1
00:00:00,000 --> 00:00:08,000
Okay. So now it turns out this bra-ket
notation is, is extremely useful for a

2
00:00:08,000 --> 00:00:18,071
number of purposes. So, let's, let's look
at how we write down the projection matrix

3
00:00:20,002 --> 00:00:32,009
projection matrix, P which projects onto
the phi. So, here's our picture, here's a

4
00:00:32,009 --> 00:00:41,036
state phi and what we want is an operator,
we want a Matrix P that given an arbitrary

5
00:00:41,036 --> 00:00:48,046
state psi tells us how to project psi
onto, onto, onto this vector so it, it

6
00:00:48,046 --> 00:00:57,070
should give us, so if we, if we apply, if
you look at P psi, the result of P psi

7
00:00:57,070 --> 00:01:06,087
should be this vector. It's the projection
of psi onto phi. Okay. So, what, what is,

8
00:01:06,087 --> 00:01:18,011
what is this matrix P which does this
projection for us? So, P is, okay, so

9
00:01:18,011 --> 00:01:28,054
again, assume that, psi was the phi was
given by this, this vector, beta not

10
00:01:28,054 --> 00:01:40,041
through beta k - one. Then P is given to
us by, by this matrix, which is beta not

11
00:01:40,041 --> 00:01:50,003
through beta k - one times the
corresponding row vector, beta north star

12
00:01:50,003 --> 00:01:59,043
through beta k - one star. So, this is
going to be a, k by k matrix and you can

13
00:01:59,043 --> 00:02:06,076
check that this, this will actually
project any given, given vector onto, onto

14
00:02:06,076 --> 00:02:17,030
phi. So, how do we write P out in the ket
notation? Well, it's just ket phi followed

15
00:02:17,030 --> 00:02:27,012
by bra phi. Okay. Now if, if I were
writing this quickly, if you were writing

16
00:02:27,012 --> 00:02:33,031
it quickly though, maybe you'd write it as
phi, and then we put an x here and phi,

17
00:02:33,031 --> 00:02:40,015
right? And then you can read is as ket bra
phi. Alright. So now, this is P. So,

18
00:02:40,015 --> 00:02:48,050
what's P times psi. So, what, what happens
if you apply P to psi? Well, we'll get

19
00:02:48,050 --> 00:02:57,015
that followed by psi. Okay, so you'll
notice, in this notation, we always omit

20
00:02:57,015 --> 00:03:05,080
the, you know, if you have two parallel
lines, we just omit one of them. And now,

21
00:03:05,080 --> 00:03:14,009
we can use associativity of you know, of
these operation, of these multiplication

22
00:03:14,009 --> 00:03:22,004
operations. And we can write this, instead
parenthesize this, like this. So, before

23
00:03:22,004 --> 00:03:33,001
we had, we were given this times phi. And
now, we can write it as this times that

24
00:03:33,001 --> 00:03:39,009
and what's this, what's this quantity?
Well, this is just an inner product. So,

25
00:03:39,009 --> 00:03:46,042
this is just a inner product between phi
and psi, which is this belongs to the comp

26
00:03:46,042 --> 00:03:52,009
lex numbers. The inner product is a
complex number times five. So, what, what

27
00:03:52,009 --> 00:03:59,003
this tells us is when you project psi,
when we apply this projection operator on

28
00:03:59,003 --> 00:04:04,079
to, onto this state psi, you'll get the
state phi. Okay. So, this is the state you

29
00:04:04,079 --> 00:04:12,001
get and, and except that it's, it's, it's
a scaled version. And what's it scaled by,

30
00:04:12,001 --> 00:04:19,006
it's scaled exactly by the inner product.
Okay. So, this is one of the nice features

31
00:04:19,006 --> 00:04:25,005
of, of this bra-ket notation. Which is
that the notation itself, you know, if

32
00:04:25,005 --> 00:04:31,007
something looks natural in the notation,
it's actually a legal move and you're,

33
00:04:31,007 --> 00:04:38,008
you're allowed to do it. So, it leads you
in the right direction. Well, let's see

34
00:04:38,008 --> 00:04:43,031
another example. So, what happens if you
apply this projection operator twice, if

35
00:04:43,031 --> 00:04:49,000
you apply this matrix projection twice?
Well, okay , so let's, let's look at the

36
00:04:49,000 --> 00:04:54,003
example. If you project the state psi and
you project it twice on to five and you

37
00:04:54,003 --> 00:04:59,005
project it the first time and you get this
vector. And if you project it again, the

38
00:04:59,005 --> 00:05:05,005
green vector will not move anywhere,
because it's already it's already

39
00:05:05,005 --> 00:05:13,054
proportional to FI. So, P^2 should equal
to P. So, let's see how, you know, why,

40
00:05:13,054 --> 00:05:22,000
why this is true? Well, what's P^2? It's,
P is that, and what's B^2? Well, we repeat

41
00:05:22,000 --> 00:05:29,005
that. And again, we use associativity. So,
what's this middle term here? Well, this

42
00:05:29,005 --> 00:05:36,008
middle, middle term is just, it's just
the, inner product of phi with itself, and

43
00:05:36,008 --> 00:05:43,081
assuming that phi is a unit vector, this
is equal to one. And so, simplifying it,

44
00:05:43,081 --> 00:05:52,096
you just get, this is equal to that which
is P. So, P^2 is equal to this, is equal

45
00:05:52,096 --> 00:06:02,095
to that, is equal to P. Okay? So, so,
that's the really nice thing about the ket

46
00:06:02,095 --> 00:06:13,033
notation. Now, let's, let's go ahead and
do one other thing. Let's, let's recall

47
00:06:13,033 --> 00:06:22,047
that U is unitary, is the same thing as U
dagger as U dagger U is that entity. And

48
00:06:22,047 --> 00:06:30,013
for finite dimensional matrices, actually
one condition will follow from the other.

49
00:06:30,013 --> 00:06:38,044
Now, what are the properties of unitary
matrices? So, the properties are that the

50
00:06:38,044 --> 00:06:51,061
rows are all phenomenal as are the colons.
Okay, what, what does this tell u S? So,

51
00:06:51,061 --> 00:07:01,030
one of the things it tells us is, let's
say, this is, this is our basis for this

52
00:07:01,030 --> 00:07:11,018
space, zero one through k - one. And now,
we perform a unitary rotation on this. So,

53
00:07:11,018 --> 00:07:24,012
we perform some unitary U. So, zero now
goes to U times zero. One state goes to U

54
00:07:24,012 --> 00:07:36,043
times one. And, and the k - one goes to U
(k - one). Okay. So, if you were to adopt

55
00:07:36,043 --> 00:07:49,043
some notations. So, let's say U with two
subscripts ., j. By this, I mean, the jth

56
00:07:49,043 --> 00:08:02,083
column of U. So, what we, what we can say
is that, this is the 0th column. And this

57
00:08:02,083 --> 00:08:14,012
is the [inaudible] minus first column. The
first column etc. Okay, so these are just

58
00:08:14,012 --> 00:08:22,042
columns of you and what we know is that
the columns are, are orthonormal, and so

59
00:08:22,042 --> 00:08:30,024
we started from orthogonal vectors and we
moved to orthogonal vectors. Okay, so in

60
00:08:30,024 --> 00:08:43,042
general, we can say that you preserves
angles or actually more precisely, inner

61
00:08:43,042 --> 00:08:53,088
products. What I mean by this is that if
you start with two vectors, phi and psi,

62
00:08:53,088 --> 00:09:02,008
and this the inner product and now let U
times phi equal to phi prime and U times

63
00:09:02,008 --> 00:09:10,027
psi equal to psi prime, then what we are
claiming is that the inner product between

64
00:09:10,027 --> 00:09:18,048
phi and psi is the same thing as the, is
the same as the inner product between phi

65
00:09:18,048 --> 00:09:27,035
prime and psi prime, okay? How do we write
this down? Well, you know, what's phi

66
00:09:27,035 --> 00:09:39,042
prime? It's, it's of course, U times phi.
So, if phi prime is zero times phi, how do

67
00:09:39,042 --> 00:09:51,076
we write and draw phi prime. Okay. It's
just a conjugate transpose and you should

68
00:09:51,076 --> 00:10:04,018
convince yourself that it is, this. It's
graph phi times U dagger, okay? So, so,

69
00:10:04,018 --> 00:10:19,027
this is really saying that graph phi, U
dagger, U psi is the same as phi psi and.

70
00:10:19,027 --> 00:10:31,004
Of course, the reason that's, that's the
case is that U dagger U is the identity.