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So now, just making some room to, to write
a little more here. Okay. So, so now,

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what, what this, what this tells us is
that if you work it out, it's as though,

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you know as the slinky is precessing, as
it's rotating. It's as though it's

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velocity is, is proportional to K. Okay,
so, why is that? Well, you see, we, we

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already said that the period of this
function, the period of this, this the

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wave function is two pi / K. And if you,
if you, if you look at this, you know,

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00:00:01,069 --> 00:00:01,077
this time dependence e^i K^2 t, the time
it takes to, for this, you know, as this

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wave function is precessing to, to precess
to one full circle, is actually two pi /

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K^2. And so, so the fond movement of the
slinky, in, in one, one such, one such

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full rotation of the phase, is two pi / K
but, but this happens in time t2 pi / K^2.

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So, the apparent velocity is this over
this, so it is two pi / K / two pi / K^2,

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which is K. Okay, so, so now, what, what,
what we've understood is if we, if our

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00:00:02,015 --> 00:00:02,022
wave function looked like this, if it
looks like e^ikx, then it would have a

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definite velocity of K. Okay, but we're
not given, you know, but, but, we are

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given just some psi (x, t). It's not, it's
not e^ikx, you know, this is those wave

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function with had the definite velocity of
K. So now, we can ask what's the, you

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know, if we were to write out psi (x, t)
in the, in the velocity basis, right? So,

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we could write it out in the velocity,
instead of in the position basis, we could

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00:00:02,060 --> 00:00:02,068
write it out in the velocity basis and we
might have some superposition which, you

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00:00:02,068 --> 00:00:02,076
know, which, which, which, which look like
that, which would say, with what amplitude

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00:00:02,076 --> 00:00:02,084
is the velocity of this, of this particle
at time t equal to, equal to v and the

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00:00:02,084 --> 00:00:02,091
answer would be, it's, it's this much,
alright. So, this is, this is amplitude.

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00:00:02,091 --> 00:00:02,099
Okay, so, let me, let me call it phi (v,
t). So, this is the amplitude which it has

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00:00:02,099 --> 00:00:03,006
this. And so, how do you compute phi (v,
t)? Well, it's exactly the component of

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00:00:03,006 --> 00:00:03,014
psi in the direction of this function. So,
how do you compute the component of psi in

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00:00:03,014 --> 00:00:03,021
this direction? Yo u take the inner
product. The inner product of this

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00:00:03,021 --> 00:00:03,028
function with psi (x, t) which is what?
It's the integral of e^-ikx, it's the

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00:00:03,028 --> 00:00:03,036
complex conjugate of this. Psi (x, t) dx
of minus infinity to infinity. Alright,

29
00:00:03,036 --> 00:00:03,044
what's the inner product is just, you, you
just, you just take the product of the

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00:00:03,044 --> 00:00:03,052
corresponding components. Well, with, with
the complex conjugate of the first one

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00:00:03,052 --> 00:00:03,060
times the component of the second one, and
then you add it over all, you know, over,

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00:00:03,060 --> 00:00:03,068
over all the dimensions. But here, we are,
we are working on continuous dimensional

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00:00:03,068 --> 00:00:03,075
space where x is what indexes these
dimensions. So, we just take complex

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00:00:03,075 --> 00:00:03,082
conjugate of the first time. So, second
and instead of summing over all the

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00:00:03,082 --> 00:00:03,090
dimensions, now since x is continuous, we
integrate this from minus infinity to

36
00:00:03,090 --> 00:00:03,098
infinity this time dx. And that's, that's
the component in this direction, but now,

37
00:00:03,098 --> 00:00:04,006
some of you might recognize this function.
What is this? Well, phi just the Fourier

38
00:00:04,006 --> 00:00:04,013
transform of psi, okay? So, while I'm
calling it velocity here just for, you

39
00:00:04,013 --> 00:00:04,021
know, just, just for intuitive purposes.
You know in quantum mechanics, we are

40
00:00:04,021 --> 00:00:04,029
really measuring momentum which is
proportional to velocity. And so velocity,

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00:00:04,029 --> 00:00:06,080
momentum is the Fourier of transform of
the, of position. So, position and

42
00:00:06,080 --> 00:00:07,051
momentum are related to each other by
Fourier transforms. This is a very

43
00:00:07,051 --> 00:00:07,051
interesting thing because what it says is,
you start with some superposition, psi.

44
00:00:07,051 --> 00:00:07,051
So, this is psi (x). This is a, this is a,
a position of a particle at, at, at

45
00:00:07,051 --> 00:00:07,051
whatever time, alight. It's, it's sum
function. Alright. This is psi (x). Now,

46
00:00:07,051 --> 00:00:07,083
we write phi (p), p for momentum. Okay,
so, this is, okay, let's, let's write it

47
00:00:07,083 --> 00:00:00,000
in blue. It's phi (p), okay? This is meant
to be some other superposition, and what,

48
00:00:00,000 --> 00:00:00,000
what, what I'm cleaning is that the way
you're getting from one to the other and

49
00:00:00,000 --> 00:00:00,000
back is by doing a Fourier transform.
Okay, so, what do you know about a Fourier

50
00:00:00,000 --> 00:00:00,000
transform? The one thing you should know
is that the more you concentrate a

51
00:00:00,000 --> 00:00:00,000
function in the primary domain, the more
it spreads out in the Fourier domain. The

52
00:00:00,000 --> 00:00:00,000
more you try to concentrate it in the
Fourier domain, the more it spreads out in

53
00:00:00,000 --> 00:00:00,000
the primary domain. So, what are you to
do? Well, it turns out that the best you

54
00:00:00,000 --> 00:00:00,000
can do is sort of make, make this, this
function on this side, be a Gaussian with

55
00:00:00,000 --> 00:00:00,000
some standard deviation sigma. And then,
you have some corresponding standard

56
00:00:00,000 --> 00:00:00,000
deviation sigma hat on this, on this side
and it turns out that the, that the

57
00:00:00,000 --> 00:00:00,000
smaller sigma is the bigger sigma hat is
and vice versa. And so, you can try to

58
00:00:00,000 --> 00:00:00,000
find the happy median between the two. And
this is what gives us this uncertainty

59
00:00:00,000 --> 00:00:00,000
principle which says that delta x delta p
is at least h bar / two. So, delta x is

60
00:00:00,000 --> 00:00:00,000
really sigma and delta p is sigma hat and
what if, you know, the, the fact about

61
00:00:00,000 --> 00:00:00,000
Fourier transforms is, is that the way you
min, minimize this product is by using

62
00:00:00,000 --> 00:00:00,000
Gaussian in the first place and in, in the
second place, if you use Gaussian, then,

63
00:00:00,000 --> 00:00:00,000
then the width of this Gaussian times the
width of this Gaussian in the Fourier,

64
00:00:00,000 --> 00:00:00,000
Fourier transform. The product of these
two bits is bounded below by some

65
00:00:00,000 --> 00:00:00,000
constant, okay? So, let me reiterate.
There is not much of this lecture that you

66
00:00:00,000 --> 00:00:00,000
really need to know. But, this is, you
know, this should help you get a picture

67
00:00:00,000 --> 00:00:00,000
of what's going on with continuous quantum
states, with the Schrodinger equation for

68
00:00:00,000 --> 00:00:00,000
the particle on the line with, where this
uncertainty principle comes from, okay? In

69
00:00:00,000 --> 00:00:00,000
the next lecture, we'll look at all of
this in a more precise way.