1
00:00:00,500 --> 00:00:08,880
PROFESSOR: We can write,
however, J left as J of A,

2
00:00:08,880 --> 00:00:21,410
minus J of B. Where J of A would
be h bar k over m A squared.

3
00:00:21,410 --> 00:00:31,090
And J of B would be h
bar k over m B squared.

4
00:00:31,090 --> 00:00:37,360
You see the current that exists
to the left of the barrier

5
00:00:37,360 --> 00:00:39,760
has two components and--

6
00:00:39,760 --> 00:00:41,180
it's very intuitive.

7
00:00:41,180 --> 00:00:45,610
It's the current that would
have been brought alone

8
00:00:45,610 --> 00:00:49,780
by the incoming A-wave.

9
00:00:49,780 --> 00:00:53,320
Minus the current that
would have existed alone

10
00:00:53,320 --> 00:00:56,530
from the reflected B-wave.

11
00:00:56,530 --> 00:01:00,510
B is the reflected wave.

12
00:01:00,510 --> 00:01:02,570
So that's very nice.

13
00:01:02,570 --> 00:01:05,330
There's no interference
between these two terms.

14
00:01:05,330 --> 00:01:07,850
We can really think
of a current that

15
00:01:07,850 --> 00:01:09,770
is associated to
the incoming wave,

16
00:01:09,770 --> 00:01:13,560
and a current that is associated
with the reflected wave.

17
00:01:13,560 --> 00:01:17,990
So this suggests how you
should define a reflection

18
00:01:17,990 --> 00:01:20,430
coefficient.

19
00:01:20,430 --> 00:01:31,960
Reflection coefficient
R would give me

20
00:01:31,960 --> 00:01:36,820
the amount of current
I get reflected,

21
00:01:36,820 --> 00:01:41,220
compared to the amount of
current that there is incident.

22
00:01:41,220 --> 00:01:43,810
You see, the incident
current is going

23
00:01:43,810 --> 00:01:48,140
to be partially reflected
and partially transmitted.

24
00:01:48,140 --> 00:01:52,630
So an idea of a
reflection is the value

25
00:01:52,630 --> 00:02:00,232
of the reflected current
divided by the incident current.

26
00:02:00,232 --> 00:02:06,270
It's a definition, but it's
a reasonable definition.

27
00:02:06,270 --> 00:02:11,200
And then, if it is this ratio--

28
00:02:11,200 --> 00:02:13,180
because of these
expressions-- it

29
00:02:13,180 --> 00:02:17,715
happens to be B over A squared.

30
00:02:23,160 --> 00:02:26,470
And that's an
interesting number.

31
00:02:26,470 --> 00:02:29,010
Now, there's some physics in it.

32
00:02:29,010 --> 00:02:32,380
It tells me how much
of the probability

33
00:02:32,380 --> 00:02:35,560
gets reflected as a function
of the probability that

34
00:02:35,560 --> 00:02:36,950
is incident.

35
00:02:36,950 --> 00:02:39,370
So that's a good measure.

36
00:02:39,370 --> 00:02:42,700
If you get a reflection
coefficient of 1/10,

37
00:02:42,700 --> 00:02:48,240
then you would expect 1/10 of
the particles to be reflected.

38
00:02:48,240 --> 00:02:50,130
Now we don't have particles yet.

39
00:02:50,130 --> 00:02:52,590
This is non-normalizable
solution.

40
00:02:52,590 --> 00:02:57,750
But, still, this will be
the intuition very soon.

41
00:02:57,750 --> 00:03:02,420
Now we could have a transmission
coefficient, as well.

42
00:03:02,420 --> 00:03:06,490
And here is something where
we sometimes make a mistake.

43
00:03:06,490 --> 00:03:10,699
T is going to be the
transmission coefficient.

44
00:03:10,699 --> 00:03:13,920
Transmission coefficient.

45
00:03:20,150 --> 00:03:22,910
And how should we define it?

46
00:03:22,910 --> 00:03:28,980
There is a temptation to define
it-- well, coefficient B over A

47
00:03:28,980 --> 00:03:29,690
gives me this.

48
00:03:29,690 --> 00:03:37,760
Then maybe it should be
C over A. But actually--

49
00:03:37,760 --> 00:03:40,070
while c over a
gives you some idea

50
00:03:40,070 --> 00:03:42,890
of how big the wave to
the right is compared

51
00:03:42,890 --> 00:03:44,480
to the wave of the left--

52
00:03:44,480 --> 00:03:50,225
that's not what we should
call a reflection coefficient.

53
00:03:50,225 --> 00:03:53,930
And the reason is that--

54
00:03:53,930 --> 00:03:59,480
I will call this
current J C. And that's

55
00:03:59,480 --> 00:04:02,480
the amount of probability--

56
00:04:02,480 --> 00:04:05,450
because it's a current
associated to the wave C.

57
00:04:05,450 --> 00:04:08,090
And that's the
amount of probability

58
00:04:08,090 --> 00:04:12,350
that is being carried
by the transmitted wave.

59
00:04:12,350 --> 00:04:14,250
That is the probability.

60
00:04:14,250 --> 00:04:16,630
Not necessarily C over A.

61
00:04:16,630 --> 00:04:18,589
So the transmission
coefficient will

62
00:04:18,589 --> 00:04:29,024
be defined to be J
C divided by J A.

63
00:04:29,024 --> 00:04:32,710
And then J C divided by J A--

64
00:04:32,710 --> 00:04:38,550
J C has an h-bar, k-bar.

65
00:04:38,550 --> 00:04:42,570
And J A has a k.

66
00:04:42,570 --> 00:04:45,600
So this is not
equal to this ratio,

67
00:04:45,600 --> 00:04:58,900
but is actually k-bar
over k, C over A.

68
00:04:58,900 --> 00:05:03,970
So it's not just this number.

69
00:05:03,970 --> 00:05:07,480
The reflection coefficient--
and transmission coefficients--

70
00:05:07,480 --> 00:05:10,000
really originate
from probabilities.

71
00:05:10,000 --> 00:05:13,900
And the probabilities
for this current.

72
00:05:13,900 --> 00:05:17,400
And, therefore, there is no--

73
00:05:17,400 --> 00:05:20,440
it would have been very
hand-wavy, and actually

74
00:05:20,440 --> 00:05:25,030
wrong to think it's C over A.

75
00:05:25,030 --> 00:05:28,510
These definitions--
because after all,

76
00:05:28,510 --> 00:05:30,380
this is a definition--

77
00:05:30,380 --> 00:05:32,320
makes some nice sense.

78
00:05:32,320 --> 00:05:33,940
Because you have J L--

79
00:05:33,940 --> 00:05:41,410
we said is equal to J right,
but J L is J A minus J

80
00:05:41,410 --> 00:05:49,740
B, is equal to J C.
And therefore R plus T.

81
00:05:49,740 --> 00:05:54,370
The reflection coefficient plus
the transmission coefficient--

82
00:05:54,370 --> 00:06:04,850
which is J B over J
A plus J C over J A

83
00:06:04,850 --> 00:06:12,020
is equal to J B
plus J C over J A.

84
00:06:12,020 --> 00:06:28,566
And you see here that J A plus
J C is indeed equal to J B.

85
00:06:28,566 --> 00:06:29,390
I'm sorry.

86
00:06:29,390 --> 00:06:30,890
I got this wrong.

87
00:06:33,410 --> 00:06:34,090
Yes.

88
00:06:34,090 --> 00:06:34,923
Where is the eraser.

89
00:06:37,310 --> 00:06:39,315
I should have passed
the B to the other side.

90
00:06:43,040 --> 00:06:49,510
This force implies J
A equals J B plus J C.

91
00:06:49,510 --> 00:06:54,080
And this ratio is equal to 1,
which is something you usually

92
00:06:54,080 --> 00:06:58,580
want when you define reflection
and transmission coefficients.

93
00:06:58,580 --> 00:07:01,950
They should add up to 1.

94
00:07:01,950 --> 00:07:05,810
So now we've got an idea.

95
00:07:05,810 --> 00:07:09,770
Yes, with this solution I
can understand the reflection

96
00:07:09,770 --> 00:07:11,060
and transmission coefficients.

97
00:07:11,060 --> 00:07:13,700
But do these apply to particles?

98
00:07:13,700 --> 00:07:16,670
Well, the good news
is that it roughly

99
00:07:16,670 --> 00:07:19,670
applies to particles-- as we
will see with the wave packets

100
00:07:19,670 --> 00:07:20,480
soon.

101
00:07:20,480 --> 00:07:24,110
If you send the
wave packet, it's

102
00:07:24,110 --> 00:07:26,270
going to have some
uncertainty and momentum.

103
00:07:26,270 --> 00:07:29,310
It's going to have some
uncertainty and energy.

104
00:07:29,310 --> 00:07:31,760
But for some energy--

105
00:07:31,760 --> 00:07:35,270
suppose the wave packet doesn't
have that much uncertainty--

106
00:07:35,270 --> 00:07:39,920
basically, the probability
that the wave packet bounces

107
00:07:39,920 --> 00:07:44,250
is the reflection at that
energy, that is the main energy

108
00:07:44,250 --> 00:07:46,250
that the wave packet sends.

109
00:07:46,250 --> 00:07:50,270
If your wave packet is
very broad over energies,

110
00:07:50,270 --> 00:07:52,440
then it's a more
complicated thing.

111
00:07:52,440 --> 00:07:55,310
But as long as the wave packet
is such that it basically

112
00:07:55,310 --> 00:08:01,340
has one narrow band of energy,
the reflection coefficient

113
00:08:01,340 --> 00:08:06,320
associated to this calculation
is the reflection coefficient

114
00:08:06,320 --> 00:08:10,150
or reflection probability
for the wave packet

115
00:08:10,150 --> 00:08:12,430
that you're sending in.