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 --> 00:00:16
Let's get going,
here.

6
00:00:16 --> 00:00:20
Remember where we were?
We were trying to figure out

7
00:00:20 --> 00:00:25
the structure of the atom.
At the beginning of the course,

8
00:00:25 --> 00:00:29
we saw classical physics,
classical mechanics fail to

9
00:00:29 --> 00:00:35
describe how that electron in
the nucleus hung together.

10
00:00:35 --> 00:00:39
Then we started talking about
this wave-particle duality of

11
00:00:39 --> 00:00:43
light and matter.
We saw that radiation and

12
00:00:43 --> 00:00:47
matter both can exhibit both
wave-like properties and

13
00:00:47 --> 00:00:51
particle-like properties.
And it was really important,

14
00:00:51 --> 00:00:56
this observation of Davisson
and Germer, and George Thompson,

15
00:00:56 --> 00:01:02
this observation that electrons
exhibited inference phenomena.

16
00:01:02 --> 00:01:07
That is when you took electrons
and scattered them from a nickel

17
00:01:07 --> 00:01:11
single crystal.
The electrons scattered back as

18
00:01:11 --> 00:01:16
if they were behaving as waves.
There were diffraction

19
00:01:16 --> 00:01:20
phenomena or interference
phenomena, bright,

20
00:01:20 --> 00:01:23
dark, bright,
dark patterns of electrons.

21
00:01:23 --> 00:01:30
Actually, that Davisson and
Germer paper is on our website.

22
00:01:30 --> 00:01:33
You are welcome to take a look
at that.

23
00:01:33 --> 00:01:39
It was just that observation,
coupled with de Broglie's

24
00:01:39 --> 00:01:45
insight into Schršdinger's
relativistic equations of motion

25
00:01:45 --> 00:01:51
that led Schršdinger to say,
well, maybe what I need to do

26
00:01:51 --> 00:01:58
is I need to treat the wave-like
properties of that electron in a

27
00:01:58 --> 00:02:03
hydrogen atom.
Maybe that is the key.

28
00:02:03 --> 00:02:07
In particular,
maybe that is the key because

29
00:02:07 --> 00:02:14
the electron has a de Broglie
wavelength that is on the order

30
00:02:14 --> 00:02:20
of the size of its environment.
Maybe, in those cases,

31
00:02:20 --> 00:02:26
I need to treat the particle as
a wave and not as a particle

32
00:02:26 --> 00:02:31
with classical mechanics.
He wrote down this wave

33
00:02:31 --> 00:02:37
equation, an equation of motion
for waves, this H Psi equals E

34
00:02:37 --> 00:02:42
Psi, where we said last time we
are going to represent the

35
00:02:42 --> 00:02:45
electron, our particle,
by this Psi,

36
00:02:45 --> 00:02:48
the wave.
We are going to call it a wavef

37
00:02:48 --> 00:02:53
unction because we are going to
put a functional form to it very

38
00:02:53 --> 00:02:56
soon.
And there was some kind of

39
00:02:56 --> 00:02:59
operator, here,
called the Hamiltonian

40
00:02:59 --> 00:03:05
operator, that operated on this
wave function.

41
00:03:05 --> 00:03:08
And, when it did,
you got back the same wave

42
00:03:08 --> 00:03:12
function times a constant E.
And this constant,

43
00:03:12 --> 00:03:16
as we are going to see,
is going to be the binding

44
00:03:16 --> 00:03:19
energy of the electron to the
nucleus.

45
00:03:19 --> 00:03:23
But then, we took a little
detour and I said,

46
00:03:23 --> 00:03:26
well, let's see if we can
derive, in a sense,

47
00:03:26 --> 00:03:32
the Schršdinger equation.
And that is what we started to

48
00:03:32 --> 00:03:35
do.
And I am really doing this for

49
00:03:35 --> 00:03:41
fun, you are not responsible for
it, but I am doing it because I

50
00:03:41 --> 00:03:45
want you to see just how easy
this is.

51
00:03:45 --> 00:03:49
To illustrate this,
I am just going to take a

52
00:03:49 --> 00:03:54
one-dimensional problem.
I am going to let my electron

53
00:03:54 --> 00:03:57
be represented by this wave,
one-dimension,

54
00:03:57 --> 00:04:02
Psi of x.
2 a cosine 2 pi x over lambda.

55
00:04:02 --> 00:04:05
And then I said,

56
00:04:05 --> 00:04:10
suppose I want an equation of
motion, I want to know how that

57
00:04:10 --> 00:04:13
Psi changes with x.
Well, you already know that if

58
00:04:13 --> 00:04:17
I take the derivative of Psi
with x, that is going to tell me

59
00:04:17 --> 00:04:21
how Psi changes with x.
And we did that last time.

60
00:04:21 --> 00:04:24
And then I said,
well, I want to know the rate

61
00:04:24 --> 00:04:27
of change of Psi with x.
I am going to take the

62
00:04:27 --> 00:04:32
derivative again.
I have the second derivative of

63
00:04:32 --> 00:04:34
Psi of x.
And that is what we got last

64
00:04:34 --> 00:04:37
time.
And then, I noticed that in the

65
00:04:37 --> 00:04:39
second derivative,
and you noticed,

66
00:04:39 --> 00:04:43
too, somebody said this was
recursive, that we have our

67
00:04:43 --> 00:04:47
original wave function back in
this expression.

68
00:04:47 --> 00:04:51
I can rewrite that whole second
derivative here just as minus 2

69
00:04:51 --> 00:04:54
pi squared, quantity squared,
over lambda,

70
00:04:54 --> 00:04:57
psi of x.

71
00:04:57 --> 00:05:00
So far,
this is just any old wave

72
00:05:00 --> 00:05:05
equation.
Nothing special about this.

73
00:05:05 --> 00:05:10
This anybody could,
and had, written down before.

74
00:05:10 --> 00:05:14
What is special is that
Schršdinger realized,

75
00:05:14 --> 00:05:19
here, that if this is going to
be a wave equation for a

76
00:05:19 --> 00:05:25
particle, then maybe this lambda
here, maybe I ought to put in

77
00:05:25 --> 00:05:30
for lambda what de Broglie told
me.

78
00:05:30 --> 00:05:34
And that is h over p.
Maybe this lambda here is the

79
00:05:34 --> 00:05:40
wavelength of a matter wave,
so let me write this expression

80
00:05:40 --> 00:05:46
in terms of the momentum of the
particle, where the momentum has

81
00:05:46 --> 00:05:50
this mass m in it.
And so when he did this,

82
00:05:50 --> 00:05:54
this became minus p squared
over h bar squared,

83
00:05:54 --> 00:05:58
we said hbar is h over 2pi,
times psi of x.

84
00:05:58 --> 00:06:04
Hey, this is getting good

85
00:06:04 --> 00:06:08
because now we have a Psi of x
over here.

86
00:06:08 --> 00:06:14
But then what he said was,
well, I want to write this

87
00:06:14 --> 00:06:17
momentum in terms of the total
energy.

88
00:06:17 --> 00:06:22
Total energy is always kinetic
plus potential.

89
00:06:22 --> 00:06:26
The kinetic energy,
we said the other day,

90
00:06:26 --> 00:06:32
can be written in terms of the
momentum.

91
00:06:32 --> 00:06:36
The kinetic energy is p squared
over 2m-- plus the potential

92
00:06:36 --> 00:06:39
energy. And I

93
00:06:39 --> 00:06:43
am going to make this as a
function of x,

94
00:06:43 --> 00:06:47
the potential energy.
Now, I am just going to solve

95
00:06:47 --> 00:06:51
this for p squared.
p squared is equal to 2m times

96
00:06:51 --> 00:06:54
the total energy minus this
potential energy.

97
00:06:54 --> 00:06:59
Now, I am going to
plug this into

98
00:06:59 --> 00:07:03
here right in there.
And, when I do that,

99
00:07:03 --> 00:07:08
I am going to get the second
derivative of Psi of x with

100
00:07:08 --> 00:07:13
respect to x equals minus 2m
over h bar squared times E minus

101
00:07:13 --> 00:07:18
U of x times Psi of x.

102
00:07:18 --> 00:07:23
Just simple
substitution for p

103
00:07:23 --> 00:07:25
squared there.
Nothing else.

104
00:07:25 --> 00:07:30
Now, I am going to do some
rearranging.

105
00:07:30 --> 00:07:33
And the rearranging,
on the right-hand side,

106
00:07:33 --> 00:07:37
is I am going to have only E
times Psi of x.

107
00:07:37 --> 00:07:42
E times Psi of x looks like the
right-hand side of the

108
00:07:42 --> 00:07:46
Schršdinger as I wrote it down.
That is good.

109
00:07:46 --> 00:07:50
When I rearrange this,
I get minus hbar squared over

110
00:07:50 --> 00:07:55
2m times the second derivative
of Psi of x with respect to x

111
00:07:55 --> 00:08:00
plus U of x times Psi of x
equals E times Psi of x.

112
00:08:00 --> 00:08:07


113
00:08:07 --> 00:08:11
And now, I am going to pull out
a Psi of x here,

114
00:08:11 --> 00:08:17
so that is minus hbar squared
over 2m second derivative with

115
00:08:17 --> 00:08:21
respect to x plus U of x,
the quantity times Psi of x

116
00:08:21 --> 00:08:27
equals E times Psi of x.

117
00:08:27 --> 00:08:30
And guess what?

118
00:08:30 --> 00:08:36
We've got it.
We got it because all of this

119
00:08:36 --> 00:08:41
is what we define as the
Hamiltonian.

120
00:08:41 --> 00:08:47
All of this is h hat.
H hat, operating on Psi of x,

121
00:08:47 --> 00:08:54
gives us E times Psi of x.
This Hamiltonian,

122
00:08:54 --> 00:09:01
as you will learn
later on, is a kinetic energy

123
00:09:01 --> 00:09:06
operator.
This is the potential energy

124
00:09:06 --> 00:09:11
operator operating on psi.
That is the Schršdinger

125
00:09:11 --> 00:09:15
equation.
It is hardly a derivation.

126
00:09:15 --> 00:09:19
It is taking derivatives.
It is a wave equation.

127
00:09:19 --> 00:09:25
The insight came right here,
this substitution of the de

128
00:09:25 --> 00:09:31
Broglie wavelength in an
ordinary wave equation.

129
00:09:31 --> 00:09:37
This is the insight,
getting that momentum in there

130
00:09:37 --> 00:09:40
with the mass,
making this,

131
00:09:40 --> 00:09:44
then, an equation for a matter
wave.

132
00:09:44 --> 00:09:48
That is it.
You just "derived the

133
00:09:48 --> 00:09:53
Schršdinger equation." Easy.

134
00:09:53 --> 00:10:03


135
00:10:03 --> 00:10:07
Bottom line here is that the
Schršdinger equation is to

136
00:10:07 --> 00:10:11
quantum mechanics like Newton's
equations are to classical

137
00:10:11 --> 00:10:13
mechanics.
When the wavelength of a

138
00:10:13 --> 00:10:17
particle is on the order of the
size of its environment,

139
00:10:17 --> 00:10:22
the equation of motion that you
have to use to describe that

140
00:10:22 --> 00:10:26
particle moving within some
potential field U of x

141
00:10:26 --> 00:10:30
or U, you have to us this
equation of motion and not

142
00:10:30 --> 00:10:35
Newton's equations.
Newton's equations don't work

143
00:10:35 --> 00:10:41
to describe the motion of any
particle whose wavelength is on

144
00:10:41 --> 00:10:45
the order of the size of the
environment.

145
00:10:45 --> 00:10:49
It just does not work.
Now, just as an aside,

146
00:10:49 --> 00:10:56
classical mechanics really is
embedded in quantum mechanics.

147
00:10:56 --> 00:11:00
That is, if you took a problem
and solved it quantum

148
00:11:00 --> 00:11:04
mechanically,
and you solved the problem,

149
00:11:04 --> 00:11:08
a problem for which the
wavelength of a particle was

150
00:11:08 --> 00:11:13
much, much greater than the size
of the environment,

151
00:11:13 --> 00:11:18
which is the classical limit,
quantum mechanics would give

152
00:11:18 --> 00:11:21
you the right answer.
In other words,

153
00:11:21 --> 00:11:26
say you took some problem where
the wavelength of the particle

154
00:11:26 --> 00:11:32
is larger than the size of the
environment.

155
00:11:32 --> 00:11:35
That is, a problem where you
would normally use classical

156
00:11:35 --> 00:11:37
mechanics.
But if you use quantum

157
00:11:37 --> 00:11:40
mechanics, you would get the
right answer,

158
00:11:40 --> 00:11:44
if you could solve the problem
because the equations are very

159
00:11:44 --> 00:11:45
difficult.
But, in principle,

160
00:11:45 --> 00:11:49
you would get the right answer.
However, if you took a quantum

161
00:11:49 --> 00:11:52
mechanical problem,
that is, a problem where the

162
00:11:52 --> 00:11:56
wavelength of the particle is on
the order of the size of the

163
00:11:56 --> 00:11:59
environment and you used
classical mechanics,

164
00:11:59 --> 00:12:03
well, you won't get the right
answer.

165
00:12:03 --> 00:12:08
Because classical mechanics is,
in a sense, a subset.

166
00:12:08 --> 00:12:12
It is contained within quantum
mechanics.

167
00:12:12 --> 00:12:15
It is a limit of quantum
mechanics.

168
00:12:15 --> 00:12:20
We have to learn a new kind of
mechanics, here,

169
00:12:20 --> 00:12:24
this mechanics for the motion
of waves.

170
00:12:24 --> 00:12:29
Now, for a hydrogen atom,
we have to think of the wave

171
00:12:29 --> 00:12:35
function in three dimensions
instead of just one dimension,

172
00:12:35 --> 00:12:39
here.
And so we are going to have to

173
00:12:39 --> 00:12:44
describe the particle in terms
of three position coordinates.

174
00:12:44 --> 00:12:47
Usually you use Cartesian
coordinates, x,

175
00:12:47 --> 00:12:50
y, and z.
But this problem is solvable

176
00:12:50 --> 00:12:54
exactly if we use spherical
coordinates.

177
00:12:54 --> 00:12:59
How many of you had spherical
coordinates before and know what

178
00:12:59 --> 00:13:03
we are talking about?
Not everybody.

179
00:13:03 --> 00:13:06
If I gave you an x,
y and z for this electron in

180
00:13:06 --> 00:13:11
this atom, where the nucleus was
pinned at the origin.

181
00:13:11 --> 00:13:14
If I gave you an x,
y and z coordinate,

182
00:13:14 --> 00:13:17
you would know where that
electron was.

183
00:13:17 --> 00:13:21
But, alternatively,
I could tell you the position

184
00:13:21 --> 00:13:24
of this electron using spherical
coordinates.

185
00:13:24 --> 00:13:29
That is, I could tell you what
the distance is of the electron

186
00:13:29 --> 00:13:34
from the nucleus.
I am going to call that r.

187
00:13:34 --> 00:13:36
That will be one of the
variables.

188
00:13:36 --> 00:13:40
I could then also tell you this
angle theta.

189
00:13:40 --> 00:13:43
Theta is the angle that r makes
from the z-axis.

190
00:13:43 --> 00:13:47
That is the second coordinate.
And then, finally,

191
00:13:47 --> 00:13:52
the third coordinate is phi.
Phi is the angle made by the

192
00:13:52 --> 00:13:55
following.
If I take the electron and drop

193
00:13:55 --> 00:14:00
it perpendicular to the
x,y-plane and then I draw a line

194
00:14:00 --> 00:14:04
here, well, the angle between
that line and the x-axis is the

195
00:14:04 --> 00:14:09
angle phi.
Instead of giving you x,

196
00:14:09 --> 00:14:12
y, and z, I am just going to
give you r, theta,

197
00:14:12 --> 00:14:15
and phi in spherical
coordinates.

198
00:14:15 --> 00:14:20
And now, this wave function is
also a function of time,

199
00:14:20 --> 00:14:25
and I will talk about that a
little bit probably next time.

200
00:14:25 --> 00:14:30
So, that is the wave function
that is in some way going to

201
00:14:30 --> 00:14:35
represent our electron.
I haven't told you exactly yet

202
00:14:35 --> 00:14:39
how Psi represents the electron,
and I won't tell you that for

203
00:14:39 --> 00:14:42
another few days.
Question?

204
00:14:42 --> 00:14:55


205
00:14:55 --> 00:14:59
No.
I actually thought this was the

206
00:14:59 --> 00:15:04
way the book set it up.
I may have it backwards.

207
00:15:04 --> 00:15:09


208
00:15:09 --> 00:15:12
I think this is the way our
book sets it up.

209
00:15:12 --> 00:15:16
It won't make a difference in
any of the problems we solve

210
00:15:16 --> 00:15:18
here.
But, in general,

211
00:15:18 --> 00:15:23
if you are given a problem to
solve, you are going to have to

212
00:15:23 --> 00:15:28
look and see how they define
their coordinate system.

213
00:15:28 --> 00:15:33
Actually, somebody else asked
me that the other day.

214
00:15:33 --> 00:15:39
Now, what we have to do is
actually set up the Hamiltonian

215
00:15:39 --> 00:15:45
for the hydrogen atom.
That is, we have to set up the

216
00:15:45 --> 00:15:50
Hamiltonian specifically for the
hydrogen atom.

217
00:15:50 --> 00:15:56
And we have to set it up in
terms of spherical coordinates,

218
00:15:56 --> 00:16:01
r, theta, and phi.
And when we do that,

219
00:16:01 --> 00:16:06
and we are not actually going
to do that, that is what the

220
00:16:06 --> 00:16:10
Hamiltonian looks like.
Actually, this is in three

221
00:16:10 --> 00:16:14
dimensions, and so if I were
doing it in the x,

222
00:16:14 --> 00:16:20
y, z, what I would have here is
a second derivative with respect

223
00:16:20 --> 00:16:24
to y, plus a second derivative
with respect to z.

224
00:16:24 --> 00:16:29
That is what it would look like
in three dimensions in Cartesian

225
00:16:29 --> 00:16:33
coordinates.
But when I transform from

226
00:16:33 --> 00:16:37
Cartesian coordinates to
spherical coordinates,

227
00:16:37 --> 00:16:41
an exercise that takes five
pages, and everybody should have

228
00:16:41 --> 00:16:45
that experience once in their
life, but maybe now is not the

229
00:16:45 --> 00:16:48
right time for that experience,
you do get this.

230
00:16:48 --> 00:16:51
And essentially,
the Hamiltonian is a sum of

231
00:16:51 --> 00:16:56
second derivatives with respect
to each one of the coordinates,

232
00:16:56 --> 00:17:00
because essentially this term
is a second derivative with

233
00:17:00 --> 00:17:03
respect to r.
This term is a second

234
00:17:03 --> 00:17:06
derivative with respect to
theta.

235
00:17:06 --> 00:17:10
This term is a second
derivative with respect to phi.

236
00:17:10 --> 00:17:12
That is the specific
Hamiltonian.

237
00:17:12 --> 00:17:17
These are all kinetic energy
operators, as you will learn

238
00:17:17 --> 00:17:20
later on.
And then there is this term

239
00:17:20 --> 00:17:22
here, U of r.
This U of r,

240
00:17:22 --> 00:17:24
what is that?
What is U(r)?

241
00:17:24 --> 00:17:27
Potential energy.
It is the Coulomb potential

242
00:17:27 --> 00:17:32
energy of interaction.
It is isotropic,

243
00:17:32 --> 00:17:35
meaning it is the same at all
angles.

244
00:17:35 --> 00:17:40
The only thing it depends on is
the distance of the electron

245
00:17:40 --> 00:17:43
from the nucleus.
It only depends on r.

246
00:17:43 --> 00:17:46
It doesn't depend on theta and
phi.

247
00:17:46 --> 00:17:50
So, that is the Schršdinger
equation for the hydrogen atom.

248
00:17:50 --> 00:17:55
It is a differential equation,
second-order ordinary

249
00:17:55 --> 00:18:00
differential equation.
In 18.03, you are going to

250
00:18:00 --> 00:18:04
learn how to solve that.
In 5.61, or 8.04 I think it is,

251
00:18:04 --> 00:18:08
in physics, in quantum
mechanics, you are going to

252
00:18:08 --> 00:18:10
solve that.
You can do that.

253
00:18:10 --> 00:18:14
It is not hard.
But what do I mean when I say

254
00:18:14 --> 00:18:16
solve?
What I mean is that we are

255
00:18:16 --> 00:18:21
going to calculate these Es,
these binding energies,

256
00:18:21 --> 00:18:23
this constant in front of the
psi.

257
00:18:23 --> 00:18:28
That is called an eigenvalue,
for those of you who might know

258
00:18:28 --> 00:18:35
something already about these
kinds of differential equations.

259
00:18:35 --> 00:18:40
This E is the binding energy of
the electron to the nucleus.

260
00:18:40 --> 00:18:45
We are going to look at those
results in just a moment.

261
00:18:45 --> 00:18:50
And the other quantity we are
going to solve for,

262
00:18:50 --> 00:18:53
here, is psi.
What we are going to want to

263
00:18:53 --> 00:19:00
find is the actual functional
form for the wave functions.

264
00:19:00 --> 00:19:04
That, we can get out of solving
this differential equation.

265
00:19:04 --> 00:19:07
The actual functional form for
psi.

266
00:19:07 --> 00:19:13
The functional form is going to
be more complicated than what I

267
00:19:13 --> 00:19:18
wrote here, but we can get that.
We will look at that in a few

268
00:19:18 --> 00:19:22
days from now.
And do you know what those wave

269
00:19:22 --> 00:19:25
functions are?
They are what you studied in

270
00:19:25 --> 00:19:31
high school as orbitals.
You talked about an s-orbital,

271
00:19:31 --> 00:19:34
p-orbital, d-orbital.
Orbitals are wave functions.

272
00:19:34 --> 00:19:38
That where they come from,
orbitals, from solving the

273
00:19:38 --> 00:19:41
Schršdinger equation.
Now, specifically,

274
00:19:41 --> 00:19:45
orbitals are the spatial part
of a wave function.

275
00:19:45 --> 00:19:48
There is also a spin part to
the wave function,

276
00:19:48 --> 00:19:52
but for all intents and
purposes, we are going to use

277
00:19:52 --> 00:19:56
the term orbital and wave
function interchangeably because

278
00:19:56 --> 00:20:00
they are the same thing.
Question?

279
00:20:00 --> 00:20:03
We are going to get to that in
just a moment,

280
00:20:03 --> 00:20:09
and if you think I don't answer
it then we will go back to it.

281
00:20:09 --> 00:20:13
We are going to solve this.
And this equation is going to

282
00:20:13 --> 00:20:18
predict binding energies,
and it is going to predict

283
00:20:18 --> 00:20:22
these wave functions in
agreement with our observations,

284
00:20:22 --> 00:20:29
and it is going to predict that
the hydrogen atom is stable.

285
00:20:29 --> 00:20:33
Remember when we tried to
predict the hydrogen atom using

286
00:20:33 --> 00:20:36
classical ideas?
We found it lived a whole

287
00:20:36 --> 00:20:40
whopping 10^-10 seconds.
Not so when we treat the

288
00:20:40 --> 00:20:45
hydrogen atom with the quantum
mechanical equations of motion.

289
00:20:45 --> 00:20:50
Let's write down the results
for solving the Schršdinger

290
00:20:50 --> 00:20:53
equation.
And the part I am going to

291
00:20:53 --> 00:20:58
concentrate on today is these
binding energies.

292
00:20:58 --> 00:21:00
And, on Friday,
we will talk about the wave

293
00:21:00 --> 00:21:03
function, solving the
Schršdinger equation for those

294
00:21:03 --> 00:21:05
wave functions.

295
00:21:05 --> 00:21:16


296
00:21:16 --> 00:21:19
What does the binding energy
look like?

297
00:21:19 --> 00:21:23
Well, the binding energies,
here, coming out of the

298
00:21:23 --> 00:21:28
Schršdinger equation look like
this, minus 1 over n squared

299
00:21:28 --> 00:21:32
times m e to the 4 over 8
epsilon nought squared time h

300
00:21:32 --> 00:21:35
squared.

301
00:21:35 --> 00:21:40
m is the mass of the electron.

302
00:21:40 --> 00:21:43
e is the charge on the
electron.

303
00:21:43 --> 00:21:48
Epsilon nought is this
permittivity of vacuum we talked

304
00:21:48 --> 00:21:52
about before.
This is a conversion between

305
00:21:52 --> 00:21:56
ESU units and SI units.
h is Planck's constant.

306
00:21:56 --> 00:21:59
Planck's constant is
ubiquitous.

307
00:21:59 --> 00:22:05
It is everywhere.
What we typically do is we lump

308
00:22:05 --> 00:22:11
these constants together.
And we call those constants a

309
00:22:11 --> 00:22:15
new constant,
the Rydberg constant,

310
00:22:15 --> 00:22:21
R sub H.
And so our expression is equal

311
00:22:21 --> 00:22:28
to minus R sub H over n squared.

312
00:22:28 --> 00:22:35
The value of R sub H is equal
to 2.17987x10^-18 joules.

313
00:22:35 --> 00:22:43
That is a number that you are
going to use a lot in the next

314
00:22:43 --> 00:22:48
few days.
But what you also see,

315
00:22:48 --> 00:22:51
here, is an n.
What is n?

316
00:22:51 --> 00:23:00
Well, n is what we call the
principle quantum number.

317
00:23:00 --> 00:23:07
And its allowed values are
integers, where the integers

318
00:23:07 --> 00:23:13
start with 1,
2, 3, all the way up to n is

319
00:23:13 --> 00:23:19
equal to infinity.
Well, let's try to understand

320
00:23:19 --> 00:23:26
this a little bit more by
looking at an energy level

321
00:23:26 --> 00:23:32
diagram again.
I am just plotting here energy.

322
00:23:32 --> 00:23:38
Here is the zero of energy.
Here is the expression for the

323
00:23:38 --> 00:23:43
energy levels that come out of
the Schršdinger equation.

324
00:23:43 --> 00:23:48
When n is equal to one,
that is the lowest allowed

325
00:23:48 --> 00:23:54
value for n, binding energy is
essentially equal to minus the

326
00:23:54 --> 00:23:58
Rydberg constant.
**E = -(R)H**

327
00:23:58 --> 00:24:02
But our equation says the
binding energy of the electron

328
00:24:02 --> 00:24:07
can also be this value because
when n is equal to 2,

329
00:24:07 --> 00:24:11
well, the binding energy now is
only a quarter of the Rydberg

330
00:24:11 --> 00:24:15
constant. It is higher

331
00:24:15 --> 00:24:17
in energy.
When n is equal to 3,

332
00:24:17 --> 00:24:22
well, the binding energy of
that electron to the nucleus is

333
00:24:22 --> 00:24:26
a ninth to the Rydberg constant.
When it is equal to 4,

334
00:24:26 --> 00:24:30
it is a sixteenth.
When it is equal to 5,

335
00:24:30 --> 00:24:34
it is a twenty-fifth.
When it is equal to six,

336
00:24:34 --> 00:24:37
it is a thirty-sixth.
So on, so on,

337
00:24:37 --> 00:24:40
and so on until it gets n equal
to infinity.

338
00:24:40 --> 00:24:45
When n is equal to infinity,
then the binding energy is

339
00:24:45 --> 00:24:47
zero.
When n is equal to infinity,

340
00:24:47 --> 00:24:51
the electron and the nucleus
are no longer bound.

341
00:24:51 --> 00:24:54
They are separated from each
other.

342
00:24:54 --> 00:25:00
They don't hang together when n
is equal to infinity.

343
00:25:00 --> 00:25:05
Now, this is rather peculiar.
This is saying that the binding

344
00:25:05 --> 00:25:10
energy of the electron to the
nucleus can have essentially an

345
00:25:10 --> 00:25:16
infinite number of values,
except it is a discrete number

346
00:25:16 --> 00:25:21
of infinite numbers of values.
In the sense that the binding

347
00:25:21 --> 00:25:27
energy can be this or this or
this, but it cannot be something

348
00:25:27 --> 00:25:31
in between, here or here or
here.

349
00:25:31 --> 00:25:34
This is the quantum nature of
the hydrogen atom.

350
00:25:34 --> 00:25:37
There are allowed energy
levels.

351
00:25:37 --> 00:25:40
Where did this quantization
come from?

352
00:25:40 --> 00:25:44
It came from solving the
Schršdinger equation.

353
00:25:44 --> 00:25:49
When you solve a differential
equation, as you will learn,

354
00:25:49 --> 00:25:54
and that differential equation
applies to some physical

355
00:25:54 --> 00:25:59
problem, in order to make that
differential equation specific

356
00:25:59 --> 00:26:03
to your physical problem,
you often have to apply

357
00:26:03 --> 00:26:09
something called boundary
conditions to the problem.

358
00:26:09 --> 00:26:13
When you do that,
that is when this quantization

359
00:26:13 --> 00:26:16
comes out.
It drops out of solving the

360
00:26:16 --> 00:26:21
differential equation.
What are boundary conditions?

361
00:26:21 --> 00:26:27
Well, remember I told you in
the spherical coordinate system,

362
00:26:27 --> 00:26:30
r, theta, phi?
Well, phi sweeps from zero to

363
00:26:30 --> 00:26:35
360 degrees.
Well, if you just went 90

364
00:26:35 --> 00:26:38
degrees further,
so if you went to 450 degrees,

365
00:26:38 --> 00:26:43
you really have the same
situation as you had when phi

366
00:26:43 --> 00:26:47
was equal to 90 degrees.
What you have to do in a

367
00:26:47 --> 00:26:51
differential equation,
which usually has a series as a

368
00:26:51 --> 00:26:54
solution, is that you have to
cut it off.

369
00:26:54 --> 00:26:57
You have to apply boundary
conditions.

370
00:26:57 --> 00:27:02
You have to cut it off at
degrees.

371
00:27:02 --> 00:27:06
Because otherwise you just have
the same problem that you had

372
00:27:06 --> 00:27:09
before.
This is a cyclical boundary

373
00:27:09 --> 00:27:12
condition.
And it is that cutting it off

374
00:27:12 --> 00:27:16
to make the equation be really
pertinent or apply to your

375
00:27:16 --> 00:27:20
physical problem,
that is what leads to these

376
00:27:20 --> 00:27:23
boundary conditions,
mathematically.

377
00:27:23 --> 00:27:27
That is where it comes from.
That is where all the

378
00:27:27 --> 00:27:32
quantization comes from.
Now, let's talk a little bit

379
00:27:32 --> 00:27:38
about the significance here of
these binding energies because

380
00:27:38 --> 00:27:41
somebody asked me about it
already.

381
00:27:41 --> 00:27:47
When the electron is bound to
the nucleus with this much

382
00:27:47 --> 00:27:52
energy, we say that the electron
is in the n equals 1 state,

383
00:27:52 --> 00:27:57
or equivalently,
we say that the hydrogen atom

384
00:27:57 --> 00:28:02
is in the n equals 1 state.
We use both kinds of

385
00:28:02 --> 00:28:07
expressions equivalently.
When the hydrogen atom or the

386
00:28:07 --> 00:28:13
electron is in the n equals 1
state, we call that the ground

387
00:28:13 --> 00:28:16
state.
The ground state is the lowest

388
00:28:16 --> 00:28:20
energy state.
The electron is most strongly

389
00:28:20 --> 00:28:23
bound there.
The binding energy is most

390
00:28:23 --> 00:28:26
negative.
And the physical significance

391
00:28:26 --> 00:28:32
of that binding energy is that
it is minus the ionization

392
00:28:32 --> 00:28:35
energy.
Or, alternatively,

393
00:28:35 --> 00:28:40
the ionization energy is minus
the binding energy.

394
00:28:40 --> 00:28:46
It is going to require this
much energy, from here to here,

395
00:28:46 --> 00:28:50
to rip the electron off of the
nucleus.

396
00:28:50 --> 00:28:56
The binding energy here is
minus the ionization energy.

397
00:28:56 --> 00:29:01
That is the physical
significance of it.

398
00:29:01 --> 00:29:04
Now, did you ask me about the
work function?

399
00:29:04 --> 00:29:07
Okay.
The work function is the

400
00:29:07 --> 00:29:11
ionization energy when we talk
about a solid.

401
00:29:11 --> 00:29:15
That is just a terminology,
that is historical.

402
00:29:15 --> 00:29:20
When we talk about ripping and
electron off of a solid,

403
00:29:20 --> 00:29:25
we call it the work function.
When we talk about ripping it

404
00:29:25 --> 00:29:30
off of an atom or a molecule,
we call it the ionization

405
00:29:30 --> 00:29:34
energy.
And the other important thing

406
00:29:34 --> 00:29:39
to know here is that when we
talk about ionization energy,

407
00:29:39 --> 00:29:44
we are usually talking about
the energy required to pull the

408
00:29:44 --> 00:29:50
electron off when the molecule
or the atom is in the lowest

409
00:29:50 --> 00:29:52
energy state,
the ground state.

410
00:29:52 --> 00:29:57
That is also important.
But our equations tell us we

411
00:29:57 --> 00:30:03
also can have a hydrogen atom in
the n equals 2 state.

412
00:30:03 --> 00:30:06
If the hydrogen atom is in the
n equals 2 state,

413
00:30:06 --> 00:30:11
it is in an excited state.
Actually, it is the first

414
00:30:11 --> 00:30:14
excited state.
The electron is bound less

415
00:30:14 --> 00:30:17
strongly.
It is bound less strongly,

416
00:30:17 --> 00:30:21
and consequently,
the ionization energy from an

417
00:30:21 --> 00:30:26
excited state hydrogen atom is
less because the electron is not

418
00:30:26 --> 00:30:30
bound so strongly.
And, of course,

419
00:30:30 --> 00:30:34
we could have a hydrogen at n
equals 3, n equals 4 and n

420
00:30:34 --> 00:30:38
equals 5, any of these allowed
energy levels.

421
00:30:38 --> 00:30:42
Not all at the same time,
but, at any given time,

422
00:30:42 --> 00:30:47
if you had a lot of hydrogen
atoms, you could have hydrogen

423
00:30:47 --> 00:30:50
atoms in all of these different
states.

424
00:30:50 --> 00:30:54
Now, the other point that I
want to make is that this

425
00:30:54 --> 00:30:58
Schršdinger result,
here, for the energy levels

426
00:30:58 --> 00:31:04
predicts the energy levels of
all one electron atoms.

427
00:31:04 --> 00:31:07
What is a one electron atom?
Well, helium plus,

428
00:31:07 --> 00:31:11
that is a one electron atom or
a one electron ion.

429
00:31:11 --> 00:31:15
Helium usually has two
electrons, but if we pull one

430
00:31:15 --> 00:31:19
off it only has one left,
so it is a one electron atom.

431
00:31:19 --> 00:31:23
Lithium double plus
is a one electron atom because

432
00:31:23 --> 00:31:26
we pulled two of its three
electrons off.

433
00:31:26 --> 00:31:30
One electron is left,
and that is a one electron atom

434
00:31:30 --> 00:31:34
or an ion.
Uranium plus 91 is

435
00:31:34 --> 00:31:38
a one electron atom.
We pulled 91 electrons off,

436
00:31:38 --> 00:31:42
we have one left,
and that is a one electron

437
00:31:42 --> 00:31:45
atom.
The Schršdinger equation will

438
00:31:45 --> 00:31:51
predict what those energy levels
are, as long as you remember the

439
00:31:51 --> 00:31:53
Z squared up here.
For hydrogen,

440
00:31:53 --> 00:31:56
Z is 1, and so it doesn't
appear.

441
00:31:56 --> 00:32:03
But Z is not one for all of
these other one electron atoms.

442
00:32:03 --> 00:32:07
Where does the Z come from?
It comes from the Coulomb

443
00:32:07 --> 00:32:10
interaction.
That potential energy of

444
00:32:10 --> 00:32:15
interaction is the charge on the
electron times the charge on the

445
00:32:15 --> 00:32:19
nucleus, which is Z times e.
That is where the Z squared

446
00:32:19 --> 00:32:22
comes from.
We have to remember that.

447
00:32:22 --> 00:32:27
Now, how do we know that the
Schršdinger's predictions for

448
00:32:27 --> 00:32:32
these energy levels are correct,
are accurate?

449
00:32:32 --> 00:32:34
Well, we have to do an
experiment.

450
00:32:34 --> 00:32:39
The experiment we are going to
do is we are going to take a

451
00:32:39 --> 00:32:43
bulb here, pump it out,
and then we are going to fill

452
00:32:43 --> 00:32:47
it up with molecular hydrogen.
And then in this bulb,

453
00:32:47 --> 00:32:51
there is a negative and a
positive electrode.

454
00:32:51 --> 00:32:55
What we are going to do is
crank up the potential energy

455
00:32:55 --> 00:33:00
difference between those two so
high until finally that gas is

456
00:33:00 --> 00:33:04
going to ignite,
just like that.

457
00:33:04 --> 00:33:08
And we are going to look at the
light coming out.

458
00:33:08 --> 00:33:13
And what we are going to do is
we are going to disperse that

459
00:33:13 --> 00:33:16
light.
We are going to send it through

460
00:33:16 --> 00:33:19
a diffraction grating,
essentially.

461
00:33:19 --> 00:33:22
And that is like an array of
slits.

462
00:33:22 --> 00:33:26
What you are going to see are
points of constructive and

463
00:33:26 --> 00:33:31
destructive interference.
But in the constructive

464
00:33:31 --> 00:33:34
interference,
you are going to see the color

465
00:33:34 --> 00:33:37
separated.
There is going to be purple,

466
00:33:37 --> 00:33:38
blue, green,
etc.

467
00:33:38 --> 00:33:41
The reason is that the
different radiation,

468
00:33:41 --> 00:33:45
the different colors have
different wavelengths.

469
00:33:45 --> 00:33:48
They have different
wavelengths, then they have

470
00:33:48 --> 00:33:52
slightly different points in
space at which constructive

471
00:33:52 --> 00:33:56
interference occurs,
and so the light is dispersed

472
00:33:56 --> 00:34:00
in space.
We are going to analyze what

473
00:34:00 --> 00:34:04
the wavelengths of the light
that are being emitted from

474
00:34:04 --> 00:34:08
these hydrogen atoms are.
See, the discharge pulls apart

475
00:34:08 --> 00:34:12
the H two and makes
hydrogen atoms.

476
00:34:12 --> 00:34:16
We have to do this by taking a
diffraction grating.

477
00:34:16 --> 00:34:20
The TAs are going to give you a
pair of diffraction grating

478
00:34:20 --> 00:34:23
glasses.
You are supposed to come and

479
00:34:23 --> 00:34:28
look at the discharge here.
You can see it.

480
00:34:28 --> 00:34:32
I will turn it a little bit for
those of you on the sides here.

481
00:34:32 --> 00:34:35
And we will see what we see.

482
00:34:35 --> 00:35:00


483
00:35:00 --> 00:35:03
If you cannot see the lamp,
you are welcome to get up and

484
00:35:03 --> 00:35:06
move around so that you can see
it.

485
00:35:06 --> 00:35:20


486
00:35:20 --> 00:35:21
Can you do this light,
too?

487
00:35:21 --> 00:35:23
This one over here,
sir.

488
00:35:23 --> 00:35:26
Can you move this one away?

489
00:35:26 --> 00:35:35


490
00:35:35 --> 00:35:38
If you look up at the lights in
the room, you can see the whole

491
00:35:38 --> 00:35:40
spectrum, because that is white
light.

492
00:35:40 --> 00:36:08


493
00:36:08 --> 00:36:13
I am going to turn the lamp
over here so that you can see

494
00:36:13 --> 00:36:18
this a little better.
The spectrum that you should

495
00:36:18 --> 00:36:21
see is what is shown on the
board.

496
00:36:21 --> 00:36:27
If you look to the right of the
lamp, here, you should see a

497
00:36:27 --> 00:36:32
purple line.
The purple line is actually

498
00:36:32 --> 00:36:36
very light, so only if you are
up close are you going to see

499
00:36:36 --> 00:36:39
the purple line.
You can see the blue line very

500
00:36:39 --> 00:36:41
well.
That is very intense.

501
00:36:41 --> 00:36:44
The green line is also very
diffuse.

502
00:36:44 --> 00:36:48
Again, only if you are close
are you going to be able to see

503
00:36:48 --> 00:36:52
the green line.
And then, the red line is very

504
00:36:52 --> 00:36:55
bright.
And now I am going to move this

505
00:36:55 --> 00:37:00
over here so you have the
opportunity to see it.

506
00:37:00 --> 00:37:09
Again, you are welcome to get
out of your seats and move

507
00:37:09 --> 00:37:15
around so that you can,
in fact, see it.

508
00:37:15 --> 00:37:24
What you should be seeing,
hey, interference phenomena

509
00:37:24 --> 00:37:29
works.
Useful pointer.

510
00:37:29 --> 00:37:40


511
00:37:40 --> 00:37:43
What you should see,
depending on which one of the

512
00:37:43 --> 00:37:48
constructive interference
patterns you are looking at,

513
00:37:48 --> 00:37:53
there should be a purple line,
there should be a blue line,

514
00:37:53 --> 00:37:55
very intense,
purple is weak,

515
00:37:55 --> 00:38:00
green is rather week,
and the red is very intense.

516
00:38:00 --> 00:38:02
What is happening,
here?

517
00:38:02 --> 00:38:06
Well, what is happening is that
in this discharge,

518
00:38:06 --> 00:38:12
there is enough energy to put
some of the hydrogen atoms in a

519
00:38:12 --> 00:38:17
high energy state.
And we will call that state E

520
00:38:17 --> 00:38:21
sub i,
the energy of the initial

521
00:38:21 --> 00:38:24
state.
Actually, that is an unstable

522
00:38:24 --> 00:38:28
situation.
That hydrogen atom wants to

523
00:38:28 --> 00:38:32
relax.
It wants to be in the lower

524
00:38:32 --> 00:38:36
energy state.
And what happens is that it

525
00:38:36 --> 00:38:38
does relax.
It relaxes.

526
00:38:38 --> 00:38:42
The electron falls into the
lower energy state,

527
00:38:42 --> 00:38:47
but, because it is lower
energy, it has to give up a

528
00:38:47 --> 00:38:50
photon.
And so the photon that is

529
00:38:50 --> 00:38:56
emitted, the energy of that
photon has to be exactly the

530
00:38:56 --> 00:39:02
difference in energy between the
energy of the initial state and

531
00:39:02 --> 00:39:07
the final state.
That is the quantum nature of

532
00:39:07 --> 00:39:12
the hydrogen atom.
The photon that comes out has

533
00:39:12 --> 00:39:15
to have that energy difference
exactly.

534
00:39:15 --> 00:39:18
And, therefore,
the frequency,

535
00:39:18 --> 00:39:24
here, of the radiation coming
out is going to correspond to

536
00:39:24 --> 00:39:29
that exact energy difference.
And, for the different

537
00:39:29 --> 00:39:33
energies, for the different
transitions, you are going to

538
00:39:33 --> 00:39:38
have very specific values of the
frequency of the radiation or

539
00:39:38 --> 00:39:41
the wavelength of the radiation.
For example,

540
00:39:41 --> 00:39:46
some of those hydrogen atoms in
this discharge have been excited

541
00:39:46 --> 00:39:49
to this excited state,
which we will call B.

542
00:39:49 --> 00:39:54
The energy difference between B
and this ground state here is

543
00:39:54 --> 00:39:58
small.
Therefore, we are going to have

544
00:39:58 --> 00:40:01
some radiation that is low
frequency because delta E,

545
00:40:01 --> 00:40:05
the difference in the energy
between the two states is small.

546
00:40:05 --> 00:40:09
And then there are going to be
some hydrogen atoms that are

547
00:40:09 --> 00:40:12
going to be excited to this
state, the A state.

548
00:40:12 --> 00:40:15
And, relatively speaking,
that energy difference is

549
00:40:15 --> 00:40:17
large.
There are going to be some

550
00:40:17 --> 00:40:21
hydrogen atoms that are going to
be relaxing to this ground

551
00:40:21 --> 00:40:23
state.
And when they do,

552
00:40:23 --> 00:40:27
since that energy difference is
large, the frequency of the

553
00:40:27 --> 00:40:31
radiation coming off is going to
be high.

554
00:40:31 --> 00:40:35
Or, correspondingly,
the wavelength of the radiation

555
00:40:35 --> 00:40:39
coming off is going to be low
because the wavelength is

556
00:40:39 --> 00:40:42
inversely proportional to the
frequency.

557
00:40:42 --> 00:40:45
And, likewise,
for this transition we are

558
00:40:45 --> 00:40:50
going to have some long
wavelength radiation emitted.

559
00:40:50 --> 00:40:54
Well, let's see if we can
understand specifically the

560
00:40:54 --> 00:40:57
spectrum, here,
in the visible range for the

561
00:40:57 --> 00:41:03
hydrogen atom.
What I have done is to draw an

562
00:41:03 --> 00:41:09
energy level diagram again.
Here is the energy of n equals

563
00:41:09 --> 00:41:12
1, n equals 2,
n equals 3, etc.

564
00:41:12 --> 00:41:16
Here is the n equals infinity,
up here.

565
00:41:16 --> 00:41:22
It turns out that this purple
line is a transition from a

566
00:41:22 --> 00:41:30
hydrogen atom in the n equals 6
state to the n equals 2 state.

567
00:41:30 --> 00:41:36
That blue line is a transition
from n equals 5 to n equals 2,

568
00:41:36 --> 00:41:43
the green line is a transition
from n equals 4 to n equals 2

569
00:41:43 --> 00:41:48
and the red line from n equals 3
to n equals 2.

570
00:41:48 --> 00:41:54
Notice that since this energy
difference here is small,

571
00:41:54 --> 00:41:59
this line has a long
wavelength.

572
00:41:59 --> 00:42:04
Since this energy difference is
larger, this line has a shorter

573
00:42:04 --> 00:42:08
wavelength.
All of these transitions that

574
00:42:08 --> 00:42:13
you are seeing in the visible
range have the final state of n

575
00:42:13 --> 00:42:16
equals 2.
Now, how do we know that the

576
00:42:16 --> 00:42:22
Schršdinger equation is making
predictions that are consistent

577
00:42:22 --> 00:42:27
with the frequencies or the
wavelengths of the radiation

578
00:42:27 --> 00:42:32
that we observe here?
Well, we have got to do a plug

579
00:42:32 --> 00:42:34
here.
This is the frequency that we

580
00:42:34 --> 00:42:37
would expect.
It is the energy difference

581
00:42:37 --> 00:42:40
between two states over H.
Here is the initial energy.

582
00:42:40 --> 00:42:44
Here is the final energy.
We said that the Schršdinger

583
00:42:44 --> 00:42:48
equation tells us that the
energies of the state are minus

584
00:42:48 --> 00:42:50
R sub H over n.

585
00:42:50 --> 00:42:54
For the initial energy level,
it is minus R sub H over n sub

586
00:42:54 --> 00:42:57
i squared. We plug

587
00:42:57 --> 00:43:01
that in.
For the final energy level,

588
00:43:01 --> 00:43:05
well, it is minus R sub H over
n sub f,

589
00:43:05 --> 00:43:08
the final quantum number
squared.

590
00:43:08 --> 00:43:11
We plug that in.
We rearrange some things.

591
00:43:11 --> 00:43:15
And then this is the prediction
for the frequency.

592
00:43:15 --> 00:43:20
I said that this level right
here is the n equals 2 level,

593
00:43:20 --> 00:43:24
so we will plug in the two for
the final quantum state.

594
00:43:24 --> 00:43:28
And so this is then the
prediction for the frequencies

595
00:43:28 --> 00:43:33
of the radiation to the n equals
2 level.

596
00:43:33 --> 00:43:37
You just plug in n equals 6,
n equals 5, n equals 4,

597
00:43:37 --> 00:43:39
n equals 3.
And you know what?

598
00:43:39 --> 00:43:43
The frequencies that are
predicted match what we observe

599
00:43:43 --> 00:43:46
to one part in 10^8.
There really is precise

600
00:43:46 --> 00:43:51
agreement between the results of
the Schršdinger equation and

601
00:43:51 --> 00:43:55
what we actually observe in
nature, therefore,

602
00:43:55 --> 00:44:01
we think it is correct.
And there are no experiments

603
00:44:01 --> 00:44:06
that cast any doubt,
so far, on the Schršdinger

604
00:44:06 --> 00:44:09
equation.
Now, the set of transitions

605
00:44:09 --> 00:44:14
that we just looked at are these
transitions.

606
00:44:14 --> 00:44:20
Plotted here is another
energy-level diagram for the

607
00:44:20 --> 00:44:24
hydrogen atom.
Here is the n equals 1 state,

608
00:44:24 --> 00:44:30
n equals 2, n equals 3,
n equals 4.

609
00:44:30 --> 00:44:34
And we looked at all the
transitions that end up in the n

610
00:44:34 --> 00:44:36
equals 2 state.
This set of transitions.

611
00:44:36 --> 00:44:41
It is called the Balmer series.
Now, the n equals 2 state is

612
00:44:41 --> 00:44:44
not the ground state.
There is a transition from n

613
00:44:44 --> 00:44:47
equals 6 to n equals 1,
the ground state.

614
00:44:47 --> 00:44:50
It is this one right here.
But, you see,

615
00:44:50 --> 00:44:53
that is a very high energy
transition.

616
00:44:53 --> 00:44:57
Actually, this transition,
from n equals 6 to n equals 1,

617
00:44:57 --> 00:45:03
is in the ultraviolet range of
the electromagnetic spectrum.

618
00:45:03 --> 00:45:06
And so you cannot see it with
the experiment that we did,

619
00:45:06 --> 00:45:09
but it is there.
And then, of course,

620
00:45:09 --> 00:45:12
the hydrogen atoms that relaxed
to n equals 2,

621
00:45:12 --> 00:45:16
well, they actually eventually
relax to n equals 1.

622
00:45:16 --> 00:45:19
And there is a transition
there, it is over here,

623
00:45:19 --> 00:45:23
n equals 2 to n equals 1.
But, again, that is a high

624
00:45:23 --> 00:45:26
energy transition.
It is also in the ultraviolet

625
00:45:26 --> 00:45:30
range of the electromagnetic
spectrum.

626
00:45:30 --> 00:45:34
And so all of these transitions
that end up in n equals 1 are in

627
00:45:34 --> 00:45:36
the UV range.
They are called the Lyman

628
00:45:36 --> 00:45:38
series.
All the transitions that end up

629
00:45:38 --> 00:45:40
in n equals 2 are in the
visible.

630
00:45:40 --> 00:45:44
They are called the Balmer.
n equals 3, the Paschen,

631
00:45:44 --> 00:45:47
is in the infrared.
Brackett is in the infrared.

632
00:45:47 --> 00:45:50
Pfund is in the infrared.
And so we cannot see these

633
00:45:50 --> 00:45:52
easily.
Now, these are named for the

634
00:45:52 --> 00:45:55
different discoverers.
The reason there are so many

635
00:45:55 --> 00:46:01
discovers is because these are
all different frequency ranges.

636
00:46:01 --> 00:46:04
Different frequencies of
radiation require different

637
00:46:04 --> 00:46:08
kinds of detectors.
And so, depending on exactly

638
00:46:08 --> 00:46:11
what kind of detector the
experimentalist had,

639
00:46:11 --> 00:46:16
that will dictate then which
one of these sets of transitions

640
00:46:16 --> 00:46:20
he was able to discover.
But not only does this work for

641
00:46:20 --> 00:46:24
emission, but this also works
for absorption.

642
00:46:24 --> 00:46:28
That is, we can have a hydrogen
atom in the ground state,

643
00:46:28 --> 00:46:33
a low energy state.
And if there is a photon around

644
00:46:33 --> 00:46:37
that is exactly the energy
difference between these two

645
00:46:37 --> 00:46:40
states, well,
that photon can be absorbed.

646
00:46:40 --> 00:46:44
If that photon is a little bit
higher in energy,

647
00:46:44 --> 00:46:47
it won't be absorbed.
If it is a little lower,

648
00:46:47 --> 00:46:50
it won't be absorbed.
It has to be exactly the

649
00:46:50 --> 00:46:54
difference in the energies
between these two states.

650
00:46:54 --> 00:47:00
Again, that is the quantum
nature of the hydrogen atom.

651
00:47:00 --> 00:47:04
And now, if you are calculating
the frequency for absorption,

652
00:47:04 --> 00:47:09
what I have done here is I have
reversed n sub i and n

653
00:47:09 --> 00:47:12
sub f.
This is 1 over n sub i squared

654
00:47:12 --> 00:47:17
instead of 1 over
n sub f squared,

655
00:47:17 --> 00:47:21
which was the case in emission.
And I have reversed this so

656
00:47:21 --> 00:47:25
that you get a positive value
for the frequency of the

657
00:47:25 --> 00:47:28
radiation absorbed.
We will have two different

658
00:47:28 --> 00:47:33
equations for absorption and for
emission.

659
00:47:33 --> 00:47:36
So, those are the Schršdinger
equation results.

660
00:47:36 --> 00:47:40
I forgot to announce that there
is a forum this evening from

661
00:47:40 --> 00:47:44
5:00 to 6:00.
You should have signed up in

662
00:47:44 --> 00:47:45
recitation.
If you didn't,

663
00:47:45 --> 00:47:49
and still want to come,
send us an email and come.

664
00:47:49 --> 00:47:54
And we need the diffraction
glasses back as you are exiting.

665
00:47:54 --> 47:57
See you Friday.