Okay. So, today's, in today's lecture, we
are going to understand how the quantum
state evolves over a period of time. Okay.
So, so, far, what we have studied is what
are the allowable states of the quantum
system. So, if the, if we have a k-level
system. So, classically it could have,
take on one of k different states. Then
the state of the quantum system is a unit
vector in a k-dimensional complex vector
space. So, this complex vector space is
called a Hilbert space. So, what, what,
what we are, what we are told is that the
state can be any point on the unit sphere
in a k-dimensional complex, complex vector
space. And it can be written as a complex
vector either in your standard notation or
in the kept notation like this. And the
second thing we, we studied so far is what
happens when we measure the system. And
what we've learned is that a measurement
corresponds to the choice of an autonomal
basis for the space. The result of the
measurement is one of k possibilities,
each corresponding to one of the
orthonormal, one of the basis vectors. The
probability of that, of each, each basis
vector outcome is, is the square of the
inner product of the state vector with
that, with that basis vector. And
moreover, if, if the outcome of the
measurement is the ith basis factor, then
the new state is exactly the ith basis
factor. So, for example in, you know, this
is easiest to picture in two dimensions,
where your amplitudes happen to be real so
that you can, you can plot everything out
in a two, two-dimensional real space, and
the state sits on the unit circle. And
then the chance that the out, if you
measure in the 0,1 basis or the standard
basis, the chance that the outcome is zero
is cosine^2 theta. And one is sine^2
theta. The new state after the measurement
is either the, either the zero state or
the one state. Okay. So now, what we want
to understand is how does this state
evolve in time? So, let's look at the case
of a qubit. So, suppose that we have our
qubit psi, which is this unit vector in a
two-dimens ional vector space. What does
this system do over time? What, what
happens when forces act on this system,
etc? The answer is actually very elegant,
very simple. So, the evolution of the
system corresponds to a rotation of the
space. So, think of it this way. We are
sitting on the outside, we don't know what
the state of this qubit is and now we, we
do something to this state to move it in
time. And what we do is we take this
Hilbert space, this two-dimensional
complex vector space and we rotate it. So,
again, it's easier to picture in the real
case so, so what we do is take this, take
the circle and spin it with, through some
angles. So, every state gets rotated by
some angle theta. Okay. That's the
evolution of the state of a, of a qubit.
What this also means is that if instead of
psi, the state of the system had been,
phi, then the new state would be some
state5' and the angle between psi and phi
would be exactly equal to the angle
between psi prime and phi prime. Okay. So,
let's try to see this in you know, more
formally. So, what does it, mean that we
are doing a, a rotation through angle
theta? So, what it means is that zero gets
transformed to cos theta zero + sine theta
one and one gets transformed to. -Sine
theta zero + cos theta one? That's what,
that's what a rotation through angle theta
does to the basis vectors. But now,
rotation is a linear transformation and
so, so, the way to write it, is as a
matrix, where what we are saying, is that
if you start with a, with a state alpha
zero + beta one and what it gets
transformed to is because, because
rotation is a linear map, it should get
transformed to alpha (cos theta zero +
sine theta one) + beta (-sine theta zero +
cos theta one). Right? And you can, you
can, of course simplify this. And you get
alpha (cosine theta - beta sine theta
zero) +( alpha sine theta + beta cosine
theta) one Okay. So, the fact that
rotation is a linear operation means that
you can, you can express it as a matrix.
And of course, what should the matrix look
like? Well, the columns of this matrix are
e xactly, they exactly correspond to what
the zero state and the one state get
transformed into. And so, the columns of
the matrix are cosine theta, sine theta.
And then the second column is -sine theta,
cosine theta. And of course you can see
this because where does, where does zero
get mapped to? Well, you know, zero, the
zero state is, in standard matrix notation
vector notation, it would be 1,0. And so,
if you, if you see where it gets mapped
to, you'd get exactly this first column of
the matrix and similarly, one gets mapped
to, exactly, the second column of the
matrix. Okay, corresponding to this, this
matrix, you can also write down the
rotation through minus theta. The
rotations through minus theta, you can
check, would look like this. It's cosine
theta sine theta - sine theta cosine
theta. Okay? So, it's just R of theta
transposed. And of course, you know, that
if you do a rotation through theta, and
follow it by a rotation through minus
theta, it's like you did nothing. So, R of
theta R of -theta is the identity. Which,
in this case, we'll, we can also say by
saying, R of theta R of theta transpose =
to R of theta transpose R of theta is the
identity. Okay. So, now actually we, we're
in a place where we can state the basic
axioms of quantum mechanics. So, there are
basically three axioms in quantum
mechanics. The first one, the
superposition principle states, what are
the allowable states of a system? The
second one, the measurement axiom says,
what happens when you measure, or look at
the system, the state of the system. And
the third axiom, unitary evolution, tells
you how the state of the system evolves in
time. And this is a very beautiful, very
elegant sort of formulation of quantum
mechanics. The first axiom says the state
of the system is the unit vector in a
k-dimensional complex vector space. So, if
it's a k-level system, you have a
k-dimensional complex vector space. And
the, the, the state of the system can, can
lie anywhere on the unit sphere. The
second axiom, the measurement axiom s ays,
if you want to measure, you have to reach
into this, into this, into this Hilbert
Space. And the way you reach in is you
pick an orthonormal basis. You measure and
the state collapses into one of the basis
vectors of this orthonormal basis. Which
one, whether it collapses onto a random
one, that jf one with probability cosine
square theta, where theta is the angle
between the state vector and the jfbasis
vector. So, it's the square of the
magnitude of the inner product between the
state vector and the jth basis vector.
Moreover, the, the result of measurement
is that the state collapses to the jf
basis vector. The last axiom the, the, of
unitary evolution says, the quantum state
evolves by rigid body rotation of the
Hilbert space. So, what you do is, you
know the state of the system is, is some
unit vector, it's some point on the unit
sphere in this k-dimensional complex
Hilbert space. What you can do is you can
take the sphere and rotate it. That's,
that's what a transformation of the, the
evolution of the system looks like. It's
some rotation of some axis. And then, what
this rotation does is, the state now
evolves because of this rotation.
Everything gets carried in some direction
and you get the new state of the system.
So, it's that simple. That's, those are
the three basic axioms of quantum
mechanics, that tell you what the states
of the system are, how they evolve in
time, what a measurement is. Now, of
course, it's a much longer story than
this.Um, to really understand what, what
all this means will take a, will take us a
lot of time. In particular, for the rest
of the lecture, we'll, we'll try to
understand formally what it means to do a
rigid body rotation of the Hilbert space.