Hello everyone. This week, we are going to
talk about continuous quantum states. By
this, I mean, for example, let's, how do
you write the quantum states of a particle
which is moving in the one dimensional
infinite line? So, the particle can be in
any position. Not just any discrete
position, but it's, it's you might specify
by a real number by a real number, its
position along the line. So, okay, as far
as these continuous quantum states are
concerned we are only going to be talking
about them this week. And the, the reason
I want to do this is, you know, is for a
number of reasons. One is so that you
understand how the formalism we have been
looking at so far, which has been confined
to discrete states actually extends to
continuous states. But also, this will
allow us to understand a particularly
simple model which is this so-called
Particle in a Box model which we'll use as
a proxy for an atom for the states of an
electron in an atom. And, and so, in this
particularly simple example, which we have
solved completely, we'll, we'll actually
understand how to implement a qubit as, as
the other electronic, the other energy
levels of a, of, of an electron in an
atom. You know, the, the example that we,
we started with. Okay now, in, in concrete
terms we are going to be talking about how
to represent the states of a continuous
quantum system. We'll talk about
Schroedinger's equation for, for, for, for
this for a particle, a free particle on a
line. And we'll talk about the uncertainty
relation to try to make sense of what
position and momentum mean, mean in this
context. Now, in terms of how these two
lectures are organized Lecture nine, this
particular lecture is going to be done in
an extremely informal style. So, what I'll
try to do is paint for you a picture of
what, what a particle on a line looks
like, what, you know, what Schrodinger's
equation looks like, why it looks the way
it does and where the uncertainty
principle comes from. So, some of you
might find this a bit you know, a, a bit
disturbing because it's, it's, it's going
to be a picture and not, you know, not
precise mathematics. But do play along
with me and in the next lecture, in
Lecture ten, we are going to make many of
these things much more precise. Okay, but
I, I, I think it's actually useful to have
this more general intuitive picture to
have a sense of why, why things are the
way are. Okay one more comment which is
that for continuous quantum system is the
math gets much more difficult. We are just
going to sort of I'll, I'll try to sweep
much of it under the rug even in the next
lecture. But, of course, a little bit of
it is essential to describe these results
more precisely. Okay, so now one thing
that you might be you, that might be
useful for you is just to understand that
much of this, this and the next lecture
for your background, for your, for your
understanding of the subject to understand
how, what we are doing is discrete system
corresponds to you know, continuous
systems of course, it ends up being much
more important for the, the extra segment
we have in the Berkeley course, which is
on how to implement qubits in there in the
lab, you know, experimental realizations
of quantum computing. This is a segment
that I might add at some later time in
the, in a later offering? But, but, you
know, so, for now, this two lecture
segment is, is sort of self-contained and,
and okay, so, in terms of the actual
things that you might, you know that, that
I'd like you to get out of this lecture
beyond the picture, you know, the, the
things that he might want to, want to keep
you know, which will actually be being
worked in the, in the, in the assignments
are. First how do you represent continuous
[sound] quantum state? [sound] And then,
what are the corresponding observables?
Okay, so, these are, these are sort of the
basic things about continuous quantum
states. And then, the other thing we,
we'll of course you, you should know by
the end of this is, is what is
Schrodinger's equation? What, what does
Schrodinger's equation look like for a
free particle, a free particle on a line?
In one, one-dimensional, in one d. And
the, the last thing is, is the uncertainty
principle [sound] between position and
momentum. So, so, what are the position
and momentum observables and what does a
uncertainty principle for these, these?
Okay. So, these are the concrete things
you might want to get out of these two
lectures but of course, the lectures are
going to be, you know, they, they will
they will have a lot more intuition in
them which is just for your for your
benefit. Okay. So, so let's, let's try to
describe the state of our particle, which
is free to move along a line and stretches
out from minus infinity to infinity. So,
let's pretend for the moment that, that
our particle is only allowed to be in one
of the discreet set of states. So, so,
only on these, on these particular points
the integer points from -K to +K. Okay,
so, this way, of course, know how to
write, we would say that the state of this
particle is given by a sum, superposition,
now for sub j, but j goes from -K2 + K.
So, it's, it's, on one of these 2K + one
points and, of course, there's this
normalization condition that, that, that
this is the unit vector, which means that
the summation of, of a j magnitude square
is one. Okay, and, you know, for by a
slight of use of notation, we could, we
could even think of alpha sub j as psi
(j). Okay. Alright and, okay, what is this
wave function look like? Well, what it
looks like is it gives a complex amplitude
for each of these points so, so, maybe you
know, maybe this, this complex amplitude
might you know, if it was real, you could
plot it out and, and perhaps it looks
like, it looks like that. Okay, so, it's,
it, it gives some sort of a function like
that. Now, what do we do? Well, first of
all, you know, to think of t his as, as,
as, if you want to think of this tending
towards a continuous function, let's
actually let these integer points not be
zero, minus, from -K to +K, but let's have
it be -K delta to +K delta so that the
distance between two successive points is
now sum delta instead of one, right? So
this is, this is not one. This is actually
now delta, two delta, -delta, -two delta,
etc. Alright? So, the, the distance
between the points becomes, becomes, where
we, we allow it to be variable. And we let
delta turn to zero and you let K turn into
infinity. Okay, so now, the picture that
we get is this, so we have our infinite
line, and now, that superposition instead
of being confined to these, these discrete
points, which are delta apart, it's, it's
now just a function from the real line to
the complex numbers. So, so we, we just
have, we, we just have some, some function
which, which goes, which goes like that.
Okay, so, so we have some function psi
(x). Okay. So it's, it's, it's, it's a
function from, it's a function from the,
the real numbers from the real line to the
complex numbers, okay. So, for every x,
which is a real number, you have a complex
amplitude, okay. So, so instead of this,
this superposition, now what we have is
what's called the wave function. [sound]
psi (x), right? It plays exactly the same
role as this vector psi. Except now, it's
continuous, you know, it, you, you don't,
it's not just a K or 2K + one dimensional
complex vector, it's actually, it's
actually an infinite dimensional vector,
right? So, for every point x, you have a
complex amplitude, alright. So, we have an
infinite dimensional vector space, a
Hilbert space, where each of these points
x corresponds to an orthogonal direction
in this Hilbert space. And then, we have
this normalization condition. So, what
would the normalization condition say?
Well, it would say, the corresponding
thing here would say that the integral of
psi (x) magnitude squared dx from
-infinity to infinity is one. Okay, what's
another way of writing it? Well, this is,
this is just the integral from minus
infinity to infinity of psi (x) conjugate
psi (x) dx, right? And this is one. Okay,
so, so, particle on a line is described by
a wave function which, which is a function
from the real, from the real line to the
complex numbers. It's normalized, it's
normalized in exactly the same way, so
this, the integral of psi (x) dx is one.
Okay, so now, that we understand, what,
you know, what the state of a, of a
particle on a line is, let's, let's pose
the basic question that we want to ask.
So, the, the most basic question we want
to ask is, suppose that we know that what
psi (x) is at time t = zero. So, psi is
now a function of both x and t and so, at
time t = zero, that's, that's the particle
is anywhere on this line. But we, we know
that, we know what the superposition is.
Let's say we know it's, it's actually
Gaussian, e^-x^2, right? Psi (x) at time
equal to zero is e^-x^2. Suitably
normalized well, maybe that normalization,
I don't know, it might be something like,
like that, okay, but doesn't matter, it's
some constant. So, what does this so what
does our wave function look like at time
equal to zero? It looks, it looks just
like, it's a Gaussian, right. So, sorry
I'm making a mess of it but, it's a
symmetrical figure like this. And now,
this is a free particle in a, on a line.
So, there's no potential and we want to
understand how does psi (x, t) evolve with
time. Okay. So, in other words, we want to
figure out, for, for some general value of
t, what is psi (x, t) looks like? Okay,
so, we want to understand how this way
function changes as a function of time.
Okay. Well, once we answer this question,
we also want to, want to answer a new, a
different question. So, can we answer,
what is the velocity of the particle at
time t? How, how quickly is this particle
moving, okay? Or well, you know that, that
in quantum mechanics, we, we already
talked about how, if you measure the, the
system at a given time, then you disturb
the system. So, how would you m easure
the, the velocity of the particle? Well,
you know, normally, classically, what you
would do is, you would see that it was at
time, time t and then you would see where
it was at time t + delta t. And you'd look
how far it went in, in that small
interval, delta t of time. And you'd,
you'd divide distance by time, by delta t,
and get the, get the velocity. Okay, but,
but now in quantum mechanics, if you, if
you measure the particle at time t, it's,
you know, you've changed the state of the
system. So, how can you measure the
velocity? So, instead, in quantum
mechanics, we'll, you know, the quantity
of interest isn't, is not velocity so much
as momentum. Which is classically is M V.
So, if the mass is M and the velocity is
V, then the momentum is M V. And so, well,
why, why is that momentum can be measured
but velocity, not? Well, to measure
momentum, you don't need to measure the
position at two different points in time.
So, if you've if ever played American
Football or Rugby, then, you know that,
you know the, that, that the player who's
ran into you, had momentum. Not because
you measured his velocity at two different
times but just from how much it knocked
you out. So, so, you can measure the
momentum of a particle by just you know,
hitting it against a detector. Okay, so,
these are the sorts of questions that we
are going to answer in the, in the next
couple of videos, but in a very informal
style. Okay.