Okay, so now let's give a geometrical interpretation for quantum states. So
here's probably we are so we are thinking about the state of an electron in a
hydrogen atom. So its energy levels are quantized, grand state first excited,
second excited state etcetera. And of course if this was a classical system we
could use it to store some information. You know if, if there are three
possibilities then it would be a treat of information zero, one, or two. And okay,
so now in general the state of the system as we saw is a, is a linear superposition.
If it's a k level system, then the state of the system is given by linear
superposition alpha zero, zero + alpha one, one + alpha key - one k - one.
Meaning that is in the ground state with amplitude alpha zero etcetera, etcetera.
Now there is a different way we could have written the state of the system. We could
have just, if you have to write out a different complex numbers, one way we
could do that is by writing it down as a key dimensional vector so we could have
instead written it by putting these, these k numbers, stacking them up like this. And
so, these are really to equivalent ways of, of writing the state of the system.
But if we write it out as a, as a vector then of course we can also you know, we
have we have a picture of it. So for example, if, we you have k = three then
the way we could represent the state is by, is by representing it as a unit
vector. It's a unit vector because some of the squares of this, of this amplitude is
one so. So we'd be representing it in three dimensional space if case equal to
three and if this is the x, y, and z axis then our state vector would actually be a
unit vector in this, in this space and we could label it, you know it would be this
vector right? And it would be a unit vector. It would be sitting on a, on a
unit bowl like this. Sorry about my drawing but it's a unit bowl because,
because of course alpha nought^2 + alpha one^2 + alpha two^2 = one. So, it sits on
unit bowl. Now we could ask, what are these three basic directions? What is the
x axis represent and what is the y-axis represent because these are also unit
vectors and of course, this, this x-axis represents, this is one, zero, zero which
means alpha zero is one and alpha one and alpha two are zero. Which means that the
system is in the state zero, the ground state. What about the y axis at 0,1,0
right? And that corresponds to the state one. In the z axis which is, which is
zero, zero, one. It corresponds to this state two. So, here is the, here is the
interesting thing to think about. So, the, the fact that this final notation we have
where we put, we, we thought of the ground state as, you know we root it as zero
inside these funny brackets and the first excited state as, as one in this funny
bracket. It's just another notation for a vector. Except that, you know usually when
you, when you, when you write on a vector in, in usual vector notation, you write it
as vector u and, and you put this, this little, little bar on top. Here what we're
doing is we have directly saying that the ground state represents it's, it's zero
state for us. And so, so we, we write the label inside this funny brackets and
that's, that's our, that's a notation. This is called the direct notation and
then by the great physicist Derek. Okay, so that's one way to interpret a quantum
state. A quantum state of k level system is just a unit vector in a key dimensional
complex in Hilbert Space. So, it's inside c to the k, okay? Okay so now of course if
we have cubit so we're sitting inside the two dimensional complex of that space. And
if we were to write the state of the system, if you were to say that it's, it's
represented by the vector sign well then maybe it represented the psi inside this
funny brackets is alpha zero, zero + alpha one, one. Now, alpha zero and alpha one
are real. I can actually draw it. This is the zero state. That's the one state and
that's the state psi. It's a unit vector so it sits on the unit circle. And now, we
can ask what's, what's alpha zero? Well, alpha zero is, is this intercept. It's the
x intercept and alpha one. Is this y intercept. And now if you do a
measurement, if you make a measurement the probability that you see that the outcome
is if zero is alpha zero^2 well, in this case that's, you know I am saying it's,
it's real so it's just alpha zero^2 and the probability is one is alpha one^2.
And, and of course the new state. In this case is just zero. In this case, it's one.
So, here's another interpretation of the measurement process. So, when you do a
measurement, so, if you think of this angle as theta. And what happens is that
the probability that the outcome is zero is cosine squared theta. It's the cosine,
cosine squared of the angle between, between psi and the zero axis. And
similarly the probability of 01, well this is really 5/2 - theta and so it's cosine
squared 5/2 - theta which is sine squared theta. So, measurement as this process
whether state vector is projected either on to the zero state or the one state and
the probability is projected on to the zero state is just cosine squared theta.
In one state is cosine squared of the angle that the state vector mix with the,
with the one state.