So now, just making some room to, to write
a little more here. Okay. So, so now,
what, what this, what this tells us is
that if you work it out, it's as though,
you know as the slinky is precessing, as
it's rotating. It's as though it's
velocity is, is proportional to K. Okay,
so, why is that? Well, you see, we, we
already said that the period of this
function, the period of this, this the
wave function is two pi / K. And if you,
if you, if you look at this, you know,
this time dependence e^i K^2 t, the time
it takes to, for this, you know, as this
wave function is precessing to, to precess
to one full circle, is actually two pi /
K^2. And so, so the fond movement of the
slinky, in, in one, one such, one such
full rotation of the phase, is two pi / K
but, but this happens in time t2 pi / K^2.
So, the apparent velocity is this over
this, so it is two pi / K / two pi / K^2,
which is K. Okay, so, so now, what, what,
what we've understood is if we, if our
wave function looked like this, if it
looks like e^ikx, then it would have a
definite velocity of K. Okay, but we're
not given, you know, but, but, we are
given just some psi (x, t). It's not, it's
not e^ikx, you know, this is those wave
function with had the definite velocity of
K. So now, we can ask what's the, you
know, if we were to write out psi (x, t)
in the, in the velocity basis, right? So,
we could write it out in the velocity,
instead of in the position basis, we could
write it out in the velocity basis and we
might have some superposition which, you
know, which, which, which, which look like
that, which would say, with what amplitude
is the velocity of this, of this particle
at time t equal to, equal to v and the
answer would be, it's, it's this much,
alright. So, this is, this is amplitude.
Okay, so, let me, let me call it phi (v,
t). So, this is the amplitude which it has
this. And so, how do you compute phi (v,
t)? Well, it's exactly the component of
psi in the direction of this function. So,
how do you compute the component of psi in
this direction? Yo u take the inner
product. The inner product of this
function with psi (x, t) which is what?
It's the integral of e^-ikx, it's the
complex conjugate of this. Psi (x, t) dx
of minus infinity to infinity. Alright,
what's the inner product is just, you, you
just, you just take the product of the
corresponding components. Well, with, with
the complex conjugate of the first one
times the component of the second one, and
then you add it over all, you know, over,
over all the dimensions. But here, we are,
we are working on continuous dimensional
space where x is what indexes these
dimensions. So, we just take complex
conjugate of the first time. So, second
and instead of summing over all the
dimensions, now since x is continuous, we
integrate this from minus infinity to
infinity this time dx. And that's, that's
the component in this direction, but now,
some of you might recognize this function.
What is this? Well, phi just the Fourier
transform of psi, okay? So, while I'm
calling it velocity here just for, you
know, just, just for intuitive purposes.
You know in quantum mechanics, we are
really measuring momentum which is
proportional to velocity. And so velocity,
momentum is the Fourier of transform of
the, of position. So, position and
momentum are related to each other by
Fourier transforms. This is a very
interesting thing because what it says is,
you start with some superposition, psi.
So, this is psi (x). This is a, this is a,
a position of a particle at, at, at
whatever time, alight. It's, it's sum
function. Alright. This is psi (x). Now,
we write phi (p), p for momentum. Okay,
so, this is, okay, let's, let's write it
in blue. It's phi (p), okay? This is meant
to be some other superposition, and what,
what, what I'm cleaning is that the way
you're getting from one to the other and
back is by doing a Fourier transform.
Okay, so, what do you know about a Fourier
transform? The one thing you should know
is that the more you concentrate a
function in the primary domain, the more
it spreads out in the Fourier domain. The
more you try to concentrate it in the
Fourier domain, the more it spreads out in
the primary domain. So, what are you to
do? Well, it turns out that the best you
can do is sort of make, make this, this
function on this side, be a Gaussian with
some standard deviation sigma. And then,
you have some corresponding standard
deviation sigma hat on this, on this side
and it turns out that the, that the
smaller sigma is the bigger sigma hat is
and vice versa. And so, you can try to
find the happy median between the two. And
this is what gives us this uncertainty
principle which says that delta x delta p
is at least h bar / two. So, delta x is
really sigma and delta p is sigma hat and
what if, you know, the, the fact about
Fourier transforms is, is that the way you
min, minimize this product is by using
Gaussian in the first place and in, in the
second place, if you use Gaussian, then,
then the width of this Gaussian times the
width of this Gaussian in the Fourier,
Fourier transform. The product of these
two bits is bounded below by some
constant, okay? So, let me reiterate.
There is not much of this lecture that you
really need to know. But, this is, you
know, this should help you get a picture
of what's going on with continuous quantum
states, with the Schrodinger equation for
the particle on the line with, where this
uncertainty principle comes from, okay? In
the next lecture, we'll look at all of
this in a more precise way.