1
00:00:00,527 --> 00:00:01,110
PROFESSOR: OK.

2
00:00:01,110 --> 00:00:03,620
So what is our
Schrodinger equation?

3
00:00:03,620 --> 00:00:07,180
Therefore, our
Schrodinger equation

4
00:00:07,180 --> 00:00:16,096
is ih bar d psi dt is
equal to 1 over 2 m.

5
00:00:16,096 --> 00:00:18,460
H bar over i grad--

6
00:00:18,460 --> 00:00:30,070
that's p-- minus q over c a
squared plus q phi on psi.

7
00:00:34,820 --> 00:00:38,970
We're going to motivate
that next time.

8
00:00:38,970 --> 00:00:43,790
But let's look at it for
a little while, at least.

9
00:00:43,790 --> 00:00:47,540
There's several things
we've done here.

10
00:00:47,540 --> 00:00:54,716
We've replaced p by
p minus q over c a.

11
00:01:04,269 --> 00:01:06,160
So this is a replacement.

12
00:01:06,160 --> 00:01:08,020
P has been replaced by that.

13
00:01:08,020 --> 00:01:10,420
We used to have p
squared over 2m.

14
00:01:10,420 --> 00:01:13,010
Now we have this quantity.

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00:01:13,010 --> 00:01:16,580
Well, we'll see why that
is the right thing to do,

16
00:01:16,580 --> 00:01:23,270
but you could ask
yourself, is this p still

17
00:01:23,270 --> 00:01:29,930
intuitively equal to mv or not?

18
00:01:29,930 --> 00:01:31,020
And the answer is no.

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00:01:31,020 --> 00:01:33,800
That's not really mv.

20
00:01:33,800 --> 00:01:36,240
We will see that from
Heisenberg's equation

21
00:01:36,240 --> 00:01:40,380
of motion, but intuitively,
when you have here--

22
00:01:40,380 --> 00:01:43,200
this is the energy
of a particle--

23
00:01:43,200 --> 00:01:45,750
kinetic energy and
potential energy--

24
00:01:45,750 --> 00:01:47,390
the kinetic energy--

25
00:01:47,390 --> 00:01:57,590
1 over 2m times m
squared v squared.

26
00:01:57,590 --> 00:01:58,700
That's p squared.

27
00:01:58,700 --> 00:02:04,640
So 1 over 2m m squared v
squared is 1/2 mv squared.

28
00:02:04,640 --> 00:02:06,050
That's the kinetic energy.

29
00:02:06,050 --> 00:02:08,220
And that should come from here.

30
00:02:08,220 --> 00:02:09,979
So what we will
see is that, if you

31
00:02:09,979 --> 00:02:13,910
want to speak about
the velocity operator,

32
00:02:13,910 --> 00:02:19,090
this whole thing is
the velocity operator.

33
00:02:19,090 --> 00:02:23,350
We don't speak, really, of
velocity operators in quantum

34
00:02:23,350 --> 00:02:26,660
mechanics before we put
electromagnetic fields,

35
00:02:26,660 --> 00:02:31,420
but here, it will be natural to
call this m times the velocity

36
00:02:31,420 --> 00:02:35,660
operator in the sense of
Heisenberg equations of motion.

37
00:02:35,660 --> 00:02:38,850
This operator is Heisenberg
equation of motion--

38
00:02:38,850 --> 00:02:43,930
is going to look like the
Lorenz force equation.

39
00:02:43,930 --> 00:02:48,130
So it will be reasonable
to think of this like that.

40
00:02:48,130 --> 00:02:54,220
This operator is nicer than
the operator p, as we will see,

41
00:02:54,220 --> 00:02:55,135
for many reasons.

42
00:02:58,020 --> 00:03:04,690
So we've emphasized gauging
variance so that, perhaps,

43
00:03:04,690 --> 00:03:06,910
the most important
thing we could say, now,

44
00:03:06,910 --> 00:03:09,610
to end up this
lecture is, what is

45
00:03:09,610 --> 00:03:13,835
the statement of gauge
invariance for this Schrodinger

46
00:03:13,835 --> 00:03:14,335
equation?

47
00:03:18,660 --> 00:03:23,010
So gauge invariance
means that the physics

48
00:03:23,010 --> 00:03:27,570
that you obtain with
one set of potentials

49
00:03:27,570 --> 00:03:30,510
should be the same
as the physics you

50
00:03:30,510 --> 00:03:34,590
obtain with a gauge-equivalent
set of potentials.

51
00:03:41,410 --> 00:03:45,780
So I will say this way--

52
00:03:45,780 --> 00:03:52,020
suppose you solve the
Schrodinger equation

53
00:03:52,020 --> 00:04:00,130
with h bar over i
grad minus q over c,

54
00:04:00,130 --> 00:04:05,160
with the new potentials--

55
00:04:05,160 --> 00:04:20,010
plus q-- or you solve
the Schrodinger equation

56
00:04:20,010 --> 00:04:21,638
with the old potentials?

57
00:04:33,490 --> 00:04:35,310
So you have here
the two Schrodinger

58
00:04:35,310 --> 00:04:38,430
equations-- one with
the new potentials, one

59
00:04:38,430 --> 00:04:39,990
with the old potentials.

60
00:04:44,620 --> 00:04:48,530
They should be the
same physical solution.

61
00:04:48,530 --> 00:04:54,610
This is not going to be too
obvious how to do, however.

62
00:04:54,610 --> 00:04:59,220
How do I guarantee they are
the same physical solution?

63
00:04:59,220 --> 00:05:03,550
I'll have to go on a
limb and try something.

64
00:05:03,550 --> 00:05:04,330
Look.

65
00:05:04,330 --> 00:05:06,310
Here are the gauge
transformation.

66
00:05:06,310 --> 00:05:08,590
That's what a prime is.

67
00:05:08,590 --> 00:05:10,495
That's what phi prime is.

68
00:05:14,600 --> 00:05:20,880
Should the same psi
be a solution of both?

69
00:05:20,880 --> 00:05:24,450
Should this equation
imply this equation,

70
00:05:24,450 --> 00:05:28,830
so that the same psi works
when you change the gauge

71
00:05:28,830 --> 00:05:30,930
potentials?

72
00:05:30,930 --> 00:05:32,380
That would not work.

73
00:05:32,380 --> 00:05:34,390
That is asking too much.

74
00:05:37,570 --> 00:05:43,460
Certainly, the same psi worked
would be simple looking,

75
00:05:43,460 --> 00:05:51,210
but that's not what you
really have to demand.

76
00:05:51,210 --> 00:05:54,020
It's not going to be
able to occur here.

77
00:05:54,020 --> 00:05:58,310
What you're going to need
is to change psi as well.

78
00:05:58,310 --> 00:06:01,850
The gauge transformation
is going to affect the wave

79
00:06:01,850 --> 00:06:03,740
function, too.

80
00:06:03,740 --> 00:06:08,330
Not only the electromagnetic
fields get gauge transform--

81
00:06:08,330 --> 00:06:11,630
the wave function must
be gauge transform.

82
00:06:11,630 --> 00:06:13,290
And you would say, OK.

83
00:06:13,290 --> 00:06:16,880
That sounds a little dangerous
because if you change the wave

84
00:06:16,880 --> 00:06:19,610
function, you're going
to change the physics.

85
00:06:19,610 --> 00:06:20,940
Could happen.

86
00:06:20,940 --> 00:06:22,880
But the change in
the wave function

87
00:06:22,880 --> 00:06:25,380
is going to be subtle enough--

88
00:06:25,380 --> 00:06:28,850
is going to be just by a phase.

89
00:06:28,850 --> 00:06:30,950
That can still
change the physics.

90
00:06:30,950 --> 00:06:33,860
If you have a complex phase,
you can change the physics.

91
00:06:33,860 --> 00:06:37,730
But will be simple
enough that we

92
00:06:37,730 --> 00:06:40,290
will check that the
physics is not changed.

93
00:06:40,290 --> 00:06:44,270
So the claim of gauge
invariance is a statement

94
00:06:44,270 --> 00:06:50,390
that this equation implies
this, or this implies that,

95
00:06:50,390 --> 00:06:57,140
if psi prime also transforms.

96
00:06:57,140 --> 00:07:00,380
And the formula
is-- at psi prime--

97
00:07:00,380 --> 00:07:09,095
should be equal to e to the
i q over hc lambda times psi.

98
00:07:16,690 --> 00:07:22,090
So that's the key to it.

99
00:07:22,090 --> 00:07:24,910
When you transform
the potentials--

100
00:07:24,910 --> 00:07:28,390
when you change a to a
prime and phi to phi prime--

101
00:07:28,390 --> 00:07:30,720
you should change
psi to psi prime.

102
00:07:30,720 --> 00:07:31,220
What?

103
00:07:31,220 --> 00:07:32,080
With what?

104
00:07:32,080 --> 00:07:35,500
Using the same lambda
that you needed

105
00:07:35,500 --> 00:07:41,170
to change the potentials,
you do a phase rotation.

106
00:07:41,170 --> 00:07:43,030
And it's not a constant phase.

107
00:07:43,030 --> 00:07:45,460
This depends on x and t.

108
00:07:45,460 --> 00:07:47,680
So it's a substantial change.

109
00:07:47,680 --> 00:07:51,610
So you now have the technical
problem of first checking

110
00:07:51,610 --> 00:07:54,040
that this is true.

111
00:07:54,040 --> 00:07:57,580
This is the statement of gauge
invariance of the Schrodinger

112
00:07:57,580 --> 00:07:58,450
equation.

113
00:07:58,450 --> 00:08:01,780
There is a way to
transform the wave function

114
00:08:01,780 --> 00:08:05,320
so that the new
Schrodinger equation

115
00:08:05,320 --> 00:08:08,350
solution is obtained from
the old Schrodinger equation

116
00:08:08,350 --> 00:08:08,990
solution.

117
00:08:08,990 --> 00:08:12,580
And then we will have to check
that the physics is the same.

118
00:08:12,580 --> 00:08:15,100
If you wanted to
compute the expectation

119
00:08:15,100 --> 00:08:20,420
value of x, on
this wave function,

120
00:08:20,420 --> 00:08:21,980
this phase factor would cancel.

121
00:08:21,980 --> 00:08:24,320
So it would give you the same.

122
00:08:24,320 --> 00:08:26,550
If you want to compute some
other expectation values,

123
00:08:26,550 --> 00:08:27,840
it's a little funny.

124
00:08:27,840 --> 00:08:32,180
So there will be operators that
are nice for wave functions

125
00:08:32,180 --> 00:08:34,760
or [INAUDIBLE] gauge
invariant operators.

126
00:08:34,760 --> 00:08:39,630
And it will be a nice story
that we will develop next time.