1
00:00:00,500 --> 00:00:02,460
BARTON ZWIEBACH: After
this long detour,
2
00:00:02,460 --> 00:00:05,190
you must think that
one is just trying
3
00:00:05,190 --> 00:00:08,720
to avoid doing the real
computation, so here comes,
4
00:00:08,720 --> 00:00:12,050
the real computation.
5
00:00:12,050 --> 00:00:15,920
The real computation is
taking that right hand side
6
00:00:15,920 --> 00:00:19,830
on the top of the blackboard
and trying to just calculate
7
00:00:19,830 --> 00:00:20,960
this right hand side.
8
00:00:20,960 --> 00:00:23,230
So back to the calculation.
9
00:00:26,500 --> 00:00:40,530
The calculation dN/dt
is equal to this thing
10
00:00:40,530 --> 00:00:45,635
over there, integral
dx i over h-bar.
11
00:00:48,570 --> 00:00:58,470
I'll still copy it here-- h
psi-star psi minus psi-star h
12
00:00:58,470 --> 00:01:00,520
psi.
13
00:01:00,520 --> 00:01:01,020
OK.
14
00:01:04,580 --> 00:01:09,242
Well, let's do this.
15
00:01:09,242 --> 00:01:24,670
This whole quantity is d-rho/dt,
and let's see how much it is.
16
00:01:24,670 --> 00:01:30,790
Well, you would have of the
following-- i over h-bar h
17
00:01:30,790 --> 00:01:33,360
psi-star.
18
00:01:33,360 --> 00:01:43,420
Well, h in detail is over
there, so I'll put it here.
19
00:01:43,420 --> 00:01:52,870
Minus h squared over 2m d
second dx squared of psi-star.
20
00:01:52,870 --> 00:01:57,040
So I'm beginning h psi star--
21
00:01:57,040 --> 00:02:00,485
that's from the first term in
the Hamiltonian-- times psi.
22
00:02:03,670 --> 00:02:06,400
And then from the other
term in the Hamiltonian
23
00:02:06,400 --> 00:02:16,512
is the potential, so it would be
plus V of x and t psi-star psi.
24
00:02:19,280 --> 00:02:24,230
V of x and t times
psi-star times psi.
25
00:02:24,230 --> 00:02:29,810
This other term would
be minus psi-star h psi,
26
00:02:29,810 --> 00:02:36,560
so it's going to be opposite
sign to here, so plus h squared
27
00:02:36,560 --> 00:02:48,140
over 2m psi-star d second
dx squared psi, and then
28
00:02:48,140 --> 00:02:56,470
minus psi-star V of x and t psi.
29
00:02:56,470 --> 00:02:59,450
Here, there was a little
thing that I probably
30
00:02:59,450 --> 00:03:02,960
should have said before
is that the potential is
31
00:03:02,960 --> 00:03:07,880
real, that's why it didn't
get complex conjugated here.
32
00:03:07,880 --> 00:03:17,330
H psi would have a term V psi
and we just conjugate the psi.
33
00:03:17,330 --> 00:03:20,930
OK, this is not so bad.
34
00:03:20,930 --> 00:03:25,740
In particular, you see that
these two terms cancel.
35
00:03:25,740 --> 00:03:27,990
So that's neat.
36
00:03:27,990 --> 00:03:34,370
And now, this becomes
the following--
37
00:03:34,370 --> 00:03:55,370
this d-rho/dt has become minus
ih over 2m d second psi-star dx
38
00:03:55,370 --> 00:04:05,480
squared times psi minus psi-star
d second psi dx squared.
39
00:04:05,480 --> 00:04:05,980
OK.
40
00:04:13,670 --> 00:04:19,339
That's what d-rho/dt is and
that's the thing that should be
41
00:04:19,339 --> 00:04:23,430
0 when you integrate--
42
00:04:23,430 --> 00:04:26,840
it doesn't look like
anything equal to 0,
43
00:04:26,840 --> 00:04:31,820
and that was pretty
much to be expected.
44
00:04:31,820 --> 00:04:35,850
So what do we have
to do with this?
45
00:04:35,850 --> 00:04:39,200
Well, we have to
simplify it more,
46
00:04:39,200 --> 00:04:43,550
and what could save us
is, and it's usually
47
00:04:43,550 --> 00:04:46,070
the same thing that saves you
all the time when you want
48
00:04:46,070 --> 00:04:51,530
to show an integral vanishes,
many times, what you show
49
00:04:51,530 --> 00:04:53,870
is that it is a
total derivative.
50
00:04:53,870 --> 00:04:57,640
So remember, we're
computing here d-rho/dt,
51
00:04:57,640 --> 00:05:02,820
which is all this
thing circled here,
52
00:05:02,820 --> 00:05:05,630
and it's to be
integrated over x.
53
00:05:05,630 --> 00:05:09,980
So if I could show this is
a derivative with respect
54
00:05:09,980 --> 00:05:14,060
to x, the total x
derivative, then the integral
55
00:05:14,060 --> 00:05:16,400
would go to the
boundaries and I would
56
00:05:16,400 --> 00:05:18,890
have a chance to make it 0.
57
00:05:18,890 --> 00:05:23,110
So what do we have?
58
00:05:23,110 --> 00:05:27,420
That derivative is
indeed at boundaries,
59
00:05:27,420 --> 00:05:34,960
so d-rho/dt is equal to
minus i h-bar bar over 2m.
60
00:05:34,960 --> 00:05:42,350
And look, this can be
written as d/dx of something
61
00:05:42,350 --> 00:05:44,660
and what is that something?
62
00:05:44,660 --> 00:05:58,130
It's d psi-star dx times psi
minus psi-star d second psi--
63
00:05:58,130 --> 00:06:00,930
no not d second--
d first psi dx.
64
00:06:05,740 --> 00:06:07,790
The nice thing that
happens here is
65
00:06:07,790 --> 00:06:10,850
that if you act
with this d/dx, you
66
00:06:10,850 --> 00:06:15,600
get the second derivative
terms that you had in there.
67
00:06:15,600 --> 00:06:19,400
But you also get derivatives
acting here on d psi
68
00:06:19,400 --> 00:06:23,580
and here on d psi-star,
but those will cancel.
69
00:06:23,580 --> 00:06:26,630
So it's a very
lucky circumstance,
70
00:06:26,630 --> 00:06:31,670
it had better happen, but
this is a total derivative
71
00:06:31,670 --> 00:06:33,530
with respect to x.
72
00:06:33,530 --> 00:06:38,340
And that's just
very a good deal.
73
00:06:38,340 --> 00:06:43,650
So we're going to
rewrite it a little more.
74
00:06:43,650 --> 00:06:50,190
I'll write it as
the following way--
75
00:06:50,190 --> 00:07:04,420
this whole factor is h over 2im,
that's with its sign, output
76
00:07:04,420 --> 00:07:06,610
the d/dx outside--
77
00:07:06,610 --> 00:07:11,470
I'll put an extra minus
sign, so I will flip
78
00:07:11,470 --> 00:07:12,970
the order of these two terms--
79
00:07:16,350 --> 00:07:24,460
psi-star d psi dx minus
psi d psi-star dx.
80
00:07:28,370 --> 00:07:28,870
OK.
81
00:07:39,120 --> 00:07:42,660
Well, in many ways,
the most difficult part
82
00:07:42,660 --> 00:07:45,180
of the calculation
is over and it's now
83
00:07:45,180 --> 00:07:48,300
a matter of giving
proper names to things.
84
00:07:51,120 --> 00:07:52,540
Why do I say that?
85
00:07:52,540 --> 00:07:55,710
Because look, want to
see the finish line?
86
00:07:55,710 --> 00:07:58,440
It's here.
87
00:07:58,440 --> 00:08:05,380
We've shown this whole integrand
is d/dx of that right hand
88
00:08:05,380 --> 00:08:06,660
side.
89
00:08:06,660 --> 00:08:09,200
Therefore, when you
do the integral,
90
00:08:09,200 --> 00:08:12,920
you will have to go to the
boundary with that thing,
91
00:08:12,920 --> 00:08:17,330
so you just need to see what
happens to these quantities
92
00:08:17,330 --> 00:08:19,920
as x goes to infinity.
93
00:08:19,920 --> 00:08:22,040
And as x goes to
infinity, we said
94
00:08:22,040 --> 00:08:27,416
that psi must go to
0 from the beginning.
95
00:08:27,416 --> 00:08:32,720
And d psi dx must not blow
up, so if psi goes to 0
96
00:08:32,720 --> 00:08:37,039
and d psi dx doesn't blow up,
this whole thing goes to 0
97
00:08:37,039 --> 00:08:43,260
and dN/dt is equal
to 0 and you're done.
98
00:08:43,260 --> 00:08:47,700
So you're done
with the conditions
99
00:08:47,700 --> 00:08:52,130
that we mention that
the wave function must
100
00:08:52,130 --> 00:08:56,020
satisfy these conditions.
101
00:08:59,540 --> 00:09:02,460
But let's clean up this,
because we've actually
102
00:09:02,460 --> 00:09:06,290
discovered an important
quantity over there that
103
00:09:06,290 --> 00:09:10,350
is going to play a role.
104
00:09:10,350 --> 00:09:16,850
So here you see that you
have a complex number
105
00:09:16,850 --> 00:09:19,170
minus its complex conjugate.
106
00:09:19,170 --> 00:09:24,530
So this is like z
minus z-star, which
107
00:09:24,530 --> 00:09:32,520
is equal to 2i I times
the imaginary part of z.
108
00:09:32,520 --> 00:09:36,090
If you subtract from a complex
number its complex conjugate,
109
00:09:36,090 --> 00:09:40,910
you get the imaginary part only
survives, but it's twice of it.
110
00:09:44,250 --> 00:09:53,380
So from here, this whole thing
is 2i times the imaginary part
111
00:09:53,380 --> 00:10:00,350
of psi-star d psi dx.
112
00:10:05,080 --> 00:10:20,150
So d-rho/dt is equal
to minus d/dx of what?
113
00:10:20,150 --> 00:10:24,800
Of 2i times the imaginary
part of that, cancels the 2i,
114
00:10:24,800 --> 00:10:34,490
you get h-bar over m imaginary
part of psi-star d psi dx.
115
00:10:39,980 --> 00:10:47,730
And this quantity is going to
be called the current density.
116
00:10:53,120 --> 00:10:58,300
So the current density, you
say, why the current density?
117
00:10:58,300 --> 00:11:01,340
We'll see in a minute.
118
00:11:01,340 --> 00:11:07,690
But let's write it here because
it'll be very important.
119
00:11:07,690 --> 00:11:13,270
J of x and t is h-bar
over m imaginary part
120
00:11:13,270 --> 00:11:16,900
of psi-star d psi dx.
121
00:11:26,860 --> 00:11:31,820
So if this is called
the current density,
122
00:11:31,820 --> 00:11:37,710
you would have an equation
here d-rho/dt is equal to minus
123
00:11:37,710 --> 00:11:54,350
dJ/dx d/dx of J dx, or d-rho/dt
plus dJ/dx is equal to 0.
124
00:11:58,320 --> 00:12:02,770
Now this is called
current conservation.
125
00:12:02,770 --> 00:12:09,040
You've seen it before
in electromagnetism
126
00:12:09,040 --> 00:12:13,090
and we'll review it here
in a second as well.
127
00:12:22,740 --> 00:12:24,330
So look what has happened.
128
00:12:24,330 --> 00:12:31,780
You began with the introduction
of a charged density, which
129
00:12:31,780 --> 00:12:36,320
was a probability density,
but you were led now
130
00:12:36,320 --> 00:12:38,155
to the existence of a current.
131
00:12:42,620 --> 00:12:46,460
And you've seen that
in three dimensions,
132
00:12:46,460 --> 00:12:48,590
more than in one dimension--
133
00:12:48,590 --> 00:12:51,110
I think probably
in one dimension
134
00:12:51,110 --> 00:12:53,930
it doesn't look that
familiar to you,
135
00:12:53,930 --> 00:12:59,240
but let me make sure you will
recognize it in a few seconds.
136
00:12:59,240 --> 00:13:04,400
So think units here first.
137
00:13:04,400 --> 00:13:06,500
Units.
138
00:13:06,500 --> 00:13:08,940
What are the units
of the wave function?
139
00:13:08,940 --> 00:13:13,820
Well, the wave function,
you integrate over x squared
140
00:13:13,820 --> 00:13:15,080
and it gives you 1.
141
00:13:15,080 --> 00:13:20,220
So the integral of psi
squared dx is equal to 1,
142
00:13:20,220 --> 00:13:25,040
so this has units of length,
this must have units of 1
143
00:13:25,040 --> 00:13:26,425
over square root of length.
144
00:13:30,820 --> 00:13:41,590
And what are therefore the
units of psi-star d/dx psi,
145
00:13:41,590 --> 00:13:44,410
which is part of the
current formula ?
146
00:13:44,410 --> 00:13:47,080
Well, 1 of the square
root of length--
147
00:13:47,080 --> 00:13:50,500
1 over square root of length is
one over length and another 1
148
00:13:50,500 --> 00:13:54,270
over length is 1
over length squared.
149
00:13:54,270 --> 00:13:55,720
OK.
150
00:13:55,720 --> 00:13:59,780
And then you have
h-bar, which has
151
00:13:59,780 --> 00:14:05,660
units of mL squared over T.
Probably done that before
152
00:14:05,660 --> 00:14:07,660
already.
153
00:14:07,660 --> 00:14:19,690
And therefore, h over m has
units of L squared over T.
154
00:14:19,690 --> 00:14:25,315
So the current has
units of h over m--
155
00:14:25,315 --> 00:14:31,070
the units of current
has units of h over m,
156
00:14:31,070 --> 00:14:34,730
which is L squared over T--
157
00:14:34,730 --> 00:14:38,250
times units of this whole
thing, which is 1 over L
158
00:14:38,250 --> 00:14:44,920
squared, so at
the end, 1 over T.
159
00:14:44,920 --> 00:14:50,460
And this means just
probability per unit time.
160
00:14:50,460 --> 00:14:52,320
That's the units of current.
161
00:14:52,320 --> 00:14:58,620
Probability has no units, so
we're dealing probability,
162
00:14:58,620 --> 00:15:03,190
those are pure numbers, but this
is probability per unit time.
163
00:15:03,190 --> 00:15:12,640
So probability per unit time.