply this projection operator on
to, onto this state psi, you'll get the state phi. Okay. So, this is the state you
get and, and except that it's, it's, it's a scaled version. And what's it scaled by,
it's scaled exactly by the inner product. Okay. So, this is one of the nice features
of, of this bra-ket notation. Which is that the notation itself, you know, if
something looks natural in the notation, it's actually a legal move and you're,
you're allowed to do it. So, it leads you in the right direction. Well, let's see
another example. So, what happens if you apply this projection operator twice, if
you apply this matrix projection twice? Well, okay , so let's, let's look at the
example. If you project the state psi and you project it twice on to five and you
project it the first time and you get this vector. And if you project it again, the
green vector will not move anywhere, because it's already it's already
proportional to FI. So, P^2 should equal to P. So, let's see how, you know, why,
why this is true? Well, what's P^2? It's, P is that, and what's B^2? Well, we repeat
that. And again, we use associativity. So, what's this middle term here? Well, this
middle, middle term is just, it's just the, inner product of phi with itself, and
assuming that phi is a unit vector, this is equal to one. And so, simplifying it,
you just get, this is equal to that which is P. So, P^2 is equal to this, is equal
to that, is equal to P. Okay? So, so, that's the really nice thing about the ket
notation. Now, let's, let's go ahead and do one other thing. Let's, let's recall
that U is unitary, is the same thing as U dagger as U dagger U is that entity. And
for finite dimensional matrices, actually one condition will follow from the other.
Now, what are the properties of unitary matrices? So, the properties are that the
rows are all phenomenal as are the colons. Okay, what, what does this tell u S? So,
one of the things it tells us is, let's say, this is, this is our basis for this
space, zero one through k - one. And now, we perform a unitary rotation on this. So,
we perform some unitary U. So, zero now goes to U times zero. One state goes to U
times one. And, and the k - one goes to U (k - one). Okay. So, if you were to adopt
some notations. So, let's say U with two subscripts ., j. By this, I mean, the jth
column of U. So, what we, what we can say is that, this is the 0th column. And this
is the [inaudible] minus first column. The first column etc. Okay, so these are just
columns of you and what we know is that the columns are, are orthonormal, and so
we started from orthogonal vectors and we moved to orthogonal vectors. Okay, so in
general, we can say that you preserves angles or actually more precisely, inner
products. What I mean by this is that if you start with two vectors, phi and psi,
and this the inner product and now let U times phi equal to phi prime and U times
psi equal to psi prime, then what we are claiming is that the inner product between
phi and psi is the same thing as the, is the same as the inner product between phi
prime and psi prime, okay? How do we write this down? Well, you know, what's phi
prime? It's, it's of course, U times phi. So, if phi prime is zero times phi, how do
we write and draw phi prime. Okay. It's just a conjugate transpose and you should
convince yourself that it is, this. It's graph phi times U dagger, okay? So, so,
this is really saying that graph phi, U dagger, U psi is the same as phi psi and.
Of course, the reason that's, that's the case is that U dagger U is the identity.