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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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00:00:11,002 --> 00:00:18,002
what, what this, what this tells us is
that if you work it out, it's as though,

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00:00:18,002 --> 00:00:24,007
you know as the slinky is precessing, as
it's rotating. It's as though it's

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00:00:24,007 --> 00:00:31,071
velocity is, is proportional to K. Okay,
so, why is that? Well, you see, we, we

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00:00:31,071 --> 00:00:40,018
already said that the period of this
function, the period of this, this the

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00:00:40,018 --> 00:00:48,080
wave function is two pi / K. And if you,
if you, if you look at this, you know,

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00:00:48,080 --> 00:00:57,001
this time dependence e^i K^2 t, the time
it takes to, for this, you know, as this

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00:00:57,001 --> 00:01:04,072
wave function is precessing to, to precess
to one full circle, is actually two pi /

9
00:01:04,072 --> 00:01:12,003
K^2. And so, so the fond movement of the
slinky, in, in one, one such, one such

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00:01:12,003 --> 00:01:21,003
full rotation of the phase, is two pi / K
but, but this happens in time t2 pi / K^2.

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00:01:21,003 --> 00:01:31,055
So, the apparent velocity is this over
this, so it is two pi / K / two pi / K^2,

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00:01:31,055 --> 00:01:41,087
which is K. Okay, so, so now, what, what,
what we've understood is if we, if our

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00:01:41,087 --> 00:01:50,034
wave function looked like this, if it
looks like e^ikx, then it would have a

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00:01:50,034 --> 00:01:59,070
definite velocity of K. Okay, but we're
not given, you know, but, but, we are

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00:01:59,070 --> 00:02:07,011
given just some psi (x, t). It's not, it's
not e^ikx, you know, this is those wave

16
00:02:07,011 --> 00:02:15,008
function with had the definite velocity of
K. So now, we can ask what's the, you

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00:02:15,008 --> 00:02:21,059
know, if we were to write out psi (x, t)
in the, in the velocity basis, right? So,

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00:02:21,059 --> 00:02:28,033
we could write it out in the velocity,
instead of in the position basis, we could

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00:02:28,033 --> 00:02:34,029
write it out in the velocity basis and we
might have some superposition which, you

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00:02:34,029 --> 00:02:39,062
know, which, which, which, which look like
that, which would say, with what amplitude

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00:02:39,062 --> 00:02:46,042
is the velocity of this, of this particle
at time t equal to, equal to v and the

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00:02:46,042 --> 00:02:52,098
answer would be, it's, it's this much,
alright. So, this is, this is amplitude.

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00:02:52,098 --> 00:03:02,040
Okay, so, let me, let me call it phi (v,
t). So, this is the amplitude which it has

24
00:03:02,040 --> 00:03:10,065
this. And so, how do you compute phi (v,
t)? Well, it's exactly the component of

25
00:03:10,065 --> 00:03:18,001
psi in the direction of this function. So,
how do you compute the component of psi in

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00:03:18,001 --> 00:03:23,004
this direction? Yo u take the inner
product. The inner product of this

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00:03:23,004 --> 00:03:33,030
function with psi (x, t) which is what?
It's the integral of e^-ikx, it's the

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00:03:33,030 --> 00:03:43,059
complex conjugate of this. Psi (x, t) dx
of minus infinity to infinity. Alright,

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00:03:43,059 --> 00:03:48,098
what's the inner product is just, you, you
just, you just take the product of the

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00:03:48,098 --> 00:03:54,010
corresponding components. Well, with, with
the complex conjugate of the first one

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00:03:54,010 --> 00:03:59,093
times the component of the second one, and
then you add it over all, you know, over,

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00:03:59,093 --> 00:04:05,014
over all the dimensions. But here, we are,
we are working on continuous dimensional

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00:04:05,014 --> 00:04:09,063
space where x is what indexes these
dimensions. So, we just take complex

34
00:04:09,063 --> 00:04:14,062
conjugate of the first time. So, second
and instead of summing over all the

35
00:04:14,062 --> 00:04:20,059
dimensions, now since x is continuous, we
integrate this from minus infinity to

36
00:04:20,059 --> 00:04:27,051
infinity this time dx. And that's, that's
the component in this direction, but now,

37
00:04:27,051 --> 00:04:37,059
some of you might recognize this function.
What is this? Well, phi just the Fourier

38
00:04:37,059 --> 00:04:47,086
transform of psi, okay? So, while I'm
calling it velocity here just for, you

39
00:04:47,086 --> 00:04:53,048
know, just, just for intuitive purposes.
You know in quantum mechanics, we are

40
00:04:53,048 --> 00:04:59,039
really measuring momentum which is
proportional to velocity. And so velocity,

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00:04:59,039 --> 00:05:05,036
momentum is the Fourier of transform of
the, of position. So, position and

42
00:05:05,036 --> 00:05:11,038
momentum are related to each other by
Fourier transforms. This is a very

43
00:05:11,038 --> 00:05:17,049
interesting thing because what it says is,
you start with some superposition, psi.

44
00:05:17,049 --> 00:05:25,006
So, this is psi (x). This is a, this is a,
a position of a particle at, at, at

45
00:05:25,006 --> 00:05:34,084
whatever time, alight. It's, it's sum
function. Alright. This is psi (x). Now,

46
00:05:34,084 --> 00:05:48,029
we write phi (p), p for momentum. Okay,
so, this is, okay, let's, let's write it

47
00:05:48,029 --> 00:05:58,071
in blue. It's phi (p), okay? This is meant
to be some other superposition, and what,

48
00:05:58,071 --> 00:06:05,054
what, what I'm cleaning is that the way
you're getting from one to the other and

49
00:06:05,054 --> 00:06:16,028
back is by doing a Fourier transform.
Okay, so, what do you know about a Fourier

50
00:06:16,028 --> 00:06:22,036
transform? The one thing you should know
is that the more you concentrate a

51
00:06:22,036 --> 00:06:29,029
function in the primary domain, the more
it spreads out in the Fourier domain. The

52
00:06:29,029 --> 00:06:34,073
more you try to concentrate it in the
Fourier domain, the more it spreads out in

53
00:06:34,073 --> 00:06:40,033
the primary domain. So, what are you to
do? Well, it turns out that the best you

54
00:06:40,033 --> 00:06:46,008
can do is sort of make, make this, this
function on this side, be a Gaussian with

55
00:06:46,008 --> 00:06:52,006
some standard deviation sigma. And then,
you have some corresponding standard

56
00:06:52,006 --> 00:06:58,061
deviation sigma hat on this, on this side
and it turns out that the, that the

57
00:06:58,061 --> 00:07:05,045
smaller sigma is the bigger sigma hat is
and vice versa. And so, you can try to

58
00:07:05,045 --> 00:07:13,016
find the happy median between the two. And
this is what gives us this uncertainty

59
00:07:13,016 --> 00:07:20,024
principle which says that delta x delta p
is at least h bar / two. So, delta x is

60
00:07:20,024 --> 00:07:27,046
really sigma and delta p is sigma hat and
what if, you know, the, the fact about

61
00:07:27,046 --> 00:07:34,006
Fourier transforms is, is that the way you
min, minimize this product is by using

62
00:07:34,006 --> 00:07:38,095
Gaussian in the first place and in, in the
second place, if you use Gaussian, then,

63
00:07:38,095 --> 00:07:44,034
then the width of this Gaussian times the
width of this Gaussian in the Fourier,

64
00:07:44,034 --> 00:07:49,094
Fourier transform. The product of these
two bits is bounded below by some

65
00:07:49,094 --> 00:07:55,072
constant, okay? So, let me reiterate.
There is not much of this lecture that you

66
00:07:55,072 --> 00:08:01,047
really need to know. But, this is, you
know, this should help you get a picture

67
00:08:01,047 --> 00:08:07,097
of what's going on with continuous quantum
states, with the Schrodinger equation for

68
00:08:07,097 --> 00:08:14,029
the particle on the line with, where this
uncertainty principle comes from, okay? In

69
00:08:14,029 --> 00:08:20,001
the next lecture, we'll look at all of
this in a more precise way.