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Hello everyone. This week, we are going to
talk about continuous quantum states. By

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this, I mean, for example, let's, how do
you write the quantum states of a particle

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which is moving in the one dimensional
infinite line? So, the particle can be in

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any position. Not just any discrete
position, but it's, it's you might specify

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by a real number by a real number, its
position along the line. So, okay, as far

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as these continuous quantum states are
concerned we are only going to be talking

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about them this week. And the, the reason
I want to do this is, you know, is for a

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number of reasons. One is so that you
understand how the formalism we have been

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looking at so far, which has been confined
to discrete states actually extends to

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continuous states. But also, this will
allow us to understand a particularly

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simple model which is this so-called
Particle in a Box model which we'll use as

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a proxy for an atom for the states of an
electron in an atom. And, and so, in this

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particularly simple example, which we have
solved completely, we'll, we'll actually

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understand how to implement a qubit as, as
the other electronic, the other energy

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levels of a, of, of an electron in an
atom. You know, the, the example that we,

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we started with. Okay now, in, in concrete
terms we are going to be talking about how

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to represent the states of a continuous
quantum system. We'll talk about

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Schroedinger's equation for, for, for, for
this for a particle, a free particle on a

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line. And we'll talk about the uncertainty
relation to try to make sense of what

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position and momentum mean, mean in this
context. Now, in terms of how these two

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lectures are organized Lecture nine, this
particular lecture is going to be done in

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an extremely informal style. So, what I'll
try to do is paint for you a picture of

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what, what a particle on a line looks
like, what, you know, what Schrodinger's

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equation looks like, why it looks the way
it does and where the uncertainty

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principle comes from. So, some of you
might find this a bit you know, a, a bit

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disturbing because it's, it's, it's going
to be a picture and not, you know, not

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precise mathematics. But do play along
with me and in the next lecture, in

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Lecture ten, we are going to make many of
these things much more precise. Okay, but

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I, I, I think it's actually useful to have
this more general intuitive picture to

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have a sense of why, why things are the
way are. Okay one more comment which is

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that for continuous quantum system is the
math gets much more difficult. We are just

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going to sort of I'll, I'll try to sweep
much of it under the rug even in the next

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lecture. But, of course, a little bit of
it is essential to describe these results

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more precisely. Okay, so now one thing
that you might be you, that might be

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useful for you is just to understand that
much of this, this and the next lecture

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for your background, for your, for your
understanding of the subject to understand

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how, what we are doing is discrete system
corresponds to you know, continuous

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systems of course, it ends up being much
more important for the, the extra segment

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we have in the Berkeley course, which is
on how to implement qubits in there in the

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lab, you know, experimental realizations
of quantum computing. This is a segment

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that I might add at some later time in
the, in a later offering? But, but, you

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know, so, for now, this two lecture
segment is, is sort of self-contained and,

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and okay, so, in terms of the actual
things that you might, you know that, that

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I'd like you to get out of this lecture
beyond the picture, you know, the, the

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things that he might want to, want to keep
you know, which will actually be being

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worked in the, in the, in the assignments
are. First how do you represent continuous

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[sound] quantum state? [sound] And then,
what are the corresponding observables?

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Okay, so, these are, these are sort of the
basic things about continuous quantum

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states. And then, the other thing we,
we'll of course you, you should know by

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the end of this is, is what is
Schrodinger's equation? What, what does

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Schrodinger's equation look like for a
free particle, a free particle on a line?

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In one, one-dimensional, in one d. And
the, the last thing is, is the uncertainty

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principle [sound] between position and
momentum. So, so, what are the position

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and momentum observables and what does a
uncertainty principle for these, these?

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Okay. So, these are the concrete things
you might want to get out of these two

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lectures but of course, the lectures are
going to be, you know, they, they will

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they will have a lot more intuition in
them which is just for your for your

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benefit. Okay. So, so let's, let's try to
describe the state of our particle, which

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is free to move along a line and stretches
out from minus infinity to infinity. So,

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let's pretend for the moment that, that
our particle is only allowed to be in one

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of the discreet set of states. So, so,
only on these, on these particular points

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the integer points from -K to +K. Okay,
so, this way, of course, know how to

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write, we would say that the state of this
particle is given by a sum, superposition,

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now for sub j, but j goes from -K2 + K.
So, it's, it's, on one of these 2K + one

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points and, of course, there's this
normalization condition that, that, that

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this is the unit vector, which means that
the summation of, of a j magnitude square

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is one. Okay, and, you know, for by a
slight of use of notation, we could, we

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could even think of alpha sub j as psi
(j). Okay. Alright and, okay, what is this

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wave function look like? Well, what it
looks like is it gives a complex amplitude

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for each of these points so, so, maybe you
know, maybe this, this complex amplitude

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might you know, if it was real, you could
plot it out and, and perhaps it looks

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like, it looks like that. Okay, so, it's,
it, it gives some sort of a function like

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that. Now, what do we do? Well, first of
all, you know, to think of t his as, as,

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as, if you want to think of this tending
towards a continuous function, let's

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actually let these integer points not be
zero, minus, from -K to +K, but let's have

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it be -K delta to +K delta so that the
distance between two successive points is

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now sum delta instead of one, right? So
this is, this is not one. This is actually

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now delta, two delta, -delta, -two delta,
etc. Alright? So, the, the distance

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between the points becomes, becomes, where
we, we allow it to be variable. And we let

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delta turn to zero and you let K turn into
infinity. Okay, so now, the picture that

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we get is this, so we have our infinite
line, and now, that superposition instead

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of being confined to these, these discrete
points, which are delta apart, it's, it's

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now just a function from the real line to
the complex numbers. So, so we, we just

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have, we, we just have some, some function
which, which goes, which goes like that.

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Okay, so, so we have some function psi
(x). Okay. So it's, it's, it's, it's a

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function from, it's a function from the,
the real numbers from the real line to the

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complex numbers, okay. So, for every x,
which is a real number, you have a complex

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amplitude, okay. So, so instead of this,
this superposition, now what we have is

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what's called the wave function. [sound]
psi (x), right? It plays exactly the same

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role as this vector psi. Except now, it's
continuous, you know, it, you, you don't,

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it's not just a K or 2K + one dimensional
complex vector, it's actually, it's

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actually an infinite dimensional vector,
right? So, for every point x, you have a

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complex amplitude, alright. So, we have an
infinite dimensional vector space, a

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Hilbert space, where each of these points
x corresponds to an orthogonal direction

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in this Hilbert space. And then, we have
this normalization condition. So, what

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would the normalization condition say?
Well, it would say, the corresponding

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thing here would say that the integral of
psi (x) magnitude squared dx from

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-infinity to infinity is one. Okay, what's
another way of writing it? Well, this is,

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this is just the integral from minus
infinity to infinity of psi (x) conjugate

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psi (x) dx, right? And this is one. Okay,
so, so, particle on a line is described by

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a wave function which, which is a function
from the real, from the real line to the

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complex numbers. It's normalized, it's
normalized in exactly the same way, so

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this, the integral of psi (x) dx is one.
Okay, so now, that we understand, what,

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you know, what the state of a, of a
particle on a line is, let's, let's pose

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the basic question that we want to ask.
So, the, the most basic question we want

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to ask is, suppose that we know that what
psi (x) is at time t = zero. So, psi is

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now a function of both x and t and so, at
time t = zero, that's, that's the particle

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is anywhere on this line. But we, we know
that, we know what the superposition is.

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Let's say we know it's, it's actually
Gaussian, e^-x^2, right? Psi (x) at time

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equal to zero is e^-x^2. Suitably
normalized well, maybe that normalization,

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I don't know, it might be something like,
like that, okay, but doesn't matter, it's

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some constant. So, what does this so what
does our wave function look like at time

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equal to zero? It looks, it looks just
like, it's a Gaussian, right. So, sorry

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I'm making a mess of it but, it's a
symmetrical figure like this. And now,

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this is a free particle in a, on a line.
So, there's no potential and we want to

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understand how does psi (x, t) evolve with
time. Okay. So, in other words, we want to

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figure out, for, for some general value of
t, what is psi (x, t) looks like? Okay,

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so, we want to understand how this way
function changes as a function of time.

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Okay. Well, once we answer this question,
we also want to, want to answer a new, a

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different question. So, can we answer,
what is the velocity of the particle at

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time t? How, how quickly is this particle
moving, okay? Or well, you know that, that

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in quantum mechanics, we, we already
talked about how, if you measure the, the

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system at a given time, then you disturb
the system. So, how would you m easure

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the, the velocity of the particle? Well,
you know, normally, classically, what you

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would do is, you would see that it was at
time, time t and then you would see where

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it was at time t + delta t. And you'd look
how far it went in, in that small

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interval, delta t of time. And you'd,
you'd divide distance by time, by delta t,

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and get the, get the velocity. Okay, but,
but now in quantum mechanics, if you, if

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you measure the particle at time t, it's,
you know, you've changed the state of the

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system. So, how can you measure the
velocity? So, instead, in quantum

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mechanics, we'll, you know, the quantity
of interest isn't, is not velocity so much

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as momentum. Which is classically is M V.
So, if the mass is M and the velocity is

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V, then the momentum is M V. And so, well,
why, why is that momentum can be measured

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but velocity, not? Well, to measure
momentum, you don't need to measure the

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position at two different points in time.
So, if you've if ever played American

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Football or Rugby, then, you know that,
you know the, that, that the player who's

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ran into you, had momentum. Not because
you measured his velocity at two different

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times but just from how much it knocked
you out. So, so, you can measure the

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momentum of a particle by just you know,
hitting it against a detector. Okay, so,

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these are the sorts of questions that we
are going to answer in the, in the next

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couple of videos, but in a very informal
style. Okay.