1
00:00:01 --> 00:00:04
The following content is
provided by MIT OpenCourseWare

2
00:00:04 --> 00:00:06
under a Creative Commons
license.

3
00:00:06 --> 00:00:10
Additional information about
our license and MIT

4
00:00:10 --> 00:00:15
OpenCourseWare in general is
available at ocw.mit.edu.

5
00:00:15 --> 00:00:17
All right.
I am going to start with

6
00:00:17 --> 00:00:23
Friday's lecture notes because
there was a significant amount

7
00:00:23 --> 00:00:26
on them that I had not finished
up yet.

8
00:00:26 --> 00:00:31
We had finally gotten to the
point where we were talking

9
00:00:31 --> 00:00:36
about what does a wave function
mean, what is the physical

10
00:00:36 --> 00:00:40
significance of it and how does
it actually represent the

11
00:00:40 --> 00:00:47
presence of an electron?
And what we saw was that the

12
00:00:47 --> 00:00:52
physically significant
representation of the wave

13
00:00:52 --> 00:01:00
function, if you have some wave
function Psi labeled by three

14
00:01:00 --> 00:01:04
quantum numbers,
n, l and m.

15
00:01:04 --> 00:01:08
And, of course,
it is a function of r,

16
00:01:08 --> 00:01:12
theta and phi.
The physically significant

17
00:01:12 --> 00:01:16
quantity was this wave function
squared.

18
00:01:16 --> 00:01:22
That wave function squared,
that was interpreted as a

19
00:01:22 --> 00:01:28
probability density.
The wave function squared has

20
00:01:28 --> 00:01:32
units.
It has units of inverse volume.

21
00:01:32 --> 00:01:37
It is a density.
It is a probability per unit

22
00:01:37 --> 00:01:39
volume.
Now, as an aside,

23
00:01:39 --> 00:01:44
because someone asked me,
I should tell you that the more

24
00:01:44 --> 00:01:50
comprehensive definition of the
probability density is Psi,

25
00:01:50 --> 00:01:54
not squared,
but Psi times Psi star,

26
00:01:54 --> 00:02:00
where Psi star is the
complex conjugate.

27
00:02:00 --> 00:02:05
Because it turns out that some
wave functions are imaginary

28
00:02:05 --> 00:02:08
functions.
And so, if you took an

29
00:02:08 --> 00:02:14
imaginary function and squared
it, then you would still get an

30
00:02:14 --> 00:02:19
imaginary function after it.
And then it is hard to

31
00:02:19 --> 00:02:25
interpret an imaginary function
as a probability density.

32
00:02:25 --> 00:02:30
And so the more comprehensive
definition is Psi times Psi

33
00:02:30 --> 00:02:37
star, where Psi star is the
complex conjugate of Psi.

34
00:02:37 --> 00:02:42
And, when you multiply Psi by
Psi star, if Psi is a complex

35
00:02:42 --> 00:02:45
function, well,
then you get a real function.

36
00:02:45 --> 00:02:50
This is the more comprehensive
definition of the probability

37
00:02:50 --> 00:02:54
density, Psi times Psi star.
We won't use that.

38
00:02:54 --> 00:02:59
I just wanted to let you know
about it.

39
00:02:59 --> 00:03:03
So, probability density.
Not only do we want to know

40
00:03:03 --> 00:03:06
something about the probability
density.

41
00:03:06 --> 00:03:12
We also want to know something
about the probability of finding

42
00:03:12 --> 00:03:16
the electron some distance away
from the nucleus.

43
00:03:16 --> 00:03:20
And, to do that,
what we were talking about was

44
00:03:20 --> 00:03:23
this quantity,
this radial distribution,

45
00:03:23 --> 00:03:26
the radial probability
distribution.

46
00:03:26 --> 00:03:30
And what that is,
is the probability of finding

47
00:03:30 --> 00:03:35
an electron in a spherical shell
of radius r and distance or

48
00:03:35 --> 00:03:38
thickness dr.
For example,

49
00:03:38 --> 00:03:43
if this gray portion here
represented the probability

50
00:03:43 --> 00:03:47
density of the 1s wave function
in our dot density diagram.

51
00:03:47 --> 00:03:52
Remember, we squared the wave
function, got the probability

52
00:03:52 --> 00:03:57
density and then represented it
with a dot density diagram,

53
00:03:57 --> 00:04:02
where the density of the dots
was proportional to the value of

54
00:04:02 --> 00:04:08
the wave function squared.
And, in the case of the 1s wave

55
00:04:08 --> 00:04:15
function, we saw that the
probability density was largest

56
00:04:15 --> 00:04:20
right at r equals 0,
and that is exponentially

57
00:04:20 --> 00:04:24
decayed in all directions
uniformly.

58
00:04:24 --> 00:04:30
That is what that gray part
represents.

59
00:04:30 --> 00:04:35
But now, this blue,
here, is my spherical shell.

60
00:04:35 --> 00:04:40
It has a radius r,
and it has a thickness,

61
00:04:40 --> 00:04:44
here, dr.
And the radial probability

62
00:04:44 --> 00:04:49
distribution is asking,
what is the probability of

63
00:04:49 --> 00:04:54
finding the electron in this
spherical shell?

64
00:04:54 --> 00:05:01
And that spherical shell has a
thickness dr.

65
00:05:01 --> 00:05:05
Another way to ask that is the
probability of finding the

66
00:05:05 --> 00:05:10
electron between r and r was dr.
That is what we wanted to know,

67
00:05:10 --> 00:05:15
and that is what the radial
probability distribution tells

68
00:05:15 --> 00:05:18
us.
Now, how do you get a value out

69
00:05:18 --> 00:05:21
of that?
How do you actually calculate

70
00:05:21 --> 00:05:24
the radial probability?
Well, to do that,

71
00:05:24 --> 00:05:27
what we have to know is this
volume, here,

72
00:05:27 --> 00:05:33
of the spherical shell.
The volume of this spherical

73
00:05:33 --> 00:05:38
shell is just the surface area
of that spherical shell,

74
00:05:38 --> 00:05:43
4 pi r squared,
and the volume is times this

75
00:05:43 --> 00:05:48
thickness, this thickness dr.
It is a very thin shell.

76
00:05:48 --> 00:05:53
It is an infinitesimally thin
shell of thickness dr.

77
00:05:53 --> 00:05:59
Well, if we know that volume,
then what we can do is take our

78
00:05:59 --> 00:06:02
probability density,
Psi squared,

79
00:06:02 --> 00:06:08
which has units of probability
per unit volume.

80
00:06:08 --> 00:06:12
And we are multiplying it,
here, by our unit volume.

81
00:06:12 --> 00:06:17
The unit volumes cancel,
and we are left with a

82
00:06:17 --> 00:06:21
probability.
So, that is our probability of

83
00:06:21 --> 00:06:27
finding that electron in a shell
of radius r and a thickness dr.

84
00:06:27 --> 00:06:32
Let's look at the result of
calculating the radial

85
00:06:32 --> 00:06:38
probability distribution for the
1s wave function.

86
00:06:38 --> 00:06:42
What did I do?
I took Psi squared for the 1s

87
00:06:42 --> 00:06:48
wave function at some value of
r, then I multiplied it by 4 pi

88
00:06:48 --> 00:06:51
r squared dr,
and I did that for many

89
00:06:51 --> 00:06:56
different values of r and
plotted the result here.

90
00:06:56 --> 00:07:01
That is what that radial
probability is as a function of

91
00:07:01 --> 00:07:06
r.
Well, the first thing you see

92
00:07:06 --> 00:07:13
is that the most probable value
of r, or the value of r where

93
00:07:13 --> 00:07:19
the electron has the highest
probability of being is at this

94
00:07:19 --> 00:07:26
value, a nought.
The most probable value of r is

95
00:07:26 --> 00:07:33
this value, a nought.
a nought is what we call

96
00:07:33 --> 00:07:38
the Bohr radius.
And today, in a moment or so,

97
00:07:38 --> 00:07:43
I will tell you why it is
called the Bohr radius.

98
00:07:43 --> 00:07:48
It has a numerical value of
0.529 angstroms.

99
00:07:48 --> 00:07:54
And so it is most likely that
the electron is about a half an

100
00:07:54 --> 00:07:59
angstrom away from the nucleus,
making, then,

101
00:07:59 --> 00:08:05
the diameter of the hydrogen
atom, on the average,

102
00:08:05 --> 00:08:12
a little bit over one angstrom.
That is how we think about the

103
00:08:12 --> 00:08:16
size of a hydrogen atom,
is to take this most probable

104
00:08:16 --> 00:08:20
value of r and double it to get
the diameter.

105
00:08:20 --> 00:08:24
The most probable value of r,
or the most probable distance

106
00:08:24 --> 00:08:28
of the electron from the
nucleus, is half an angstrom

107
00:08:28 --> 00:08:33
away.
The most probable distance of

108
00:08:33 --> 00:08:39
the electron from the nucleus is
not r equals 0 because the

109
00:08:39 --> 00:08:44
radial probability here is zero
at r equals 0.

110
00:08:44 --> 00:08:51
That seems a little strange
because the other day we plotted

111
00:08:51 --> 00:08:58
the probability density for the
1s wave function.

112
00:08:58 --> 00:09:02
And, when we did that,
here is Psi(1,

113
00:09:02 --> 00:09:07
0, 0) squared versus r,
what we saw was that the

114
00:09:07 --> 00:09:14
probability density was some
maximum value at r equals 0 and

115
00:09:14 --> 00:09:19
that it exponentially decayed
with increasing r.

116
00:09:19 --> 00:09:25
And that is the case.
Probability density for the s

117
00:09:25 --> 00:09:31
wave functions is a maximum at r
equals 0.

118
00:09:31 --> 00:09:37
But the radial probability here
is actually zero at r equals 0.

119
00:09:37 --> 00:09:40
Why?
Look at how we defined that

120
00:09:40 --> 00:09:42
radial probability,
here.

121
00:09:42 --> 00:09:47
It is Psi squared times this
volume element.

122
00:09:47 --> 00:09:51
Our volume element is this
spherical shell.

123
00:09:51 --> 00:09:56
And, at r equals 0,
the spherical shell goes to a

124
00:09:56 --> 00:10:02
volume of zero.
So, our radial probability here

125
00:10:02 --> 00:10:07
is equal to zero at r equals 0.
That is really important,

126
00:10:07 --> 00:10:12
that you understand that this
radial probability here is

127
00:10:12 --> 00:10:18
always going to be zero at r
equals 0 for all of the wave

128
00:10:18 --> 00:10:22
functions that we are going to
look at.

129
00:10:22 --> 00:10:26
And we will talk about this a
little bit more,

130
00:10:26 --> 00:10:32
the fact that the electron is
about a half an angstrom away

131
00:10:32 --> 00:10:36
from the nucleus.
But before I do that,

132
00:10:36 --> 00:10:40
I also just want to point out
that in your textbook,

133
00:10:40 --> 00:10:43
and sometimes in the notes,
that sometimes that radial

134
00:10:43 --> 00:10:47
probability is actually written
as the following.

135
00:10:47 --> 00:10:50
It is written as the r squared,
the distance variable,

136
00:10:50 --> 00:10:53
times the radial part of the
wave function.

137
00:10:53 --> 00:10:56
That is the radial part
squared.

138
00:10:56 --> 00:11:01
We talked about the radial and
the angular part last time.

139
00:11:01 --> 00:11:06
And the radial part is labeled
only by two quantum numbers,

140
00:11:06 --> 00:11:09
n and l.
And so, for the 1s,

141
00:11:09 --> 00:11:12
that is n equals 1,
l equals 0.

142
00:11:12 --> 00:11:18
Where does this come from?
Well, let me just emphasize or

143
00:11:18 --> 00:11:23
explain where this comes from.
This radial probability

144
00:11:23 --> 00:11:28
distribution here,
we said for the s wave

145
00:11:28 --> 00:11:34
functions, was Psi squared.
You could take Psi squared,

146
00:11:34 --> 00:11:39
the probability density,
and multiply it by this unit

147
00:11:39 --> 00:11:44
volume or the volume of the
shell, 4 pi r squared dr.

148
00:11:44 --> 00:11:48
Let's write that out again,

149
00:11:48 --> 00:11:54
but write it out now so that we
write out Psi squared in terms

150
00:11:54 --> 00:11:59
of the radial part and the
angular part.

151
00:11:59 --> 00:12:05
Remember, we said last time,
for the hydrogen atom wave

152
00:12:05 --> 00:12:11
functions, that Psi is always a
product of a factor only an r,

153
00:12:11 --> 00:12:18
which was the radial part,
and a factor only in theta and

154
00:12:18 --> 00:12:21
phi, which are the angular
parts.

155
00:12:21 --> 00:12:28
Now, what you also have to
remember in looking at this is

156
00:12:28 --> 00:12:33
that the angular part for the 1s
wave functions,

157
00:12:33 --> 00:12:40
2s, 3s, all s wave functions,
was equal to 1 over 4 pi to the

158
00:12:40 --> 00:12:46
1/2.
If you square that,

159
00:12:46 --> 00:12:49
you are going to get 1 over 4
pi.

160
00:12:49 --> 00:12:55
Therefore, the 4pi's here are
going to cancel for the 1s wave

161
00:12:55 --> 00:12:58
functions.
And what you are going to have

162
00:12:58 --> 00:13:03
left is this r squared times
just the radial part dr.

163
00:13:03 --> 00:13:07
That is why the y-axis in your

164
00:13:07 --> 00:13:13
book is sometimes labeled this
way for the radial probability

165
00:13:13 --> 00:13:17
distribution.
But this is also important

166
00:13:17 --> 00:13:23
because if you were calculating
the radial distribution function

167
00:13:23 --> 00:13:29
for something other than an s
wave function.

168
00:13:29 --> 00:13:34
The way you would do it is to
take just the radial part of

169
00:13:34 --> 00:13:40
that wave function times r
squared, or just the radial part

170
00:13:40 --> 00:13:45
of that wave function and
evaluate it at that value of r

171
00:13:45 --> 00:13:48
times r squared dr.
You could not,

172
00:13:48 --> 00:13:54
for the other wave functions,
take psi squared times 4 pi r

173
00:13:54 --> 00:13:58
squared dr.
And that is because the angular

174
00:13:58 --> 00:14:02
part for the other wave
functions that are not

175
00:14:02 --> 00:14:10
spherically symmetric is not the
square root of 1 over 4 pi.

176
00:14:10 --> 00:14:14
This is a broader definition
for what the radial probability

177
00:14:14 --> 00:14:18
distribution function is.
It just works out,

178
00:14:18 --> 00:14:22
in the case for the s wave
functions, these 4pi's cancel.

179
00:14:22 --> 00:14:26
And so you can write the radial
probability for the s wave

180
00:14:26 --> 00:14:30
functions like that.
So, those are just some

181
00:14:30 --> 00:14:35
definitions.
I want to talk some more about

182
00:14:35 --> 00:14:40
this radial probability
distribution function,

183
00:14:40 --> 00:14:47
here, for the 1s wave function.
I want to talk about it and

184
00:14:47 --> 00:14:54
also explain why a nought
is called the Bohr radius.

185
00:14:54 --> 00:15:00
The reason for that is the
following.

186
00:15:00 --> 00:15:04
The nucleus was discovered in
1911, the electron was known

187
00:15:04 --> 00:15:09
before that, and Schrödinger did
not write down his wave equation

188
00:15:09 --> 00:15:12
until 1926.
And, in between that,

189
00:15:12 --> 00:15:15
1911 to 1926,
the scientific community was

190
00:15:15 --> 00:15:20
really working very hard to try
to understand the structure of

191
00:15:20 --> 00:15:23
the atom.
And we saw how the classical

192
00:15:23 --> 00:15:27
ideas, as predicted,
would live a whopping 10^-10

193
00:15:27 --> 00:15:32
seconds.
And one of the people who were

194
00:15:32 --> 00:15:35
working on that problem was
Niels Bohr.

195
00:15:35 --> 00:15:40
And, in 1919,
Niels Bohr of course realized

196
00:15:40 --> 00:15:46
that classical physics fails
this kind of planetary model for

197
00:15:46 --> 00:15:52
the atom where you put the
nucleus in the center and the

198
00:15:52 --> 00:15:58
electron is going around that
nucleus with some fixed orbit.

199
00:15:58 --> 00:16:03
We will call it r.
Well, he knew that it was not

200
00:16:03 --> 00:16:06
going to work,
that those classical ideas

201
00:16:06 --> 00:16:11
predicted that this would
plummet into the nucleus in

202
00:16:11 --> 00:16:13
10^-10 seconds.
But, he said,

203
00:16:13 --> 00:16:18
obviously, that does not
happen, so let me just forget

204
00:16:18 --> 00:16:23
classical physics at the moment.
Then, what he did was to impose

205
00:16:23 --> 00:16:28
some quantization on this
classical model for the hydrogen

206
00:16:28 --> 00:16:32
atom.
And the reason he got this idea

207
00:16:32 --> 00:16:36
of quantization is because he
already knew the hydrogen atom

208
00:16:36 --> 00:16:39
emission spectrum.
He knew that in the hydrogen

209
00:16:39 --> 00:16:43
atom emission spectrum that
light of only certain

210
00:16:43 --> 00:16:47
frequencies was emitted.
That is, there was some idea

211
00:16:47 --> 00:16:51
that there was something about
this hydrogen atom that is

212
00:16:51 --> 00:16:52
quantized.
He said, well,

213
00:16:52 --> 00:16:56
let me just ignore classical
physics for a moment.

214
00:16:56 --> 00:17:00
Let me give this a circular
orbit.

215
00:17:00 --> 00:17:05
But let me quantize something
about this hydrogen atom.

216
00:17:05 --> 00:17:09
And, in particular,
what he went and did was

217
00:17:09 --> 00:17:14
quantized the angular momentum
of that electron.

218
00:17:14 --> 00:17:19
He kind of just pasted the
quantization onto a classical

219
00:17:19 --> 00:17:24
model for the atom,
because he is trying to work

220
00:17:24 --> 00:17:30
toward explaining what the
observations were.

221
00:17:30 --> 00:17:35
When he pasted that
quantization onto this classic

222
00:17:35 --> 00:17:40
model, he was able to calculate
a value of r.

223
00:17:40 --> 00:17:46
And that value of r is what we
call the Bohr radius,

224
00:17:46 --> 00:17:51
a nought, and has the value
0.529 angstroms.

225
00:17:51 --> 00:17:59
That came out of it.
And if you calculate for the

226
00:17:59 --> 00:18:06
radial probability distribution
function for this model,

227
00:18:06 --> 00:18:15
which is called the Bohr atom,
would be one where that radial

228
00:18:15 --> 00:18:23
probability is 1 right here at r
equals a nought.

229
00:18:23 --> 00:18:28
In Bohr's model,
the electron had a

230
00:18:28 --> 00:18:34
well-defined,
precise orbit.

231
00:18:34 --> 00:18:39
The value of r at which it went
around the nucleus was given by

232
00:18:39 --> 00:18:43
a nought.
He knew exactly where the

233
00:18:43 --> 00:18:47
electron was in his model.
This kind of model,

234
00:18:47 --> 00:18:51
which is this classical model,
really, is what we call

235
00:18:51 --> 00:18:55
deterministic.
It is deterministic because we

236
00:18:55 --> 00:19:00
know exactly where the particle,
in this case the electron,

237
00:19:00 --> 00:19:04
is.
I want you to contrast it with

238
00:19:04 --> 00:19:09
the quantum mechanical result
from the Schrödinger equation.

239
00:19:09 --> 00:19:13
What you see,
in the quantum mechanical

240
00:19:13 --> 00:19:18
result, is that we don't really
know where the electron is,

241
00:19:18 --> 00:19:21
so to speak.
The best we can tell you is a

242
00:19:21 --> 00:19:26
probability of finding the
electron at some value r to r

243
00:19:26 --> 00:19:32
plus dr.
That is the best we can do

244
00:19:32 --> 00:19:36
because quantum mechanics is
non-deterministic.

245
00:19:36 --> 00:19:40
There is a limit to which we
can know the position of a

246
00:19:40 --> 00:19:43
particle.
That limit is given by

247
00:19:43 --> 00:19:47
something called the uncertainty
principle.

248
00:19:47 --> 00:19:52
The uncertainty principle is
not something we are going to

249
00:19:52 --> 00:19:57
discuss, but it tells us that
there is a limit to which we can

250
00:19:57 --> 00:20:03
know both the position and the
momentum of a particle.

251
00:20:03 --> 00:20:09
And that is the basis for why,
here, we have a probability

252
00:20:09 --> 00:20:15
distribution and knowing sort of
where the electron is.

253
00:20:15 --> 00:20:20
We don't exactly know where the
electron is, here.

254
00:20:20 --> 00:20:27
This is the classical model on
which Bohr just kind of pasted

255
00:20:27 --> 00:20:33
the quantization of the angular
momentum of the electron onto

256
00:20:33 --> 00:20:37
it.
In the case of the Schrödinger

257
00:20:37 --> 00:20:41
equation, the quantization drops
out when you solve the

258
00:20:41 --> 00:20:44
differential equation.
It comes out of the equation

259
00:20:44 --> 00:20:47
just naturally.
We did not paste it onto it.

260
00:20:47 --> 00:20:51
We did not make an ad hoc kind
of representation.

261
00:20:51 --> 00:20:55
That is the big difference here
between quantum mechanics and

262
00:20:55 --> 00:21:00
classical mechanics.
In quantum mechanics,

263
00:21:00 --> 00:21:04
it can only tell you about a
probability.

264
00:21:04 --> 00:21:09
It cannot tell you exactly
where the particle is going to

265
00:21:09 --> 00:21:11
be.
Questions on that?

266
00:21:11 --> 00:21:15
Okay.
Anyway, this value a nought,

267
00:21:15 --> 00:21:20
that is why it is called
the Bohr radius.

268
00:21:20 --> 00:21:24
And then it turns out,
quantum mechanically,

269
00:21:24 --> 00:21:29
that this value of r,
the most probable value of r

270
00:21:29 --> 00:21:35
is, in fact, exactly a nought.

271
00:21:35 --> 00:21:37
In a sense, Bohr was pretty
lucky.

272
00:21:37 --> 00:21:42
And this is kind of an accident
that he got a nought out

273
00:21:42 --> 00:21:46
of this, and it has to do with
the actual form of the Coulomb

274
00:21:46 --> 00:21:49
interaction.
But, of course,

275
00:21:49 --> 00:21:53
this doesn't work for anything
else, other than a hydrogen

276
00:21:53 --> 00:21:55
atom.
Whereas, the Schrödinger

277
00:21:55 --> 00:22:00
equation, as we are going to see
in a moment, is applicable to

278
00:22:00 --> 00:22:04
all the atoms that we know
about.

279
00:22:04 --> 00:22:08
So that is the radial
probability distribution

280
00:22:08 --> 00:22:11
function for the 1s atom,
for the 1s state.

281
00:22:11 --> 00:22:17
We want to take a look at the
radial probability distribution

282
00:22:17 --> 00:22:20
for 2s and for 3s.
Let me plot those.

283
00:22:20 --> 00:22:24
And you can actually put these
lights on here.

284
00:22:24 --> 00:22:28
That is okay.
I am going to use this board

285
00:22:28 --> 00:22:33
for a moment.
Here is the radial probability

286
00:22:33 --> 00:22:38
distribution function.
I can write it as little r

287
00:22:38 --> 00:22:42
times R(2,0) squared of r,

288
00:22:42 --> 00:22:45
or RPD.
This is for 2s versus r.

289
00:22:45 --> 00:22:50
And when I do that I get a
function that looks like this.

290
00:22:50 --> 00:22:55
And, if I evaluate it here,
what is this value of r at

291
00:22:55 --> 00:23:00
which the probability is a
maximum?

292
00:23:00 --> 00:23:04
Well, this most probable value
of r is 6 a nought.

293
00:23:04 --> 00:23:06
Look at that.

294
00:23:06 --> 00:23:10
The most probable value of r
for 1s was a nought.

295
00:23:10 --> 00:23:13
In the case of the 2s state

296
00:23:13 --> 00:23:17
here, the electron,
the most probable value is 6 a

297
00:23:17 --> 00:23:20
nought, six times as far from
the nucleus.

298
00:23:20 --> 00:23:24
If you have a hydrogen atom in
the first excited state,

299
00:23:24 --> 00:23:29
in a sense that hydrogen atom
is bigger.

300
00:23:29 --> 00:23:34
It is bigger in the sense that
the probability of you finding

301
00:23:34 --> 00:23:39
the electron at a larger
distance away from the nucleus

302
00:23:39 --> 00:23:42
is larger.
And that, in general,

303
00:23:42 --> 00:23:45
is the case.
The radial probability

304
00:23:45 --> 00:23:48
distribution,
here, also reflects the radial

305
00:23:48 --> 00:23:52
node that we talked about last
time.

306
00:23:52 --> 00:23:55
That radial node is r equals a
nought.

307
00:23:55 --> 00:24:00
Radial node is the value of r
that makes your wave function go

308
00:24:00 --> 00:24:04
to zero.
Notice, again,

309
00:24:04 --> 00:24:10
that this radial probability
distribution function right here

310
00:24:10 --> 00:24:14
is zero at r equals 0.
This is not a node.

311
00:24:14 --> 00:24:19
This is not a radial node.
This is a consequence,

312
00:24:19 --> 00:24:25
right here, of our definition
for the radial probability.

313
00:24:25 --> 00:24:30
Our volume element has gone to
zero.

314
00:24:30 --> 00:24:36
r equals 0 is never a radial
node in any wave function.

315
00:24:36 --> 00:24:40
What about 3s?
Well, let's plot 3s.

316
00:24:40 --> 00:24:44
Here is 3s.
This is the radial probability

317
00:24:44 --> 00:24:49
distribution.
I take Psi for 3s and square

318
00:24:49 --> 00:24:55
it, multiply by 4 pi r squared
dr,

319
00:24:55 --> 00:25:02
and do so for all the values of
r, and I am going to get

320
00:25:02 --> 00:25:10
something that looks like this.
Now this most probable value of

321
00:25:10 --> 00:25:16
r here, where the 3s wave
function is equal to 11.468 a

322
00:25:16 --> 00:25:22
nought.
For the second excited state of

323
00:25:22 --> 00:25:25
a hydrogen atom,
that electron,

324
00:25:25 --> 00:25:30
on the average,
is 11.5 times farther out from

325
00:25:30 --> 00:25:38
the nucleus than it is in the
case of the 1s state right here.

326
00:25:38 --> 00:25:43
Again, for that second excited
state, that hydrogen atom is

327
00:25:43 --> 00:25:49
bigger in the sense that the
probability of it being farther

328
00:25:49 --> 00:25:54
away from the nucleus is larger.
That radial probability

329
00:25:54 --> 00:26:00
distribution of the 3s also
reflects the two radial nodes in

330
00:26:00 --> 00:26:07
the 3s wave function.
The radial nodes are at 1.9 a

331
00:26:07 --> 00:26:11
nought, here,
and 7.1 a nought.

332
00:26:11 --> 00:26:19
Again, the value here at r
equals 0 is not a radial node.

333
00:26:19 --> 00:26:27
Now, as you look at this,
it is tempting to ask the

334
00:26:27 --> 00:26:33
following question.
You might want to ask,

335
00:26:33 --> 00:26:40
if the electron can be at these
values of r, and it can be at

336
00:26:40 --> 00:26:46
these values of r,
and it can be at these values

337
00:26:46 --> 00:26:53
of r, how does the electron
actually get from here to here

338
00:26:53 --> 00:27:00
to here if right at r equals 1.9
a nought and 7.1 a nought the

339
00:27:00 --> 00:27:08
probability is equal to zero?
Well, you might say maybe this

340
00:27:08 --> 00:27:14
probability isn't exactly zero.
It is something small.

341
00:27:14 --> 00:27:20
But I am telling you that it is
zero, goose egg,

342
00:27:20 --> 00:27:23
zilch, zippo,
nada, cipher,

343
00:27:23 --> 00:27:28
nix, nought.
Anybody else have another name?

344
00:27:28 --> 00:27:31
Nil.
It is nothing.

345
00:27:31 --> 00:27:34
It is zero.
How do you answer that

346
00:27:34 --> 00:27:37
question?
Well, it turns out,

347
00:27:37 --> 00:27:42
of course, that it isn't an
appropriate question.

348
00:27:42 --> 00:27:48
And the reason it is not is
because that question is asked

349
00:27:48 --> 00:27:52
in the framework of classical
mechanics.

350
00:27:52 --> 00:27:56
When you ask,
how does a particle get from

351
00:27:56 --> 00:28:01
one place to another,
you are asking about a

352
00:28:01 --> 00:28:06
trajectory.
You are asking about a path.

353
00:28:06 --> 00:28:10
Particles over here,
over here, over here,

354
00:28:10 --> 00:28:14
how does it get from one place
to another?

355
00:28:14 --> 00:28:18
And, in quantum mechanics,
we don't have the concept of

356
00:28:18 --> 00:28:22
trajectories.
Instead, what we have to think

357
00:28:22 --> 00:28:28
of is the electron as a wave.
And we already know that a wave

358
00:28:28 --> 00:28:32
can have amplitude
simultaneously at many different

359
00:28:32 --> 00:28:37
positions.
And so it has simultaneous

360
00:28:37 --> 00:28:42
amplitude or probability here,
here, and here,

361
00:28:42 --> 00:28:46
all at the same time.
We cannot talk about

362
00:28:46 --> 00:28:50
trajectories anymore.
And that, again,

363
00:28:50 --> 00:28:56
ties into the uncertainty
principle, our inability to know

364
00:28:56 --> 00:29:02
exactly the position and the
momentum of a particle at any

365
00:29:02 --> 00:29:07
given instance.
The best we can tell you is a

366
00:29:07 --> 00:29:11
probability.
We have to change the way we

367
00:29:11 --> 00:29:16
think about electrons.
You cannot cast them in the

368
00:29:16 --> 00:29:19
framework of your everyday
world.

369
00:29:19 --> 00:29:23
This is part of our world,
but you have to go do a

370
00:29:23 --> 00:29:30
specific type of experiment to
see this part of the world.

371
00:29:30 --> 00:29:35
That is why it seems so strange
to you, because it is not part

372
00:29:35 --> 00:29:40
of your everyday experience.
But this world works with

373
00:29:40 --> 00:29:45
different rules that you really
do have to accept that it just

374
00:29:45 --> 00:29:49
works differently.
Questions?

375
00:29:49 --> 00:30:00


376
00:30:00 --> 00:30:06
Now, I am going to stop talking
about the s wave functions and

377
00:30:06 --> 00:30:11
move on to talk about the p wave
functions.

378
00:30:11 --> 00:30:16
With the s wave functions,
we talked about the

379
00:30:16 --> 00:30:22
significance of the wave
function, probability density,

380
00:30:22 --> 00:30:26
radial probability
distribution.

381
00:30:26 --> 00:30:32
We talked about what a radial
node was.

382
00:30:32 --> 00:30:36
Now it is time to move onto the
p wave functions.

383
00:30:36 --> 00:30:41
And the p wave functions,
of course, are not spherically

384
00:30:41 --> 00:30:44
symmetric.
And to represent them,

385
00:30:44 --> 00:30:48
we are going to do our dot
density diagram again.

386
00:30:48 --> 00:30:54
We are going to take the wave
function and square it to get

387
00:30:54 --> 00:31:00
the probability density and then
plot that probability density as

388
00:31:00 --> 00:31:05
a density of dots.
We the dots are most dense,

389
00:31:05 --> 00:31:10
well, that means the highest
probability density.

390
00:31:10 --> 00:31:14
Here is the result for the pz
wave function.

391
00:31:14 --> 00:31:19
It is pz because you can see
the highest probability,

392
00:31:19 --> 00:31:23
here, is along the z-axis.
It is symmetric along the

393
00:31:23 --> 00:31:27
z-axis.
Here is the probability density

394
00:31:27 --> 00:31:32
for the px wave function.
You can see that the

395
00:31:32 --> 00:31:36
probability density is greatest
along the x-axis.

396
00:31:36 --> 00:31:39
It is symmetric along the
x-axis.

397
00:31:39 --> 00:31:43
And, if you look really
carefully, you can see that

398
00:31:43 --> 00:31:48
there is no probability density
in the y,z-plane for the px wave

399
00:31:48 --> 00:31:49
function.
And, over here,

400
00:31:49 --> 00:31:53
if you look carefully,
you can see that there is no

401
00:31:53 --> 00:31:58
probability density in the
x,y-plane for the pz wave

402
00:31:58 --> 00:32:03
function.
And here is a py wave function,

403
00:32:03 --> 00:32:08
the probability density of it.
The probability density is

404
00:32:08 --> 00:32:13
concentrated along the y-axis.
It is symmetric along the

405
00:32:13 --> 00:32:16
y-axis.
And, if you look very

406
00:32:16 --> 00:32:20
carefully, there is no
probability density,

407
00:32:20 --> 00:32:25
here, in the x,z-plane.
Well, the fact that there is no

408
00:32:25 --> 00:32:30
probability density,
here, in the x,y-plane,

409
00:32:30 --> 00:32:34
in the case of pz,
indicates that we have an

410
00:32:34 --> 00:32:39
angular node.
An angular node at theta equal

411
00:32:39 --> 00:32:42
90 degrees.
An angular node is the same

412
00:32:42 --> 00:32:46
thing as a radial node in the
sense that it is the value of

413
00:32:46 --> 00:32:51
the angle that makes the wave
function be equal to zero.

414
00:32:51 --> 00:32:53
Here is the wave function for
pz.

415
00:32:53 --> 00:32:57
You can see that when theta is
equal to zero,

416
00:32:57 --> 00:33:02
this wave function is going to
be equal to zero.

417
00:33:02 --> 00:33:08
An angular node is the value of
theta or phi that makes the wave

418
00:33:08 --> 00:33:12
function be zero.
And the consequence,

419
00:33:12 --> 00:33:19
then, is that we have a nodal
plane, because everywhere on the

420
00:33:19 --> 00:33:23
x,y-plane, theta is equal to 90
degrees.

421
00:33:23 --> 00:33:29
For the px wave function,
the value of the angle that

422
00:33:29 --> 00:33:35
gives you that nodal plane is
phi equals 90.

423
00:33:35 --> 00:33:41
That means everywhere in the
y,z-plane is phi equal to 90.

424
00:33:41 --> 00:33:46
In the case of py,
when phi is equal to zero,

425
00:33:46 --> 00:33:50
well, that is everywhere in the
x,z-plane.

426
00:33:50 --> 00:33:55
Everywhere in the x,z-plane,
phi is equal to zero.

427
00:33:55 --> 00:34:02
So, that is the angular nodes.
In general, and this is

428
00:34:02 --> 00:34:07
something you do have to know,
an orbital has n minus 1

429
00:34:07 --> 00:34:12
total nodes.
And what I mean by total nodes

430
00:34:12 --> 00:34:17
is angular plus radial nodes.
The number of angular nodes is

431
00:34:17 --> 00:34:20
given by this quantity,
l.

432
00:34:20 --> 00:34:25
The quantum number l that
labels your wave function always

433
00:34:25 --> 00:34:30
gives you the number of angular
nodes.

434
00:34:30 --> 00:34:34
Therefore, if n minus 1 is the
total and l is the number of

435
00:34:34 --> 00:34:38
angular, well then,
the number of radial nodes is n

436
00:34:38 --> 00:34:42
minus 1 minus l. This is

437
00:34:42 --> 00:34:44
something that you do have to
know.

438
00:34:44 --> 00:34:49
If I give you a wave function
and ask you how many radial and

439
00:34:49 --> 00:34:52
angular nodes it has,
you need to be able to

440
00:34:52 --> 00:34:56
calculate that,
and vice versa.

441
00:34:56 --> 00:35:02
Sometimes I will tell you a
function has three radial nodes

442
00:35:02 --> 00:35:07
and six or seven angular nodes
or something,

443
00:35:07 --> 00:35:12
what is the wave function?
So, we go both ways.

444
00:35:12 --> 00:35:19
Well, I also want to take a
look at the radial probability

445
00:35:19 --> 00:35:24
distribution functions for the p
wave functions.

446
00:35:24 --> 00:35:31
We looked at it for the s wave
functions already.

447
00:35:31 --> 00:35:36
I actually want to contrast the
radial probability distribution,

448
00:35:36 --> 00:35:40
say, for 2p,
here it is, with that of 2s

449
00:35:40 --> 00:35:45
that we looked at a moment ago.
Remember, how do you get the

450
00:35:45 --> 00:35:50
radial probability distribution
function here for 2p?

451
00:35:50 --> 00:35:55
It is the radial part of the 2p
wave function times r squared

452
00:35:55 --> 00:35:58
dr.
It gives me the probability of

453
00:35:58 --> 00:36:05
finding the electron a distance
between r and r plus dr.

454
00:36:05 --> 00:36:10
Again, what you see is that at
r equals 0, that is zero.

455
00:36:10 --> 00:36:15
That is not a radial node.
But what I really want to point

456
00:36:15 --> 00:36:20
out here is that the most
probable value of r,

457
00:36:20 --> 00:36:25
for the 2p wave function,
is actually smaller than it is

458
00:36:25 --> 00:36:31
for the 2s wave function.
That is, it is more likely for

459
00:36:31 --> 00:36:37
the electron in a 2p state to be
a little closer in to the

460
00:36:37 --> 00:36:40
nucleus than it is for the 2s
state.

461
00:36:40 --> 00:36:46
In general, as you increase the
angular momentum quantum number,

462
00:36:46 --> 00:36:51
the most probable value of r
gets smaller for the same value

463
00:36:51 --> 00:36:54
of n.
Similarly, here is the 3s

464
00:36:54 --> 00:37:01
radial probability distribution
function that we looked at.

465
00:37:01 --> 00:37:05
Here is a radial probability
distribution for 3p.

466
00:37:05 --> 00:37:08
Now, with the 3p,
you can see the value of the

467
00:37:08 --> 00:37:11
radial node.
You can see the radial

468
00:37:11 --> 00:37:15
probability distribution
reflects a radial node,

469
00:37:15 --> 00:37:17
here.
And here is the radial

470
00:37:17 --> 00:37:21
probability distribution
function for 3d.

471
00:37:21 --> 00:37:25
We did not look at the
probability density of 3d.

472
00:37:25 --> 00:37:30
You will do that with Professor
Cummins when you talk about

473
00:37:30 --> 00:37:35
transition metals.
But here, I just drew in the

474
00:37:35 --> 00:37:39
radial probability distribution
for 3d.

475
00:37:39 --> 00:37:44
But the point again that I want
to make is here is the most

476
00:37:44 --> 00:37:48
probable value of r for 3s,
here it is for 3p,

477
00:37:48 --> 00:37:53
here it is for 3d,
again, the most probable value

478
00:37:53 --> 00:37:59
for 3d is smaller than it is for
3p, than it is for 3s.

479
00:37:59 --> 00:38:05
Again, as you increase the
angular momentum quantum number,

480
00:38:05 --> 00:38:09
that most probable value gets
smaller.

481
00:38:09 --> 00:38:14
However, ironically,
if you actually look at the

482
00:38:14 --> 00:38:20
probability of the electron
being very, very close to the

483
00:38:20 --> 00:38:26
nucleus, that probability is
only significant for the s wave

484
00:38:26 --> 00:38:31
functions.
Look at the 3s wave function.

485
00:38:31 --> 00:38:36
Here, you see that you really
do have some probability very

486
00:38:36 --> 00:38:40
close to the nucleus.
You don't see that in the 3p

487
00:38:40 --> 00:38:43
wave function.
You certainly don't see that in

488
00:38:43 --> 00:38:48
the 3d wave function.
Again, in the 2s wave function,

489
00:38:48 --> 00:38:52
you have some significant
probability of the electron

490
00:38:52 --> 00:38:55
being really close to the
nucleus in 2s,

491
00:38:55 --> 00:39:00
but you don't in 2p.
That is important.

492
00:39:00 --> 00:39:05
And it seems in contradiction
to the fact that on the average,

493
00:39:05 --> 00:39:10
the most probable value of r
gets smaller as l gets larger.

494
00:39:10 --> 00:39:14
These two facts that look
contradictory are important.

495
00:39:14 --> 00:39:17
They dictate the behavior of
atoms.

496
00:39:17 --> 00:39:22
These two facts seem like kind
of loose threads at the moment

497
00:39:22 --> 00:39:27
in the sense that you are
probably wondering why I am

498
00:39:27 --> 00:39:31
telling you what I am telling
you.

499
00:39:31 --> 00:39:36
But we are going to use that
information in a few days,

500
00:39:36 --> 00:39:41
and you will see really the
significance of this plot.

501
00:39:41 --> 00:39:47
And this plot will be an
important one for you to refer

502
00:39:47 --> 00:39:49
back to.
Yes?

503
00:39:49 --> 00:40:07


504
00:40:07 --> 00:40:09
Probably.
I am not exactly sure of the

505
00:40:09 --> 00:40:12
picture you drew in high school,
but yes.

506
00:40:12 --> 00:40:17
If the electron in general is
further out from the nucleus,

507
00:40:17 --> 00:40:21
that is a higher energy state.
The electron is less strongly

508
00:40:21 --> 00:40:27
bound, as we are going to see in
the multi-electron atoms here.

509
00:40:27 --> 00:40:35


510
00:40:35 --> 00:40:38
Oh, no.
For the hydrogen no.

511
00:40:38 --> 00:40:42
Let me explain that.
For the hydrogen atom,

512
00:40:42 --> 00:40:48
the energies are only dictated
by the n quantum number,

513
00:40:48 --> 00:40:53
so 3s, 3p, 3d all have the same
energies.

514
00:40:53 --> 00:40:58
Where the energies become
degenerate is with a

515
00:40:58 --> 00:41:05
multi-electron atom.
And we are going to talk about

516
00:41:05 --> 00:41:11
that and how that reflects here,
these wave functions in the

517
00:41:11 --> 00:41:16
next day.
That is all I am going to say

518
00:41:16 --> 00:41:21
about the hydrogen atom.
Now it is time to move on,

519
00:41:21 --> 00:41:24
to helium.
And, of course,

520
00:41:24 --> 00:41:30
the Schrödinger equation
predicts the binding energies of

521
00:41:30 --> 00:41:39
the electrons to the nucleus in
a helium atom also very well.

522
00:41:39 --> 00:41:42
But, of course,
it is a much more complicated

523
00:41:42 --> 00:41:46
Schrödinger equation.
And I am not even going to

524
00:41:46 --> 00:41:50
write out the Hamiltonian in
this case, but I want to show

525
00:41:50 --> 00:41:54
you the wave function here.
See the wave function?

526
00:41:54 --> 00:41:59
The wave function is a function
of six variables.

527
00:41:59 --> 00:42:04
It is a function of two r's,
two distances from the nucleus,

528
00:42:04 --> 00:42:07
one for electron one,
one for electron two,

529
00:42:07 --> 00:42:12
two theta's and two phi's.
We have six variables for the

530
00:42:12 --> 00:42:16
wave function.
And the consequence of this is

531
00:42:16 --> 00:42:20
that our solutions for the
binding energies for the

532
00:42:20 --> 00:42:26
electrons in helium or any other
atoms are not going to be nice

533
00:42:26 --> 00:42:31
analytical forms.
We are no longer going to have

534
00:42:31 --> 00:42:35
e sub n equal minus the Rydberg
constant over n squared.

535
00:42:35 --> 00:42:39
If you
actually solve for those

536
00:42:39 --> 00:42:43
energies, and you have to do it
numerically, you are just going

537
00:42:43 --> 00:42:46
to get a list of numbers,
a table of numbers,

538
00:42:46 --> 00:42:50
but not a nice analytical form.
If you solve for the wave

539
00:42:50 --> 00:42:55
function, you are not going to
get a nice analytical form,

540
00:42:55 --> 00:42:59
like we got for hydrogen.
Instead, what you will get is a

541
00:42:59 --> 00:43:02
value for the amplitude of Psi
as a function of r,

542
00:43:02 --> 00:43:08
theta and phi.
But if you get actually much

543
00:43:08 --> 00:43:12
above three electrons,
it turns out that even

544
00:43:12 --> 00:43:18
numerically, you cannot solve
the Schrödinger equation,

545
00:43:18 --> 00:43:23
exactly.
You have to use approximations.

546
00:43:23 --> 00:43:30
And we are going to look at the
most basic approximation that is

547
00:43:30 --> 00:43:34
used that works,
amazingly.

548
00:43:34 --> 00:43:39
It works well enough for us to
have a framework in which to

549
00:43:39 --> 00:43:42
understand the reactions of
these atoms.

550
00:43:42 --> 00:43:47
And what is that approximation?
Well, that approximation is

551
00:43:47 --> 00:43:52
called the one-electron wave
approximation or the

552
00:43:52 --> 00:43:55
one-electron orbital
approximation.

553
00:43:55 --> 00:44:00
What does that mean?
Well, that means this.

554
00:44:00 --> 00:44:05
I am going to take my wave
function here for the helium

555
00:44:05 --> 00:44:10
atom, which strictly is a wave
function that is a function of

556
00:44:10 --> 00:44:15
six variables,
and I am going to separate it.

557
00:44:15 --> 00:44:20
I am going to let electron one
have its own wave function and

558
00:44:20 --> 00:44:24
electron two have its own wave
function.

559
00:44:24 --> 00:44:28
That is an approximation.
In addition,

560
00:44:28 --> 00:44:33
what I am going to do is let
the wave function for electron

561
00:44:33 --> 00:44:39
one have a hydrogen-like wave
function.

562
00:44:39 --> 00:44:43
I am going to say that it has
the 1s wave function,

563
00:44:43 --> 00:44:46
or the Psi(1,
0, 0) wave function of a

564
00:44:46 --> 00:44:49
hydrogen atom.
And I am going to let electron

565
00:44:49 --> 00:44:53
two have the Psi(1,
0, 0) wave function of a

566
00:44:53 --> 00:44:56
hydrogen atom.
Or, I am going to write it as

567
00:44:56 --> 00:45:00
1s of 1, for electron one,
times 1s of 2,

568
00:45:00 --> 00:45:04
for electron two.

569
00:45:04 --> 00:45:07
Or, another shorthand,
I am going to write it as 1s.

570
00:45:07 --> 00:45:09
squared.
And, if I continued on,

571
00:45:09 --> 00:45:13
here, it is for lithium.
Lithium, the wave function

572
00:45:13 --> 00:45:17
strictly has nine coordinates,
but I am going to let every one

573
00:45:17 --> 00:45:19
of those electrons,
in the one electron wave

574
00:45:19 --> 00:45:22
approximation,
have its own wave function.

575
00:45:22 --> 00:45:26
And I am going to let electron
one have a wave function that

576
00:45:26 --> 00:45:30
looks like a hydrogen atom
wavefunction.

577
00:45:30 --> 00:45:33
The 1s wave function.
The same thing with electron

578
00:45:33 --> 00:45:35
two.
And then I am going to let

579
00:45:35 --> 00:45:40
electron three have the 2s wave
function of the hydrogen atom.

580
00:45:40 --> 00:45:44
And in simplified notation,
that is just 1s squared 2s.

581
00:45:44 --> 00:45:47
And here is
beryllium, 16 variables,

582
00:45:47 --> 00:45:51
but I am going to let every
electron have its own wave

583
00:45:51 --> 00:45:54
function.
And I am going to give electron

584
00:45:54 --> 00:45:56
one the 1s wave function,
electron two,

585
00:45:56 --> 00:46:00
the 1s, electron three,
the 2s, electron four,

586
00:46:00 --> 00:46:03
the 2s.
I can also write that,

587
00:46:03 --> 00:46:07
as you have already done,
1s 2 2s 2.

588
00:46:07 --> 00:46:10
And I can keep going.
And these electron

589
00:46:10 --> 00:46:14
configurations that you have
been writing down in high

590
00:46:14 --> 00:46:18
school, that is what they are,
electron configurations,

591
00:46:18 --> 00:46:23
well, they are nothing more
than our shorthand notation for

592
00:46:23 --> 00:46:27
the electron wave functions
within this one-electron wave

593
00:46:27 --> 00:46:32
approximation.
That is what those were,

594
00:46:32 --> 00:46:37
that you were writing down.
Those were a shorthand notation

595
00:46:37 --> 00:46:42
for the wave functions in
Schrödinger's equation within

596
00:46:42 --> 00:46:46
this one-electron wave
approximation.

597
00:46:46 --> 00:46:50
Now, one thing you do notice is
that I did not,

598
00:46:50 --> 00:46:55
in the case of boron here,
let all five electrons be in

599
00:46:55 --> 00:46:59
the 1s state,
or let all five electrons be

600
00:46:59 --> 00:47:05
represented by a 1s hydrogen
atom wave function.

601
00:47:05 --> 00:47:10
I didn't because of a quantity
that you already know about,

602
00:47:10 --> 00:47:14
called spin.
You already know that if you

603
00:47:14 --> 00:47:19
are going to put electrons in
the 1s state here that one

604
00:47:19 --> 00:47:25
electron has to go in with spin
up and the other spin down.

605
00:47:25 --> 00:47:30
And the 2s, spin up and spin
down, etc.

606
00:47:30 --> 00:47:34
What is the phenomenon called
spin?

607
00:47:34 --> 00:47:42
Well, spin is entirely a
quantum mechanical phenomenon.

608
00:47:42 --> 00:47:48
There is no correct classical
analogy to spin.

609
00:47:48 --> 00:47:53
Spin is intrinsic angular
momentum.

610
00:47:53 --> 00:48:00
It is angular momentum that is
just part of a particle,

611
00:48:00 --> 00:48:07
such as an electron.
The spin quantum numbers

612
00:48:07 --> 00:48:11
actually come from solving the
relativistic Schrödinger

613
00:48:11 --> 00:48:15
equation, which we did not even
write down.

614
00:48:15 --> 00:48:20
When you solve the relativistic
Schrödinger equation,

615
00:48:20 --> 00:48:23
out drops a fourth quantum
number.

616
00:48:23 --> 00:48:28
That fourth quantum number we
are going to call m sub s.

617
00:48:28 --> 00:48:31
And we find that m sub s

618
00:48:31 --> 00:48:36
has two allowed values.
One of those values is one-half

619
00:48:36 --> 00:48:41
and the other is minus one-half.
Here, we have a case where the

620
00:48:41 --> 00:48:44
quantum number is not an
integer.

621
00:48:44 --> 00:48:47
It is one-half and it is minus
one-half.

622
00:48:47 --> 00:48:51
Now, if it helps you to think
about the electron spinning

623
00:48:51 --> 00:48:54
around its own axis,
like I depict here,

624
00:48:54 --> 00:48:58
well, if that is the case,
then the angular momentum

625
00:48:58 --> 00:49:03
quantum number is perpendicular,
here ,to this plane in which it

626
00:49:03 --> 00:49:08
is rotating.
And you might want to call that

627
00:49:08 --> 00:49:10
spin up.
And, of course,

628
00:49:10 --> 00:49:14
if it is spinning in the other
direction, well,

629
00:49:14 --> 00:49:17
then the angular momentum
vector is pointed in the

630
00:49:17 --> 00:49:21
opposite direction.
You might want to call this

631
00:49:21 --> 00:49:24
spin down.
If it helps for you to think

632
00:49:24 --> 00:49:28
about this, okay,
but remember that this is not

633
00:49:28 --> 00:49:32
correct.
This is a classical analogy

634
00:49:32 --> 00:49:37
that we are trying to draw here.
We are trying to say that this

635
00:49:37 --> 00:49:40
electron is rotating around its
own axis.

636
00:49:40 --> 00:49:44
That is not true.
This angular momentum is just

637
00:49:44 --> 00:49:47
an intrinsic part,
the intrinsic nature of a

638
00:49:47 --> 00:49:52
particular such as an electron.
Next time, I will tell you

639
00:49:52.099 --> 49:55
about Uhlenbeck and Goudsmith.
See you Wednesday.