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Okay. So, today's, in today's lecture, we
are going to understand how the quantum

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state evolves over a period of time. Okay.
So, so, far, what we have studied is what

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are the allowable states of the quantum
system. So, if the, if we have a k-level

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system. So, classically it could have,
take on one of k different states. Then

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the state of the quantum system is a unit
vector in a k-dimensional complex vector

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space. So, this complex vector space is
called a Hilbert space. So, what, what,

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what we are, what we are told is that the
state can be any point on the unit sphere

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in a k-dimensional complex, complex vector
space. And it can be written as a complex

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vector either in your standard notation or
in the kept notation like this. And the

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second thing we, we studied so far is what
happens when we measure the system. And

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what we've learned is that a measurement
corresponds to the choice of an autonomal

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basis for the space. The result of the
measurement is one of k possibilities,

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each corresponding to one of the
orthonormal, one of the basis vectors. The

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probability of that, of each, each basis
vector outcome is, is the square of the

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inner product of the state vector with
that, with that basis vector. And

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moreover, if, if the outcome of the
measurement is the ith basis factor, then

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the new state is exactly the ith basis
factor. So, for example in, you know, this

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is easiest to picture in two dimensions,
where your amplitudes happen to be real so

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that you can, you can plot everything out
in a two, two-dimensional real space, and

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the state sits on the unit circle. And
then the chance that the out, if you

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measure in the 0,1 basis or the standard
basis, the chance that the outcome is zero

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is cosine^2 theta. And one is sine^2
theta. The new state after the measurement

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is either the, either the zero state or
the one state. Okay. So now, what we want

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to understand is how does this state
evolve in time? So, let's look at the case

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of a qubit. So, suppose that we have our
qubit psi, which is this unit vector in a

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two-dimens ional vector space. What does
this system do over time? What, what

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happens when forces act on this system,
etc? The answer is actually very elegant,

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very simple. So, the evolution of the
system corresponds to a rotation of the

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space. So, think of it this way. We are
sitting on the outside, we don't know what

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the state of this qubit is and now we, we
do something to this state to move it in

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time. And what we do is we take this
Hilbert space, this two-dimensional

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complex vector space and we rotate it. So,
again, it's easier to picture in the real

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case so, so what we do is take this, take
the circle and spin it with, through some

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angles. So, every state gets rotated by
some angle theta. Okay. That's the

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evolution of the state of a, of a qubit.
What this also means is that if instead of

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psi, the state of the system had been,
phi, then the new state would be some

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state5' and the angle between psi and phi
would be exactly equal to the angle

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between psi prime and phi prime. Okay. So,
let's try to see this in you know, more

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formally. So, what does it, mean that we
are doing a, a rotation through angle

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theta? So, what it means is that zero gets
transformed to cos theta zero + sine theta

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one and one gets transformed to. -Sine
theta zero + cos theta one? That's what,

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that's what a rotation through angle theta
does to the basis vectors. But now,

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rotation is a linear transformation and
so, so, the way to write it, is as a

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matrix, where what we are saying, is that
if you start with a, with a state alpha

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zero + beta one and what it gets
transformed to is because, because

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rotation is a linear map, it should get
transformed to alpha (cos theta zero +

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sine theta one) + beta (-sine theta zero +
cos theta one). Right? And you can, you

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can, of course simplify this. And you get
alpha (cosine theta - beta sine theta

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zero) +( alpha sine theta + beta cosine
theta) one Okay. So, the fact that

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rotation is a linear operation means that
you can, you can express it as a matrix.

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And of course, what should the matrix look
like? Well, the columns of this matrix are

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e xactly, they exactly correspond to what
the zero state and the one state get

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transformed into. And so, the columns of
the matrix are cosine theta, sine theta.

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And then the second column is -sine theta,
cosine theta. And of course you can see

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this because where does, where does zero
get mapped to? Well, you know, zero, the

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zero state is, in standard matrix notation
vector notation, it would be 1,0. And so,

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if you, if you see where it gets mapped
to, you'd get exactly this first column of

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the matrix and similarly, one gets mapped
to, exactly, the second column of the

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matrix. Okay, corresponding to this, this
matrix, you can also write down the

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rotation through minus theta. The
rotations through minus theta, you can

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check, would look like this. It's cosine
theta sine theta - sine theta cosine

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theta. Okay? So, it's just R of theta
transposed. And of course, you know, that

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if you do a rotation through theta, and
follow it by a rotation through minus

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theta, it's like you did nothing. So, R of
theta R of -theta is the identity. Which,

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in this case, we'll, we can also say by
saying, R of theta R of theta transpose =

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to R of theta transpose R of theta is the
identity. Okay. So, now actually we, we're

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in a place where we can state the basic
axioms of quantum mechanics. So, there are

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basically three axioms in quantum
mechanics. The first one, the

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superposition principle states, what are
the allowable states of a system? The

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second one, the measurement axiom says,
what happens when you measure, or look at

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the system, the state of the system. And
the third axiom, unitary evolution, tells

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you how the state of the system evolves in
time. And this is a very beautiful, very

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elegant sort of formulation of quantum
mechanics. The first axiom says the state

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of the system is the unit vector in a
k-dimensional complex vector space. So, if

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it's a k-level system, you have a
k-dimensional complex vector space. And

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the, the, the state of the system can, can
lie anywhere on the unit sphere. The

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second axiom, the measurement axiom s ays,
if you want to measure, you have to reach

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into this, into this, into this Hilbert
Space. And the way you reach in is you

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pick an orthonormal basis. You measure and
the state collapses into one of the basis

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vectors of this orthonormal basis. Which
one, whether it collapses onto a random

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one, that jf one with probability cosine
square theta, where theta is the angle

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between the state vector and the jfbasis
vector. So, it's the square of the

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magnitude of the inner product between the
state vector and the jth basis vector.

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Moreover, the, the result of measurement
is that the state collapses to the jf

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basis vector. The last axiom the, the, of
unitary evolution says, the quantum state

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evolves by rigid body rotation of the
Hilbert space. So, what you do is, you

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know the state of the system is, is some
unit vector, it's some point on the unit

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sphere in this k-dimensional complex
Hilbert space. What you can do is you can

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take the sphere and rotate it. That's,
that's what a transformation of the, the

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evolution of the system looks like. It's
some rotation of some axis. And then, what

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this rotation does is, the state now
evolves because of this rotation.

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Everything gets carried in some direction
and you get the new state of the system.

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So, it's that simple. That's, those are
the three basic axioms of quantum

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mechanics, that tell you what the states
of the system are, how they evolve in

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time, what a measurement is. Now, of
course, it's a much longer story than

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this.Um, to really understand what, what
all this means will take a, will take us a

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lot of time. In particular, for the rest
of the lecture, we'll, we'll try to

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understand formally what it means to do a
rigid body rotation of the Hilbert space.