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Okay, so now let's give a geometrical
interpretation for quantum states. So

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here's probably we are so we are thinking
about the state of an electron in a

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hydrogen atom. So its energy levels are
quantized, grand state first excited,

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second excited state etcetera. And of
course if this was a classical system we

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could use it to store some information.
You know if, if there are three

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possibilities then it would be a treat of
information zero, one, or two. And okay,

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so now in general the state of the system
as we saw is a, is a linear superposition.

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If it's a k level system, then the state
of the system is given by linear

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superposition alpha zero, zero + alpha
one, one + alpha key - one k - one.

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Meaning that is in the ground state with
amplitude alpha zero etcetera, etcetera.

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Now there is a different way we could have
written the state of the system. We could

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have just, if you have to write out a
different complex numbers, one way we

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could do that is by writing it down as a
key dimensional vector so we could have

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instead written it by putting these, these
k numbers, stacking them up like this. And

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so, these are really to equivalent ways
of, of writing the state of the system.

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But if we write it out as a, as a vector
then of course we can also you know, we

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have we have a picture of it. So for
example, if, we you have k = three then

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the way we could represent the state is
by, is by representing it as a unit

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vector. It's a unit vector because some of
the squares of this, of this amplitude is

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one so. So we'd be representing it in
three dimensional space if case equal to

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three and if this is the x, y, and z axis
then our state vector would actually be a

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unit vector in this, in this space and we
could label it, you know it would be this

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vector right? And it would be a unit
vector. It would be sitting on a, on a

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unit bowl like this. Sorry about my
drawing but it's a unit bowl because,

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because of course alpha nought^2 + alpha
one^2 + alpha two^2 = one. So, it sits on

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unit bowl. Now we could ask, what are
these three basic directions? What is the

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x axis represent and what is the y-axis
represent because these are also unit

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vectors and of course, this, this x-axis
represents, this is one, zero, zero which

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means alpha zero is one and alpha one and
alpha two are zero. Which means that the

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system is in the state zero, the ground
state. What about the y axis at 0,1,0

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right? And that corresponds to the state
one. In the z axis which is, which is

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zero, zero, one. It corresponds to this
state two. So, here is the, here is the

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interesting thing to think about. So, the,
the fact that this final notation we have

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where we put, we, we thought of the ground
state as, you know we root it as zero

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inside these funny brackets and the first
excited state as, as one in this funny

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bracket. It's just another notation for a
vector. Except that, you know usually when

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you, when you, when you write on a vector
in, in usual vector notation, you write it

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as vector u and, and you put this, this
little, little bar on top. Here what we're

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doing is we have directly saying that the
ground state represents it's, it's zero

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state for us. And so, so we, we write the
label inside this funny brackets and

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that's, that's our, that's a notation.
This is called the direct notation and

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then by the great physicist Derek. Okay,
so that's one way to interpret a quantum

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state. A quantum state of k level system
is just a unit vector in a key dimensional

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complex in Hilbert Space. So, it's inside
c to the k, okay? Okay so now of course if

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we have cubit so we're sitting inside the
two dimensional complex of that space. And

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if we were to write the state of the
system, if you were to say that it's, it's

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represented by the vector sign well then
maybe it represented the psi inside this

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funny brackets is alpha zero, zero + alpha
one, one. Now, alpha zero and alpha one

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are real. I can actually draw it. This is
the zero state. That's the one state and

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that's the state psi. It's a unit vector
so it sits on the unit circle. And now, we

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can ask what's, what's alpha zero? Well,
alpha zero is, is this intercept. It's the

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x intercept and alpha one. Is this y
intercept. And now if you do a

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measurement, if you make a measurement the
probability that you see that the outcome

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is if zero is alpha zero^2 well, in this
case that's, you know I am saying it's,

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it's real so it's just alpha zero^2 and
the probability is one is alpha one^2.

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And, and of course the new state. In this
case is just zero. In this case, it's one.

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So, here's another interpretation of the
measurement process. So, when you do a

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measurement, so, if you think of this
angle as theta. And what happens is that

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the probability that the outcome is zero
is cosine squared theta. It's the cosine,

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cosine squared of the angle between,
between psi and the zero axis. And

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similarly the probability of 01, well this
is really 5/2 - theta and so it's cosine

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squared 5/2 - theta which is sine squared
theta. So, measurement as this process

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whether state vector is projected either
on to the zero state or the one state and

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the probability is projected on to the
zero state is just cosine squared theta.

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In one state is cosine squared of the
angle that the state vector mix with the,

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with the one state.