1
00:00:00,300 --> 00:00:01,674
PROFESSOR: Let's
do a work check.

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00:00:04,200 --> 00:00:05,845
So main check.

3
00:00:11,970 --> 00:00:26,270
If integral psi star x t0,
psi x t0 dx is equal to 1

4
00:00:26,270 --> 00:00:32,680
at t equal to t0,
as we say there,

5
00:00:32,680 --> 00:00:45,160
then it must hold for later
times, t greater than t0.

6
00:00:45,160 --> 00:00:49,240
This is what we want to
check, or verify, or prove.

7
00:00:49,240 --> 00:00:53,860
Now, to do it, we're
going to take our time.

8
00:00:53,860 --> 00:00:57,970
So it's not going to happen in
five minutes, not 10 minutes,

9
00:00:57,970 --> 00:01:01,450
maybe not even half an hour.

10
00:01:01,450 --> 00:01:04,209
Not because it's
so difficult. It's

11
00:01:04,209 --> 00:01:05,980
because there's so
many things that one

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00:01:05,980 --> 00:01:08,620
can say in between
that teach you

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00:01:08,620 --> 00:01:10,270
a lot about quantum mechanics.

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00:01:10,270 --> 00:01:13,700
So we're going to
take our time here.

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So we're going to first rewrite
it with better notation.

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00:01:21,840 --> 00:01:30,090
So we'll define rho
of x and t, which

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00:01:30,090 --> 00:01:35,155
is going to be called
the probability density.

18
00:01:38,590 --> 00:01:42,280
And it's nothing else than
what you would expect, psi star

19
00:01:42,280 --> 00:01:46,960
of x and t, psi of x and t.

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It's a probability density.

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00:01:56,520 --> 00:01:59,690
You know that has the
right interpretation,

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00:01:59,690 --> 00:02:02,120
it's psi squared.

23
00:02:02,120 --> 00:02:05,820
And that's the kind of thing
that integrated over space

24
00:02:05,820 --> 00:02:08,250
gives you the total probability.

25
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So this is a
positive number given

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00:02:14,150 --> 00:02:17,490
by this quantity is called
the probability density.

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00:02:17,490 --> 00:02:18,680
Fine.

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00:02:18,680 --> 00:02:21,500
What do we know about
this probability density

29
00:02:21,500 --> 00:02:26,180
that we're trying to
find about its integral?

30
00:02:26,180 --> 00:02:36,500
So define next N of t to be the
integral of rho of x and t dx.

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00:02:39,800 --> 00:02:42,830
Integrate this
probability density

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00:02:42,830 --> 00:02:50,130
throughout space, and that's
going to give you N of t.

33
00:02:50,130 --> 00:02:51,730
Now, what do we know?

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00:02:51,730 --> 00:02:57,980
We know that N of t, or
let's assume that N of t0

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is equal to 1.

36
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N is that normalization.

37
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It's that total integral
of the probability

38
00:03:05,160 --> 00:03:07,620
what had to be equal to 1.

39
00:03:07,620 --> 00:03:13,290
Well, let's assume N
at t0 is equal to 1.

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That's good.

41
00:03:15,930 --> 00:03:22,270
The question is, will
the Schrodinger equation

42
00:03:22,270 --> 00:03:28,400
guarantee that--

43
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and here's the claim--

44
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dN dt is equal to 0?

45
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Will the Schrodinger
equation guarantee this?

46
00:03:46,470 --> 00:03:49,730
If the Schrodinger
equation guarantees

47
00:03:49,730 --> 00:03:52,980
that this derivative
is, indeed, zero,

48
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then we're in good business.

49
00:03:54,960 --> 00:03:59,580
Because the derivative
is zero, the value's 1,

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00:03:59,580 --> 00:04:01,600
will remain 1 forever.

51
00:04:01,600 --> 00:04:02,344
Yes?

52
00:04:02,344 --> 00:04:03,885
AUDIENCE: May I ask
why you specified

53
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for t greater than t0?

54
00:04:08,160 --> 00:04:13,380
Well, I don't have to specify
for t greater than t naught.

55
00:04:13,380 --> 00:04:19,029
I could do it for all t
different than t naught.

56
00:04:19,029 --> 00:04:29,160
But if I say this way, as
imagining that somebody

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00:04:29,160 --> 00:04:31,500
prepares the system
at some time,

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00:04:31,500 --> 00:04:35,310
t naught, and maybe the system
didn't exist for other times

59
00:04:35,310 --> 00:04:36,180
below.

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00:04:36,180 --> 00:04:39,750
Now, if a system
existed for long time

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00:04:39,750 --> 00:04:44,340
and you look at it at t naught,
then certainly the Schrodinger

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00:04:44,340 --> 00:04:49,740
equation should imply
that it works later

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00:04:49,740 --> 00:04:52,050
and it works before.

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So it's not really necessary,
but no loss of generality.

65
00:05:01,240 --> 00:05:02,750
OK, so that's it.

66
00:05:02,750 --> 00:05:04,510
Will it guarantee that?

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00:05:04,510 --> 00:05:06,380
Well, that's our thing to do.

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00:05:06,380 --> 00:05:15,020
So let's begin the work by doing
a little bit of a calculation.

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00:05:15,020 --> 00:05:18,190
And so what do we need to do?

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00:05:18,190 --> 00:05:22,610
We need to find the
derivative of this quantity.

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00:05:22,610 --> 00:05:34,690
So what is this derivative of N
dN dt will be the integral d dt

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00:05:34,690 --> 00:05:43,160
of rho of x and t dx.

73
00:05:43,160 --> 00:05:49,040
So I went here and
brought in the d dt, which

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00:05:49,040 --> 00:05:50,990
became a partial derivative.

75
00:05:50,990 --> 00:05:55,910
Because this is just a
function of t, but inside here,

76
00:05:55,910 --> 00:05:58,160
there's a function of
t and a function of x.

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00:05:58,160 --> 00:06:03,740
So I must make clear that
I'm just differentiating t.

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00:06:03,740 --> 00:06:07,490
So is d dt of rho.

79
00:06:07,490 --> 00:06:12,680
And now we can write
it as integral dx.

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00:06:12,680 --> 00:06:14,120
What this rho?

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00:06:14,120 --> 00:06:15,860
Psi star psi.

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00:06:15,860 --> 00:06:28,240
So we would have d dt of psi
star times psi plus psi star

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00:06:28,240 --> 00:06:30,620
d dt of psi.

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00:06:40,240 --> 00:06:40,740
OK.

85
00:06:44,480 --> 00:06:49,120
And here you see, if you
were waiting for that,

86
00:06:49,120 --> 00:06:52,220
that the Schrodinger
equation has to be necessary.

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Because we have the psi dt.

88
00:06:54,950 --> 00:06:59,550
And that information is there
with Schrodinger's equation.

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So let's do that.

90
00:07:03,540 --> 00:07:04,730
So what do we have?

91
00:07:04,730 --> 00:07:12,780
ih bar d psi dt equal h psi.

92
00:07:12,780 --> 00:07:15,590
We'll write it like
that for the time being

93
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without copying all what h is.

94
00:07:18,500 --> 00:07:20,090
That would take a lot of time.

95
00:07:23,490 --> 00:07:28,200
And from this equation,
you can find immediately

96
00:07:28,200 --> 00:07:39,440
that d psi dt is minus
i over h bar h hat psi.

97
00:07:44,220 --> 00:07:47,270
Now we need to complex
conjugate this equation,

98
00:07:47,270 --> 00:07:49,620
and that is always
a little more scary.

99
00:07:59,650 --> 00:08:02,140
Actually, the way
to do this in a way

100
00:08:02,140 --> 00:08:05,530
that you never get into
scary or strange things.

101
00:08:05,530 --> 00:08:09,280
So let me take the complex
conjugate of this equation.

102
00:08:09,280 --> 00:08:14,810
Here I would have i
goes to minus i h bar,

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00:08:14,810 --> 00:08:16,880
and now I would have--

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00:08:16,880 --> 00:08:18,860
we can go very slow--

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00:08:18,860 --> 00:08:29,480
d psi dt star equals, and then
I'll be simple minded here.

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00:08:29,480 --> 00:08:32,030
I think it's the best.

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00:08:32,030 --> 00:08:36,350
I'll just start the
right hand side.

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00:08:36,350 --> 00:08:41,340
I start the left hand side
and start the right hand side.

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00:08:41,340 --> 00:08:48,310
Now here, the complex conjugate
of a derivative, in this case

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I want to clarify what it is.

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It's just the derivative
of the complex conjugate.

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So this is minus ih
bar d/dt of psi star

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00:09:04,140 --> 00:09:12,540
equals h hat psi
star, that's fine.

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00:09:12,540 --> 00:09:21,450
And from here, if I multiply
again by i divided by h bar,

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00:09:21,450 --> 00:09:36,042
we get d psi star dt is equal
to i over h star h hat psi star.

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00:09:41,560 --> 00:09:47,040
We obtain this useful formula
and this useful formula,

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and both go into our
calculation of dN dt.

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So what do we have here?

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dN dt equals integral dx, and
I will put an i over h bar,

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I think, here.

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00:10:15,356 --> 00:10:15,855
Yes.

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00:10:18,990 --> 00:10:20,400
i over h bar.

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00:10:26,400 --> 00:10:28,055
Look at this term first.

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00:10:30,630 --> 00:10:40,840
We have i over h
bar, h psi star psi.

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And the second term
involves a d psi dt that

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comes with an opposite sign.

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Same factor of i over h bar,
so minus psi star h psi.

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00:11:06,610 --> 00:11:10,680
So the virtue of what
we've done so far

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00:11:10,680 --> 00:11:13,530
is that it doesn't
look so bad yet.

130
00:11:13,530 --> 00:11:21,900
And looks relatively clean, and
it's very suggestive, actually.

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00:11:21,900 --> 00:11:23,350
So what's happening?

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00:11:23,350 --> 00:11:27,295
We want to show that
dN dt is equal to 0.

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00:11:31,220 --> 00:11:35,210
Now, are we going to be
able to show that simply

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00:11:35,210 --> 00:11:39,080
that to do a lot of algebra
and say, oh, it's 0?

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00:11:39,080 --> 00:11:41,480
Well, it's kind of
going to work that way,

136
00:11:41,480 --> 00:11:43,880
but we're going to
do the work and we're

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00:11:43,880 --> 00:11:50,870
going to get to dN dt being
an integral of something.

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00:11:50,870 --> 00:11:53,410
And it's just not
going to look like 0,

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00:11:53,410 --> 00:11:56,900
but it will be
manipulated in such a way

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that you can argue it's 0
using the boundary condition.

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00:12:01,550 --> 00:12:05,210
So it's kind of interesting
how it's going to work.

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00:12:05,210 --> 00:12:08,930
But here structurally,
you see what

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00:12:08,930 --> 00:12:13,470
must happen for this
calculation to succeed.

144
00:12:13,470 --> 00:12:15,670
So we need for this to be 0.

145
00:12:22,630 --> 00:12:27,250
We need the following
thing to happen.

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00:12:27,250 --> 00:12:36,580
The integral of h
hat psi star psi

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00:12:36,580 --> 00:12:42,776
be equal to the integral
of psi star h psi.

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00:12:45,692 --> 00:12:48,510
And I should write the dx's.

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00:12:48,510 --> 00:12:49,260
They are there.

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00:12:53,030 --> 00:12:59,310
So this would guarantee
that dN dt is equal to 0.

151
00:12:59,310 --> 00:13:05,500
So that's a very nice
statement, and it's kind of nice

152
00:13:05,500 --> 00:13:09,890
is that you have one
function starred,

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00:13:09,890 --> 00:13:12,540
one function non-starred.

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00:13:12,540 --> 00:13:16,110
The h is where the function
needs to be starred,

155
00:13:16,110 --> 00:13:18,520
but on the other
side of the equation,

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00:13:18,520 --> 00:13:21,520
the h is on the other side.

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00:13:21,520 --> 00:13:27,300
So you've kind of moved the
h from the complex conjugated

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00:13:27,300 --> 00:13:30,570
function to the non-complex
conjugated function.

159
00:13:30,570 --> 00:13:34,720
From the first function
to this second function.

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00:13:34,720 --> 00:13:40,320
And that's a very nice thing
to demand of the Hamiltonian.

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00:13:40,320 --> 00:13:43,290
So actually what
seems to be happening

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00:13:43,290 --> 00:13:47,190
is that this conservation
of probability

163
00:13:47,190 --> 00:13:50,490
will work if your
Hamiltonian is good enough

164
00:13:50,490 --> 00:13:52,500
to do something like this.

165
00:13:55,530 --> 00:13:59,295
And this is a nice formula,
it's a famous formula.

166
00:14:05,670 --> 00:14:15,600
This is true if H is
a Hermitian operator.

167
00:14:22,990 --> 00:14:25,990
It's a very interesting
new name that

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00:14:25,990 --> 00:14:29,750
shows up that an
operator being Hermitian.

169
00:14:29,750 --> 00:14:33,650
So this is what I
was promising you,

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00:14:33,650 --> 00:14:35,420
that we're going to
do this, and we're

171
00:14:35,420 --> 00:14:40,500
going to be learning all kinds
of funny things as it happens.

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00:14:40,500 --> 00:14:45,200
So what is it for a
Hermitian operator?

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00:14:45,200 --> 00:14:56,650
Well, a Hermitian
operator, H, would actually

174
00:14:56,650 --> 00:14:58,310
satisfy the following.

175
00:15:06,810 --> 00:15:17,590
That the integral,
H psi 1 star psi 2

176
00:15:17,590 --> 00:15:29,100
is equal to the integral
of psi 1 star H psi 2.

177
00:15:29,100 --> 00:15:37,230
So an operator is said to be
Hermitian if you can move it

178
00:15:37,230 --> 00:15:44,310
from the first part to the
second part in this sense,

179
00:15:44,310 --> 00:15:47,950
and with two
different functions.

180
00:15:47,950 --> 00:15:51,720
So this should be possible
to do if an operator is

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00:15:51,720 --> 00:15:53,370
to be called Hermitian.

182
00:15:55,940 --> 00:16:00,640
Now, of course, if it holds
for two arbitrary functions,

183
00:16:00,640 --> 00:16:05,400
it holds when the two functions
are the same, in this case.

184
00:16:05,400 --> 00:16:09,110
So what we need is
a particular case

185
00:16:09,110 --> 00:16:12,350
of the condition of hermiticity.

186
00:16:12,350 --> 00:16:16,700
Hermiticity simply means that
the operator does this thing.

187
00:16:20,410 --> 00:16:27,080
Any two functions that you put
here, this equality is true.

188
00:16:27,080 --> 00:16:31,300
Now if you ask yourself, how
do I even understand that?

189
00:16:31,300 --> 00:16:35,550
What allows me to move the H
from one side to the other?

190
00:16:35,550 --> 00:16:37,130
We'll see it very soon.

191
00:16:37,130 --> 00:16:41,380
But it's the fact that H
has second derivatives,

192
00:16:41,380 --> 00:16:44,240
and maybe you can
integrate them by parts

193
00:16:44,240 --> 00:16:47,980
and move the derivatives
from the psi 1 to the psi 2,

194
00:16:47,980 --> 00:16:50,210
and do all kinds of things.

195
00:16:50,210 --> 00:16:55,220
But you should try to think at
this moment structurally, what

196
00:16:55,220 --> 00:16:58,610
kind of objects you have, what
kind of properties you have.

197
00:16:58,610 --> 00:17:02,840
And the objects
are this operator

198
00:17:02,840 --> 00:17:05,540
that controls the
time evolution, called

199
00:17:05,540 --> 00:17:07,170
the Hamiltonian.

200
00:17:07,170 --> 00:17:12,420
And if I want probability
interpretation to make sense,

201
00:17:12,420 --> 00:17:17,420
we need this equality, which is
a consequence of hermiticity.

202
00:17:17,420 --> 00:17:23,170
Now, I'll maybe use a
little of this blackboard.

203
00:17:23,170 --> 00:17:28,280
I haven't used it much before.

204
00:17:28,280 --> 00:17:31,460
In terms of Hermitian
operators, I'm

205
00:17:31,460 --> 00:17:35,540
almost there with a definition
of a Hermitian operator.

206
00:17:35,540 --> 00:17:40,550
I haven't quite given it to
you, but let's let state it,

207
00:17:40,550 --> 00:17:46,750
given that we're already in
this discussion of hermiticity.

208
00:17:46,750 --> 00:17:53,690
So this is what is called the
Hermitian operator, does that.

209
00:17:53,690 --> 00:18:11,580
But in general, rho,
given an operator T,

210
00:18:11,580 --> 00:18:29,980
one defines its hermitian
conjugate P dagger as follows.

211
00:18:29,980 --> 00:18:37,135
So you have the
integral of psi 1 star T

212
00:18:37,135 --> 00:18:44,990
psi 2, and that must be
rearranged until it looks

213
00:18:44,990 --> 00:18:56,000
like T dagger psi 1 star psi 2.

214
00:18:56,000 --> 00:18:59,270
Now, these things
are the beginning

215
00:18:59,270 --> 00:19:04,970
of a whole set of ideas that are
terribly important in quantum

216
00:19:04,970 --> 00:19:05,910
mechanics.

217
00:19:05,910 --> 00:19:09,960
Hermitian operators, or
eigenvalues and eigenvectors.

218
00:19:09,960 --> 00:19:12,170
So it's going to take
a little time for you

219
00:19:12,170 --> 00:19:14,040
to get accustomed to them.

220
00:19:14,040 --> 00:19:15,770
But this is the beginning.

221
00:19:15,770 --> 00:19:17,990
You will explore a little
bit of these things

222
00:19:17,990 --> 00:19:20,690
in future homework, and
start getting familiar.

223
00:19:20,690 --> 00:19:24,950
For now, it looks very
strange and unmotivated.

224
00:19:24,950 --> 00:19:28,550
Maybe you will see that
that will change soon, even

225
00:19:28,550 --> 00:19:32,190
throughout today's lecture.

226
00:19:32,190 --> 00:19:35,900
So this is the
Hermitian conjugate.

227
00:19:35,900 --> 00:19:39,320
So if you want to calculate
the Hermitian conjugate,

228
00:19:39,320 --> 00:19:42,710
you must start with this thing,
and start doing manipulations

229
00:19:42,710 --> 00:19:48,410
to clean up the psi 2,
have nothing at the psi 2,

230
00:19:48,410 --> 00:19:51,680
everything acting on
psi 1, and that thing

231
00:19:51,680 --> 00:19:53,780
is called the dagger.

232
00:19:53,780 --> 00:20:06,830
And then finally, T is Hermitian
if T dagger is equal to T.

233
00:20:06,830 --> 00:20:10,880
So its Hermitian
conjugate is itself.

234
00:20:10,880 --> 00:20:13,730
It's almost like people
say a real number is

235
00:20:13,730 --> 00:20:17,300
a number whose complex
conjugate is equal to itself.

236
00:20:17,300 --> 00:20:23,240
So a Hermitian operator is
one whose Hermitian conjugate

237
00:20:23,240 --> 00:20:28,400
is equal to itself, and
you see if T is Hermitian,

238
00:20:28,400 --> 00:20:34,370
well then it's back to T
and T in both places, which

239
00:20:34,370 --> 00:20:36,800
is what we've been saying here.

240
00:20:36,800 --> 00:20:39,910
This is a Hermitian operator.