1
00:00:01,000 --> 00:00:04,000
The following content is
provided by MIT OpenCourseWare

2
00:00:04,000 --> 00:00:06,000
under a Creative Commons
license.

3
00:00:06,000 --> 00:00:10,000
Additional information about
our license and MIT

4
00:00:10,000 --> 00:00:15,000
OpenCourseWare in general is
available at ocw.mit.edu.

5
00:00:15,000 --> 00:00:16,000
Let's get going,
here.

6
00:00:16,000 --> 00:00:20,000
Remember where we were?
We were trying to figure out

7
00:00:20,000 --> 00:00:25,000
the structure of the atom.
At the beginning of the course,

8
00:00:25,000 --> 00:00:29,000
we saw classical physics,
classical mechanics fail to

9
00:00:29,000 --> 00:00:35,000
describe how that electron in
the nucleus hung together.

10
00:00:35,000 --> 00:00:39,000
Then we started talking about
this wave-particle duality of

11
00:00:39,000 --> 00:00:43,000
light and matter.
We saw that radiation and

12
00:00:43,000 --> 00:00:47,000
matter both can exhibit both
wave-like properties and

13
00:00:47,000 --> 00:00:51,000
particle-like properties.
And it was really important,

14
00:00:51,000 --> 00:00:56,000
this observation of Davisson
and Germer, and George Thompson,

15
00:00:56,000 --> 00:01:02,000
this observation that electrons
exhibited inference phenomena.

16
00:01:02,000 --> 00:01:07,000
That is when you took electrons
and scattered them from a nickel

17
00:01:07,000 --> 00:01:11,000
single crystal.
The electrons scattered back as

18
00:01:11,000 --> 00:01:16,000
if they were behaving as waves.
There were diffraction

19
00:01:16,000 --> 00:01:20,000
phenomena or interference
phenomena, bright,

20
00:01:20,000 --> 00:01:23,000
dark, bright,
dark patterns of electrons.

21
00:01:23,000 --> 00:01:30,000
Actually, that Davisson and
Germer paper is on our website.

22
00:01:30,000 --> 00:01:33,000
You are welcome to take a look
at that.

23
00:01:33,000 --> 00:01:39,000
It was just that observation,
coupled with de Broglie's

24
00:01:39,000 --> 00:01:45,000
insight into Schrödinger's
relativistic equations of motion

25
00:01:45,000 --> 00:01:51,000
that led Schrödinger to say,
well, maybe what I need to do

26
00:01:51,000 --> 00:01:58,000
is I need to treat the wave-like
properties of that electron in a

27
00:01:58,000 --> 00:02:03,000
hydrogen atom.
Maybe that is the key.

28
00:02:03,000 --> 00:02:07,000
In particular,
maybe that is the key because

29
00:02:07,000 --> 00:02:14,000
the electron has a de Broglie
wavelength that is on the order

30
00:02:14,000 --> 00:02:20,000
of the size of its environment.
Maybe, in those cases,

31
00:02:20,000 --> 00:02:26,000
I need to treat the particle as
a wave and not as a particle

32
00:02:26,000 --> 00:02:31,000
with classical mechanics.
He wrote down this wave

33
00:02:31,000 --> 00:02:37,000
equation, an equation of motion
for waves, this H Psi equals E

34
00:02:37,000 --> 00:02:42,000
Psi, where we said last time we
are going to represent the

35
00:02:42,000 --> 00:02:45,000
electron, our particle,
by this Psi,

36
00:02:45,000 --> 00:02:48,000
the wave.
We are going to call it a wavef

37
00:02:48,000 --> 00:02:53,000
unction because we are going to
put a functional form to it very

38
00:02:53,000 --> 00:02:56,000
soon.
And there was some kind of

39
00:02:56,000 --> 00:02:59,000
operator, here,
called the Hamiltonian

40
00:02:59,000 --> 00:03:05,000
operator, that operated on this
wave function.

41
00:03:05,000 --> 00:03:08,000
And, when it did,
you got back the same wave

42
00:03:08,000 --> 00:03:12,000
function times a constant E.
And this constant,

43
00:03:12,000 --> 00:03:16,000
as we are going to see,
is going to be the binding

44
00:03:16,000 --> 00:03:19,000
energy of the electron to the
nucleus.

45
00:03:19,000 --> 00:03:23,000
But then, we took a little
detour and I said,

46
00:03:23,000 --> 00:03:26,000
well, let's see if we can
derive, in a sense,

47
00:03:26,000 --> 00:03:32,000
the Schrödinger equation.
And that is what we started to

48
00:03:32,000 --> 00:03:35,000
do.
And I am really doing this for

49
00:03:35,000 --> 00:03:41,000
fun, you are not responsible for
it, but I am doing it because I

50
00:03:41,000 --> 00:03:45,000
want you to see just how easy
this is.

51
00:03:45,000 --> 00:03:49,000
To illustrate this,
I am just going to take a

52
00:03:49,000 --> 00:03:54,000
one-dimensional problem.
I am going to let my electron

53
00:03:54,000 --> 00:03:57,000
be represented by this wave,
one-dimension,

54
00:03:57,000 --> 00:04:02,000
Psi of x.
2 a cosine 2 pi x over lambda.

55
00:04:02,000 --> 00:04:05,000
And then I said,

56
00:04:05,000 --> 00:04:10,000
suppose I want an equation of
motion, I want to know how that

57
00:04:10,000 --> 00:04:13,000
Psi changes with x.
Well, you already know that if

58
00:04:13,000 --> 00:04:17,000
I take the derivative of Psi
with x, that is going to tell me

59
00:04:17,000 --> 00:04:21,000
how Psi changes with x.
And we did that last time.

60
00:04:21,000 --> 00:04:24,000
And then I said,
well, I want to know the rate

61
00:04:24,000 --> 00:04:27,000
of change of Psi with x.
I am going to take the

62
00:04:27,000 --> 00:04:32,000
derivative again.
I have the second derivative of

63
00:04:32,000 --> 00:04:34,000
Psi of x.
And that is what we got last

64
00:04:34,000 --> 00:04:37,000
time.
And then, I noticed that in the

65
00:04:37,000 --> 00:04:39,000
second derivative,
and you noticed,

66
00:04:39,000 --> 00:04:43,000
too, somebody said this was
recursive, that we have our

67
00:04:43,000 --> 00:04:47,000
original wave function back in
this expression.

68
00:04:47,000 --> 00:04:51,000
I can rewrite that whole second
derivative here just as minus 2

69
00:04:51,000 --> 00:04:54,000
pi squared, quantity squared,
over lambda,

70
00:04:54,000 --> 00:04:57,000
psi of x.

71
00:04:57,000 --> 00:05:00,000
So far,
this is just any old wave

72
00:05:00,000 --> 00:05:05,000
equation.
Nothing special about this.

73
00:05:05,000 --> 00:05:10,000
This anybody could,
and had, written down before.

74
00:05:10,000 --> 00:05:14,000
What is special is that
Schrödinger realized,

75
00:05:14,000 --> 00:05:19,000
here, that if this is going to
be a wave equation for a

76
00:05:19,000 --> 00:05:25,000
particle, then maybe this lambda
here, maybe I ought to put in

77
00:05:25,000 --> 00:05:30,000
for lambda what de Broglie told
me.

78
00:05:30,000 --> 00:05:34,000
And that is h over p.
Maybe this lambda here is the

79
00:05:34,000 --> 00:05:40,000
wavelength of a matter wave,
so let me write this expression

80
00:05:40,000 --> 00:05:46,000
in terms of the momentum of the
particle, where the momentum has

81
00:05:46,000 --> 00:05:50,000
this mass m in it.
And so when he did this,

82
00:05:50,000 --> 00:05:54,000
this became minus p squared
over h bar squared,

83
00:05:54,000 --> 00:05:58,000
we said hbar is h over 2pi,
times psi of x.

84
00:05:58,000 --> 00:06:04,000
Hey, this is getting good

85
00:06:04,000 --> 00:06:08,000
because now we have a Psi of x
over here.

86
00:06:08,000 --> 00:06:14,000
But then what he said was,
well, I want to write this

87
00:06:14,000 --> 00:06:17,000
momentum in terms of the total
energy.

88
00:06:17,000 --> 00:06:22,000
Total energy is always kinetic
plus potential.

89
00:06:22,000 --> 00:06:26,000
The kinetic energy,
we said the other day,

90
00:06:26,000 --> 00:06:32,000
can be written in terms of the
momentum.

91
00:06:32,000 --> 00:06:36,000
The kinetic energy is p squared
over 2m-- plus the potential

92
00:06:36,000 --> 00:06:39,000
energy. And I

93
00:06:39,000 --> 00:06:43,000
am going to make this as a
function of x,

94
00:06:43,000 --> 00:06:47,000
the potential energy.
Now, I am just going to solve

95
00:06:47,000 --> 00:06:51,000
this for p squared.
p squared is equal to 2m times

96
00:06:51,000 --> 00:06:54,000
the total energy minus this
potential energy.

97
00:06:54,000 --> 00:06:59,000
Now, I am going to
plug this into

98
00:06:59,000 --> 00:07:03,000
here right in there.
And, when I do that,

99
00:07:03,000 --> 00:07:08,000
I am going to get the second
derivative of Psi of x with

100
00:07:08,000 --> 00:07:13,000
respect to x equals minus 2m
over h bar squared times E minus

101
00:07:13,000 --> 00:07:18,000
U of x times Psi of x.

102
00:07:18,000 --> 00:07:23,000
Just simple
substitution for p

103
00:07:23,000 --> 00:07:25,000
squared there.
Nothing else.

104
00:07:25,000 --> 00:07:30,000
Now, I am going to do some
rearranging.

105
00:07:30,000 --> 00:07:33,000
And the rearranging,
on the right-hand side,

106
00:07:33,000 --> 00:07:37,000
is I am going to have only E
times Psi of x.

107
00:07:37,000 --> 00:07:42,000
E times Psi of x looks like the
right-hand side of the

108
00:07:42,000 --> 00:07:46,000
Schrödinger as I wrote it down.
That is good.

109
00:07:46,000 --> 00:07:50,000
When I rearrange this,
I get minus hbar squared over

110
00:07:50,000 --> 00:07:55,000
2m times the second derivative
of Psi of x with respect to x

111
00:07:55,000 --> 00:08:00,000
plus U of x times Psi of x
equals E times Psi of x.

112
00:08:07,000 --> 00:08:11,000
And now, I am going to pull out
a Psi of x here,

113
00:08:11,000 --> 00:08:17,000
so that is minus hbar squared
over 2m second derivative with

114
00:08:17,000 --> 00:08:21,000
respect to x plus U of x,
the quantity times Psi of x

115
00:08:21,000 --> 00:08:27,000
equals E times Psi of x.

116
00:08:27,000 --> 00:08:30,000
And guess what?

117
00:08:30,000 --> 00:08:36,000
We've got it.
We got it because all of this

118
00:08:36,000 --> 00:08:41,000
is what we define as the
Hamiltonian.

119
00:08:41,000 --> 00:08:47,000
All of this is h hat.
H hat, operating on Psi of x,

120
00:08:47,000 --> 00:08:54,000
gives us E times Psi of x.
This Hamiltonian,

121
00:08:54,000 --> 00:09:01,000
as you will learn
later on, is a kinetic energy

122
00:09:01,000 --> 00:09:06,000
operator.
This is the potential energy

123
00:09:06,000 --> 00:09:11,000
operator operating on psi.
That is the Schrödinger

124
00:09:11,000 --> 00:09:15,000
equation.
It is hardly a derivation.

125
00:09:15,000 --> 00:09:19,000
It is taking derivatives.
It is a wave equation.

126
00:09:19,000 --> 00:09:25,000
The insight came right here,
this substitution of the de

127
00:09:25,000 --> 00:09:31,000
Broglie wavelength in an
ordinary wave equation.

128
00:09:31,000 --> 00:09:37,000
This is the insight,
getting that momentum in there

129
00:09:37,000 --> 00:09:40,000
with the mass,
making this,

130
00:09:40,000 --> 00:09:44,000
then, an equation for a matter
wave.

131
00:09:44,000 --> 00:09:48,000
That is it.
You just "derived the

132
00:09:48,000 --> 00:09:53,000
Schrödinger equation." Easy.

133
00:10:03,000 --> 00:10:07,000
Bottom line here is that the
Schrödinger equation is to

134
00:10:07,000 --> 00:10:11,000
quantum mechanics like Newton's
equations are to classical

135
00:10:11,000 --> 00:10:13,000
mechanics.
When the wavelength of a

136
00:10:13,000 --> 00:10:17,000
particle is on the order of the
size of its environment,

137
00:10:17,000 --> 00:10:22,000
the equation of motion that you
have to use to describe that

138
00:10:22,000 --> 00:10:26,000
particle moving within some
potential field U of x

139
00:10:26,000 --> 00:10:30,000
or U, you have to us this
equation of motion and not

140
00:10:30,000 --> 00:10:35,000
Newton's equations.
Newton's equations don't work

141
00:10:35,000 --> 00:10:41,000
to describe the motion of any
particle whose wavelength is on

142
00:10:41,000 --> 00:10:45,000
the order of the size of the
environment.

143
00:10:45,000 --> 00:10:49,000
It just does not work.
Now, just as an aside,

144
00:10:49,000 --> 00:10:56,000
classical mechanics really is
embedded in quantum mechanics.

145
00:10:56,000 --> 00:11:00,000
That is, if you took a problem
and solved it quantum

146
00:11:00,000 --> 00:11:04,000
mechanically,
and you solved the problem,

147
00:11:04,000 --> 00:11:08,000
a problem for which the
wavelength of a particle was

148
00:11:08,000 --> 00:11:13,000
much, much greater than the size
of the environment,

149
00:11:13,000 --> 00:11:18,000
which is the classical limit,
quantum mechanics would give

150
00:11:18,000 --> 00:11:21,000
you the right answer.
In other words,

151
00:11:21,000 --> 00:11:26,000
say you took some problem where
the wavelength of the particle

152
00:11:26,000 --> 00:11:32,000
is larger than the size of the
environment.

153
00:11:32,000 --> 00:11:35,000
That is, a problem where you
would normally use classical

154
00:11:35,000 --> 00:11:37,000
mechanics.
But if you use quantum

155
00:11:37,000 --> 00:11:40,000
mechanics, you would get the
right answer,

156
00:11:40,000 --> 00:11:44,000
if you could solve the problem
because the equations are very

157
00:11:44,000 --> 00:11:45,000
difficult.
But, in principle,

158
00:11:45,000 --> 00:11:49,000
you would get the right answer.
However, if you took a quantum

159
00:11:49,000 --> 00:11:52,000
mechanical problem,
that is, a problem where the

160
00:11:52,000 --> 00:11:56,000
wavelength of the particle is on
the order of the size of the

161
00:11:56,000 --> 00:11:59,000
environment and you used
classical mechanics,

162
00:11:59,000 --> 00:12:03,000
well, you won't get the right
answer.

163
00:12:03,000 --> 00:12:08,000
Because classical mechanics is,
in a sense, a subset.

164
00:12:08,000 --> 00:12:12,000
It is contained within quantum
mechanics.

165
00:12:12,000 --> 00:12:15,000
It is a limit of quantum
mechanics.

166
00:12:15,000 --> 00:12:20,000
We have to learn a new kind of
mechanics, here,

167
00:12:20,000 --> 00:12:24,000
this mechanics for the motion
of waves.

168
00:12:24,000 --> 00:12:29,000
Now, for a hydrogen atom,
we have to think of the wave

169
00:12:29,000 --> 00:12:35,000
function in three dimensions
instead of just one dimension,

170
00:12:35,000 --> 00:12:39,000
here.
And so we are going to have to

171
00:12:39,000 --> 00:12:44,000
describe the particle in terms
of three position coordinates.

172
00:12:44,000 --> 00:12:47,000
Usually you use Cartesian
coordinates, x,

173
00:12:47,000 --> 00:12:50,000
y, and z.
But this problem is solvable

174
00:12:50,000 --> 00:12:54,000
exactly if we use spherical
coordinates.

175
00:12:54,000 --> 00:12:59,000
How many of you had spherical
coordinates before and know what

176
00:12:59,000 --> 00:13:03,000
we are talking about?
Not everybody.

177
00:13:03,000 --> 00:13:06,000
If I gave you an x,
y and z for this electron in

178
00:13:06,000 --> 00:13:11,000
this atom, where the nucleus was
pinned at the origin.

179
00:13:11,000 --> 00:13:14,000
If I gave you an x,
y and z coordinate,

180
00:13:14,000 --> 00:13:17,000
you would know where that
electron was.

181
00:13:17,000 --> 00:13:21,000
But, alternatively,
I could tell you the position

182
00:13:21,000 --> 00:13:24,000
of this electron using spherical
coordinates.

183
00:13:24,000 --> 00:13:29,000
That is, I could tell you what
the distance is of the electron

184
00:13:29,000 --> 00:13:34,000
from the nucleus.
I am going to call that r.

185
00:13:34,000 --> 00:13:36,000
That will be one of the
variables.

186
00:13:36,000 --> 00:13:40,000
I could then also tell you this
angle theta.

187
00:13:40,000 --> 00:13:43,000
Theta is the angle that r makes
from the z-axis.

188
00:13:43,000 --> 00:13:47,000
That is the second coordinate.
And then, finally,

189
00:13:47,000 --> 00:13:52,000
the third coordinate is phi.
Phi is the angle made by the

190
00:13:52,000 --> 00:13:55,000
following.
If I take the electron and drop

191
00:13:55,000 --> 00:14:00,000
it perpendicular to the
x,y-plane and then I draw a line

192
00:14:00,000 --> 00:14:04,000
here, well, the angle between
that line and the x-axis is the

193
00:14:04,000 --> 00:14:09,000
angle phi.
Instead of giving you x,

194
00:14:09,000 --> 00:14:12,000
y, and z, I am just going to
give you r, theta,

195
00:14:12,000 --> 00:14:15,000
and phi in spherical
coordinates.

196
00:14:15,000 --> 00:14:20,000
And now, this wave function is
also a function of time,

197
00:14:20,000 --> 00:14:25,000
and I will talk about that a
little bit probably next time.

198
00:14:25,000 --> 00:14:30,000
So, that is the wave function
that is in some way going to

199
00:14:30,000 --> 00:14:35,000
represent our electron.
I haven't told you exactly yet

200
00:14:35,000 --> 00:14:39,000
how Psi represents the electron,
and I won't tell you that for

201
00:14:39,000 --> 00:14:42,000
another few days.
Question?

202
00:14:55,000 --> 00:14:59,000
No.
I actually thought this was the

203
00:14:59,000 --> 00:15:04,000
way the book set it up.
I may have it backwards.

204
00:15:09,000 --> 00:15:12,000
I think this is the way our
book sets it up.

205
00:15:12,000 --> 00:15:16,000
It won't make a difference in
any of the problems we solve

206
00:15:16,000 --> 00:15:18,000
here.
But, in general,

207
00:15:18,000 --> 00:15:23,000
if you are given a problem to
solve, you are going to have to

208
00:15:23,000 --> 00:15:28,000
look and see how they define
their coordinate system.

209
00:15:28,000 --> 00:15:33,000
Actually, somebody else asked
me that the other day.

210
00:15:33,000 --> 00:15:39,000
Now, what we have to do is
actually set up the Hamiltonian

211
00:15:39,000 --> 00:15:45,000
for the hydrogen atom.
That is, we have to set up the

212
00:15:45,000 --> 00:15:50,000
Hamiltonian specifically for the
hydrogen atom.

213
00:15:50,000 --> 00:15:56,000
And we have to set it up in
terms of spherical coordinates,

214
00:15:56,000 --> 00:16:01,000
r, theta, and phi.
And when we do that,

215
00:16:01,000 --> 00:16:06,000
and we are not actually going
to do that, that is what the

216
00:16:06,000 --> 00:16:10,000
Hamiltonian looks like.
Actually, this is in three

217
00:16:10,000 --> 00:16:14,000
dimensions, and so if I were
doing it in the x,

218
00:16:14,000 --> 00:16:20,000
y, z, what I would have here is
a second derivative with respect

219
00:16:20,000 --> 00:16:24,000
to y, plus a second derivative
with respect to z.

220
00:16:24,000 --> 00:16:29,000
That is what it would look like
in three dimensions in Cartesian

221
00:16:29,000 --> 00:16:33,000
coordinates.
But when I transform from

222
00:16:33,000 --> 00:16:37,000
Cartesian coordinates to
spherical coordinates,

223
00:16:37,000 --> 00:16:41,000
an exercise that takes five
pages, and everybody should have

224
00:16:41,000 --> 00:16:45,000
that experience once in their
life, but maybe now is not the

225
00:16:45,000 --> 00:16:48,000
right time for that experience,
you do get this.

226
00:16:48,000 --> 00:16:51,000
And essentially,
the Hamiltonian is a sum of

227
00:16:51,000 --> 00:16:56,000
second derivatives with respect
to each one of the coordinates,

228
00:16:56,000 --> 00:17:00,000
because essentially this term
is a second derivative with

229
00:17:00,000 --> 00:17:03,000
respect to r.
This term is a second

230
00:17:03,000 --> 00:17:06,000
derivative with respect to
theta.

231
00:17:06,000 --> 00:17:10,000
This term is a second
derivative with respect to phi.

232
00:17:10,000 --> 00:17:12,000
That is the specific
Hamiltonian.

233
00:17:12,000 --> 00:17:17,000
These are all kinetic energy
operators, as you will learn

234
00:17:17,000 --> 00:17:20,000
later on.
And then there is this term

235
00:17:20,000 --> 00:17:22,000
here, U of r.
This U of r,

236
00:17:22,000 --> 00:17:24,000
what is that?
What is U(r)?

237
00:17:24,000 --> 00:17:27,000
Potential energy.
It is the Coulomb potential

238
00:17:27,000 --> 00:17:32,000
energy of interaction.
It is isotropic,

239
00:17:32,000 --> 00:17:35,000
meaning it is the same at all
angles.

240
00:17:35,000 --> 00:17:40,000
The only thing it depends on is
the distance of the electron

241
00:17:40,000 --> 00:17:43,000
from the nucleus.
It only depends on r.

242
00:17:43,000 --> 00:17:46,000
It doesn't depend on theta and
phi.

243
00:17:46,000 --> 00:17:50,000
So, that is the Schrödinger
equation for the hydrogen atom.

244
00:17:50,000 --> 00:17:55,000
It is a differential equation,
second-order ordinary

245
00:17:55,000 --> 00:18:00,000
differential equation.
In 18.03, you are going to

246
00:18:00,000 --> 00:18:04,000
learn how to solve that.
In 5.61, or 8.04 I think it is,

247
00:18:04,000 --> 00:18:08,000
in physics, in quantum
mechanics, you are going to

248
00:18:08,000 --> 00:18:10,000
solve that.
You can do that.

249
00:18:10,000 --> 00:18:14,000
It is not hard.
But what do I mean when I say

250
00:18:14,000 --> 00:18:16,000
solve?
What I mean is that we are

251
00:18:16,000 --> 00:18:21,000
going to calculate these Es,
these binding energies,

252
00:18:21,000 --> 00:18:23,000
this constant in front of the
psi.

253
00:18:23,000 --> 00:18:28,000
That is called an eigenvalue,
for those of you who might know

254
00:18:28,000 --> 00:18:35,000
something already about these
kinds of differential equations.

255
00:18:35,000 --> 00:18:40,000
This E is the binding energy of
the electron to the nucleus.

256
00:18:40,000 --> 00:18:45,000
We are going to look at those
results in just a moment.

257
00:18:45,000 --> 00:18:50,000
And the other quantity we are
going to solve for,

258
00:18:50,000 --> 00:18:53,000
here, is psi.
What we are going to want to

259
00:18:53,000 --> 00:19:00,000
find is the actual functional
form for the wave functions.

260
00:19:00,000 --> 00:19:04,000
That, we can get out of solving
this differential equation.

261
00:19:04,000 --> 00:19:07,000
The actual functional form for
psi.

262
00:19:07,000 --> 00:19:13,000
The functional form is going to
be more complicated than what I

263
00:19:13,000 --> 00:19:18,000
wrote here, but we can get that.
We will look at that in a few

264
00:19:18,000 --> 00:19:22,000
days from now.
And do you know what those wave

265
00:19:22,000 --> 00:19:25,000
functions are?
They are what you studied in

266
00:19:25,000 --> 00:19:31,000
high school as orbitals.
You talked about an s-orbital,

267
00:19:31,000 --> 00:19:34,000
p-orbital, d-orbital.
Orbitals are wave functions.

268
00:19:34,000 --> 00:19:38,000
That where they come from,
orbitals, from solving the

269
00:19:38,000 --> 00:19:41,000
Schrödinger equation.
Now, specifically,

270
00:19:41,000 --> 00:19:45,000
orbitals are the spatial part
of a wave function.

271
00:19:45,000 --> 00:19:48,000
There is also a spin part to
the wave function,

272
00:19:48,000 --> 00:19:52,000
but for all intents and
purposes, we are going to use

273
00:19:52,000 --> 00:19:56,000
the term orbital and wave
function interchangeably because

274
00:19:56,000 --> 00:20:00,000
they are the same thing.
Question?

275
00:20:00,000 --> 00:20:03,000
We are going to get to that in
just a moment,

276
00:20:03,000 --> 00:20:09,000
and if you think I don't answer
it then we will go back to it.

277
00:20:09,000 --> 00:20:13,000
We are going to solve this.
And this equation is going to

278
00:20:13,000 --> 00:20:18,000
predict binding energies,
and it is going to predict

279
00:20:18,000 --> 00:20:22,000
these wave functions in
agreement with our observations,

280
00:20:22,000 --> 00:20:29,000
and it is going to predict that
the hydrogen atom is stable.

281
00:20:29,000 --> 00:20:33,000
Remember when we tried to
predict the hydrogen atom using

282
00:20:33,000 --> 00:20:36,000
classical ideas?
We found it lived a whole

283
00:20:36,000 --> 00:20:40,000
whopping 10^-10 seconds.
Not so when we treat the

284
00:20:40,000 --> 00:20:45,000
hydrogen atom with the quantum
mechanical equations of motion.

285
00:20:45,000 --> 00:20:50,000
Let's write down the results
for solving the Schrödinger

286
00:20:50,000 --> 00:20:53,000
equation.
And the part I am going to

287
00:20:53,000 --> 00:20:58,000
concentrate on today is these
binding energies.

288
00:20:58,000 --> 00:21:00,000
And, on Friday,
we will talk about the wave

289
00:21:00,000 --> 00:21:03,000
function, solving the
Schrödinger equation for those

290
00:21:03,000 --> 00:21:05,000
wave functions.

291
00:21:16,000 --> 00:21:19,000
What does the binding energy
look like?

292
00:21:19,000 --> 00:21:23,000
Well, the binding energies,
here, coming out of the

293
00:21:23,000 --> 00:21:28,000
Schrödinger equation look like
this, minus 1 over n squared

294
00:21:28,000 --> 00:21:32,000
times m e to the 4 over 8
epsilon nought squared time h

295
00:21:32,000 --> 00:21:35,000
squared.

296
00:21:35,000 --> 00:21:40,000
m is the mass of the electron.

297
00:21:40,000 --> 00:21:43,000
e is the charge on the
electron.

298
00:21:43,000 --> 00:21:48,000
Epsilon nought is this
permittivity of vacuum we talked

299
00:21:48,000 --> 00:21:52,000
about before.
This is a conversion between

300
00:21:52,000 --> 00:21:56,000
ESU units and SI units.
h is Planck's constant.

301
00:21:56,000 --> 00:21:59,000
Planck's constant is
ubiquitous.

302
00:21:59,000 --> 00:22:05,000
It is everywhere.
What we typically do is we lump

303
00:22:05,000 --> 00:22:11,000
these constants together.
And we call those constants a

304
00:22:11,000 --> 00:22:15,000
new constant,
the Rydberg constant,

305
00:22:15,000 --> 00:22:21,000
R sub H.
And so our expression is equal

306
00:22:21,000 --> 00:22:28,000
to minus R sub H over n squared.

307
00:22:28,000 --> 00:22:35,000
The value of R sub H is equal
to 2.17987x10^-18 joules.

308
00:22:35,000 --> 00:22:43,000
That is a number that you are
going to use a lot in the next

309
00:22:43,000 --> 00:22:48,000
few days.
But what you also see,

310
00:22:48,000 --> 00:22:51,000
here, is an n.
What is n?

311
00:22:51,000 --> 00:23:00,000
Well, n is what we call the
principle quantum number.

312
00:23:00,000 --> 00:23:07,000
And its allowed values are
integers, where the integers

313
00:23:07,000 --> 00:23:13,000
start with 1,
2, 3, all the way up to n is

314
00:23:13,000 --> 00:23:19,000
equal to infinity.
Well, let's try to understand

315
00:23:19,000 --> 00:23:26,000
this a little bit more by
looking at an energy level

316
00:23:26,000 --> 00:23:32,000
diagram again.
I am just plotting here energy.

317
00:23:32,000 --> 00:23:38,000
Here is the zero of energy.
Here is the expression for the

318
00:23:38,000 --> 00:23:43,000
energy levels that come out of
the Schrödinger equation.

319
00:23:43,000 --> 00:23:48,000
When n is equal to one,
that is the lowest allowed

320
00:23:48,000 --> 00:23:54,000
value for n, binding energy is
essentially equal to minus the

321
00:23:54,000 --> 00:23:58,000
Rydberg constant.
**E = -(R)H**

322
00:23:58,000 --> 00:24:02,000
But our equation says the
binding energy of the electron

323
00:24:02,000 --> 00:24:07,000
can also be this value because
when n is equal to 2,

324
00:24:07,000 --> 00:24:11,000
well, the binding energy now is
only a quarter of the Rydberg

325
00:24:11,000 --> 00:24:15,000
constant. It is higher

326
00:24:15,000 --> 00:24:17,000
in energy.
When n is equal to 3,

327
00:24:17,000 --> 00:24:22,000
well, the binding energy of
that electron to the nucleus is

328
00:24:22,000 --> 00:24:26,000
a ninth to the Rydberg constant.
When it is equal to 4,

329
00:24:26,000 --> 00:24:30,000
it is a sixteenth.
When it is equal to 5,

330
00:24:30,000 --> 00:24:34,000
it is a twenty-fifth.
When it is equal to six,

331
00:24:34,000 --> 00:24:37,000
it is a thirty-sixth.
So on, so on,

332
00:24:37,000 --> 00:24:40,000
and so on until it gets n equal
to infinity.

333
00:24:40,000 --> 00:24:45,000
When n is equal to infinity,
then the binding energy is

334
00:24:45,000 --> 00:24:47,000
zero.
When n is equal to infinity,

335
00:24:47,000 --> 00:24:51,000
the electron and the nucleus
are no longer bound.

336
00:24:51,000 --> 00:24:54,000
They are separated from each
other.

337
00:24:54,000 --> 00:25:00,000
They don't hang together when n
is equal to infinity.

338
00:25:00,000 --> 00:25:05,000
Now, this is rather peculiar.
This is saying that the binding

339
00:25:05,000 --> 00:25:10,000
energy of the electron to the
nucleus can have essentially an

340
00:25:10,000 --> 00:25:16,000
infinite number of values,
except it is a discrete number

341
00:25:16,000 --> 00:25:21,000
of infinite numbers of values.
In the sense that the binding

342
00:25:21,000 --> 00:25:27,000
energy can be this or this or
this, but it cannot be something

343
00:25:27,000 --> 00:25:31,000
in between, here or here or
here.

344
00:25:31,000 --> 00:25:34,000
This is the quantum nature of
the hydrogen atom.

345
00:25:34,000 --> 00:25:37,000
There are allowed energy
levels.

346
00:25:37,000 --> 00:25:40,000
Where did this quantization
come from?

347
00:25:40,000 --> 00:25:44,000
It came from solving the
Schrödinger equation.

348
00:25:44,000 --> 00:25:49,000
When you solve a differential
equation, as you will learn,

349
00:25:49,000 --> 00:25:54,000
and that differential equation
applies to some physical

350
00:25:54,000 --> 00:25:59,000
problem, in order to make that
differential equation specific

351
00:25:59,000 --> 00:26:03,000
to your physical problem,
you often have to apply

352
00:26:03,000 --> 00:26:09,000
something called boundary
conditions to the problem.

353
00:26:09,000 --> 00:26:13,000
When you do that,
that is when this quantization

354
00:26:13,000 --> 00:26:16,000
comes out.
It drops out of solving the

355
00:26:16,000 --> 00:26:21,000
differential equation.
What are boundary conditions?

356
00:26:21,000 --> 00:26:27,000
Well, remember I told you in
the spherical coordinate system,

357
00:26:27,000 --> 00:26:30,000
r, theta, phi?
Well, phi sweeps from zero to

358
00:26:30,000 --> 00:26:35,000
360 degrees.
Well, if you just went 90

359
00:26:35,000 --> 00:26:38,000
degrees further,
so if you went to 450 degrees,

360
00:26:38,000 --> 00:26:43,000
you really have the same
situation as you had when phi

361
00:26:43,000 --> 00:26:47,000
was equal to 90 degrees.
What you have to do in a

362
00:26:47,000 --> 00:26:51,000
differential equation,
which usually has a series as a

363
00:26:51,000 --> 00:26:54,000
solution, is that you have to
cut it off.

364
00:26:54,000 --> 00:26:57,000
You have to apply boundary
conditions.

365
00:26:57,000 --> 00:27:02,000
You have to cut it off at
degrees.

366
00:27:02,000 --> 00:27:06,000
Because otherwise you just have
the same problem that you had

367
00:27:06,000 --> 00:27:09,000
before.
This is a cyclical boundary

368
00:27:09,000 --> 00:27:12,000
condition.
And it is that cutting it off

369
00:27:12,000 --> 00:27:16,000
to make the equation be really
pertinent or apply to your

370
00:27:16,000 --> 00:27:20,000
physical problem,
that is what leads to these

371
00:27:20,000 --> 00:27:23,000
boundary conditions,
mathematically.

372
00:27:23,000 --> 00:27:27,000
That is where it comes from.
That is where all the

373
00:27:27,000 --> 00:27:32,000
quantization comes from.
Now, let's talk a little bit

374
00:27:32,000 --> 00:27:38,000
about the significance here of
these binding energies because

375
00:27:38,000 --> 00:27:41,000
somebody asked me about it
already.

376
00:27:41,000 --> 00:27:47,000
When the electron is bound to
the nucleus with this much

377
00:27:47,000 --> 00:27:52,000
energy, we say that the electron
is in the n equals 1 state,

378
00:27:52,000 --> 00:27:57,000
or equivalently,
we say that the hydrogen atom

379
00:27:57,000 --> 00:28:02,000
is in the n equals 1 state.
We use both kinds of

380
00:28:02,000 --> 00:28:07,000
expressions equivalently.
When the hydrogen atom or the

381
00:28:07,000 --> 00:28:13,000
electron is in the n equals 1
state, we call that the ground

382
00:28:13,000 --> 00:28:16,000
state.
The ground state is the lowest

383
00:28:16,000 --> 00:28:20,000
energy state.
The electron is most strongly

384
00:28:20,000 --> 00:28:23,000
bound there.
The binding energy is most

385
00:28:23,000 --> 00:28:26,000
negative.
And the physical significance

386
00:28:26,000 --> 00:28:32,000
of that binding energy is that
it is minus the ionization

387
00:28:32,000 --> 00:28:35,000
energy.
Or, alternatively,

388
00:28:35,000 --> 00:28:40,000
the ionization energy is minus
the binding energy.

389
00:28:40,000 --> 00:28:46,000
It is going to require this
much energy, from here to here,

390
00:28:46,000 --> 00:28:50,000
to rip the electron off of the
nucleus.

391
00:28:50,000 --> 00:28:56,000
The binding energy here is
minus the ionization energy.

392
00:28:56,000 --> 00:29:01,000
That is the physical
significance of it.

393
00:29:01,000 --> 00:29:04,000
Now, did you ask me about the
work function?

394
00:29:04,000 --> 00:29:07,000
Okay.
The work function is the

395
00:29:07,000 --> 00:29:11,000
ionization energy when we talk
about a solid.

396
00:29:11,000 --> 00:29:15,000
That is just a terminology,
that is historical.

397
00:29:15,000 --> 00:29:20,000
When we talk about ripping and
electron off of a solid,

398
00:29:20,000 --> 00:29:25,000
we call it the work function.
When we talk about ripping it

399
00:29:25,000 --> 00:29:30,000
off of an atom or a molecule,
we call it the ionization

400
00:29:30,000 --> 00:29:34,000
energy.
And the other important thing

401
00:29:34,000 --> 00:29:39,000
to know here is that when we
talk about ionization energy,

402
00:29:39,000 --> 00:29:44,000
we are usually talking about
the energy required to pull the

403
00:29:44,000 --> 00:29:50,000
electron off when the molecule
or the atom is in the lowest

404
00:29:50,000 --> 00:29:52,000
energy state,
the ground state.

405
00:29:52,000 --> 00:29:57,000
That is also important.
But our equations tell us we

406
00:29:57,000 --> 00:30:03,000
also can have a hydrogen atom in
the n equals 2 state.

407
00:30:03,000 --> 00:30:06,000
If the hydrogen atom is in the
n equals 2 state,

408
00:30:06,000 --> 00:30:11,000
it is in an excited state.
Actually, it is the first

409
00:30:11,000 --> 00:30:14,000
excited state.
The electron is bound less

410
00:30:14,000 --> 00:30:17,000
strongly.
It is bound less strongly,

411
00:30:17,000 --> 00:30:21,000
and consequently,
the ionization energy from an

412
00:30:21,000 --> 00:30:26,000
excited state hydrogen atom is
less because the electron is not

413
00:30:26,000 --> 00:30:30,000
bound so strongly.
And, of course,

414
00:30:30,000 --> 00:30:34,000
we could have a hydrogen at n
equals 3, n equals 4 and n

415
00:30:34,000 --> 00:30:38,000
equals 5, any of these allowed
energy levels.

416
00:30:38,000 --> 00:30:42,000
Not all at the same time,
but, at any given time,

417
00:30:42,000 --> 00:30:47,000
if you had a lot of hydrogen
atoms, you could have hydrogen

418
00:30:47,000 --> 00:30:50,000
atoms in all of these different
states.

419
00:30:50,000 --> 00:30:54,000
Now, the other point that I
want to make is that this

420
00:30:54,000 --> 00:30:58,000
Schrödinger result,
here, for the energy levels

421
00:30:58,000 --> 00:31:04,000
predicts the energy levels of
all one electron atoms.

422
00:31:04,000 --> 00:31:07,000
What is a one electron atom?
Well, helium plus,

423
00:31:07,000 --> 00:31:11,000
that is a one electron atom or
a one electron ion.

424
00:31:11,000 --> 00:31:15,000
Helium usually has two
electrons, but if we pull one

425
00:31:15,000 --> 00:31:19,000
off it only has one left,
so it is a one electron atom.

426
00:31:19,000 --> 00:31:23,000
Lithium double plus
is a one electron atom because

427
00:31:23,000 --> 00:31:26,000
we pulled two of its three
electrons off.

428
00:31:26,000 --> 00:31:30,000
One electron is left,
and that is a one electron atom

429
00:31:30,000 --> 00:31:34,000
or an ion.
Uranium plus 91 is

430
00:31:34,000 --> 00:31:38,000
a one electron atom.
We pulled 91 electrons off,

431
00:31:38,000 --> 00:31:42,000
we have one left,
and that is a one electron

432
00:31:42,000 --> 00:31:45,000
atom.
The Schrödinger equation will

433
00:31:45,000 --> 00:31:51,000
predict what those energy levels
are, as long as you remember the

434
00:31:51,000 --> 00:31:53,000
Z squared up here.
For hydrogen,

435
00:31:53,000 --> 00:31:56,000
Z is 1, and so it doesn't
appear.

436
00:31:56,000 --> 00:32:03,000
But Z is not one for all of
these other one electron atoms.

437
00:32:03,000 --> 00:32:07,000
Where does the Z come from?
It comes from the Coulomb

438
00:32:07,000 --> 00:32:10,000
interaction.
That potential energy of

439
00:32:10,000 --> 00:32:15,000
interaction is the charge on the
electron times the charge on the

440
00:32:15,000 --> 00:32:19,000
nucleus, which is Z times e.
That is where the Z squared

441
00:32:19,000 --> 00:32:22,000
comes from.
We have to remember that.

442
00:32:22,000 --> 00:32:27,000
Now, how do we know that the
Schrödinger's predictions for

443
00:32:27,000 --> 00:32:32,000
these energy levels are correct,
are accurate?

444
00:32:32,000 --> 00:32:34,000
Well, we have to do an
experiment.

445
00:32:34,000 --> 00:32:39,000
The experiment we are going to
do is we are going to take a

446
00:32:39,000 --> 00:32:43,000
bulb here, pump it out,
and then we are going to fill

447
00:32:43,000 --> 00:32:47,000
it up with molecular hydrogen.
And then in this bulb,

448
00:32:47,000 --> 00:32:51,000
there is a negative and a
positive electrode.

449
00:32:51,000 --> 00:32:55,000
What we are going to do is
crank up the potential energy

450
00:32:55,000 --> 00:33:00,000
difference between those two so
high until finally that gas is

451
00:33:00,000 --> 00:33:04,000
going to ignite,
just like that.

452
00:33:04,000 --> 00:33:08,000
And we are going to look at the
light coming out.

453
00:33:08,000 --> 00:33:13,000
And what we are going to do is
we are going to disperse that

454
00:33:13,000 --> 00:33:16,000
light.
We are going to send it through

455
00:33:16,000 --> 00:33:19,000
a diffraction grating,
essentially.

456
00:33:19,000 --> 00:33:22,000
And that is like an array of
slits.

457
00:33:22,000 --> 00:33:26,000
What you are going to see are
points of constructive and

458
00:33:26,000 --> 00:33:31,000
destructive interference.
But in the constructive

459
00:33:31,000 --> 00:33:34,000
interference,
you are going to see the color

460
00:33:34,000 --> 00:33:37,000
separated.
There is going to be purple,

461
00:33:37,000 --> 00:33:38,000
blue, green,
etc.

462
00:33:38,000 --> 00:33:41,000
The reason is that the
different radiation,

463
00:33:41,000 --> 00:33:45,000
the different colors have
different wavelengths.

464
00:33:45,000 --> 00:33:48,000
They have different
wavelengths, then they have

465
00:33:48,000 --> 00:33:52,000
slightly different points in
space at which constructive

466
00:33:52,000 --> 00:33:56,000
interference occurs,
and so the light is dispersed

467
00:33:56,000 --> 00:34:00,000
in space.
We are going to analyze what

468
00:34:00,000 --> 00:34:04,000
the wavelengths of the light
that are being emitted from

469
00:34:04,000 --> 00:34:08,000
these hydrogen atoms are.
See, the discharge pulls apart

470
00:34:08,000 --> 00:34:12,000
the H two and makes
hydrogen atoms.

471
00:34:12,000 --> 00:34:16,000
We have to do this by taking a
diffraction grating.

472
00:34:16,000 --> 00:34:20,000
The TAs are going to give you a
pair of diffraction grating

473
00:34:20,000 --> 00:34:23,000
glasses.
You are supposed to come and

474
00:34:23,000 --> 00:34:28,000
look at the discharge here.
You can see it.

475
00:34:28,000 --> 00:34:32,000
I will turn it a little bit for
those of you on the sides here.

476
00:34:32,000 --> 00:34:35,000
And we will see what we see.

477
00:35:00,000 --> 00:35:03,000
If you cannot see the lamp,
you are welcome to get up and

478
00:35:03,000 --> 00:35:06,000
move around so that you can see
it.

479
00:35:20,000 --> 00:35:21,000
Can you do this light,
too?

480
00:35:21,000 --> 00:35:23,000
This one over here,
sir.

481
00:35:23,000 --> 00:35:26,000
Can you move this one away?

482
00:35:35,000 --> 00:35:38,000
If you look up at the lights in
the room, you can see the whole

483
00:35:38,000 --> 00:35:40,000
spectrum, because that is white
light.

484
00:36:08,000 --> 00:36:13,000
I am going to turn the lamp
over here so that you can see

485
00:36:13,000 --> 00:36:18,000
this a little better.
The spectrum that you should

486
00:36:18,000 --> 00:36:21,000
see is what is shown on the
board.

487
00:36:21,000 --> 00:36:27,000
If you look to the right of the
lamp, here, you should see a

488
00:36:27,000 --> 00:36:32,000
purple line.
The purple line is actually

489
00:36:32,000 --> 00:36:36,000
very light, so only if you are
up close are you going to see

490
00:36:36,000 --> 00:36:39,000
the purple line.
You can see the blue line very

491
00:36:39,000 --> 00:36:41,000
well.
That is very intense.

492
00:36:41,000 --> 00:36:44,000
The green line is also very
diffuse.

493
00:36:44,000 --> 00:36:48,000
Again, only if you are close
are you going to be able to see

494
00:36:48,000 --> 00:36:52,000
the green line.
And then, the red line is very

495
00:36:52,000 --> 00:36:55,000
bright.
And now I am going to move this

496
00:36:55,000 --> 00:37:00,000
over here so you have the
opportunity to see it.

497
00:37:00,000 --> 00:37:09,000
Again, you are welcome to get
out of your seats and move

498
00:37:09,000 --> 00:37:15,000
around so that you can,
in fact, see it.

499
00:37:15,000 --> 00:37:24,000
What you should be seeing,
hey, interference phenomena

500
00:37:24,000 --> 00:37:29,000
works.
Useful pointer.

501
00:37:40,000 --> 00:37:43,000
What you should see,
depending on which one of the

502
00:37:43,000 --> 00:37:48,000
constructive interference
patterns you are looking at,

503
00:37:48,000 --> 00:37:53,000
there should be a purple line,
there should be a blue line,

504
00:37:53,000 --> 00:37:55,000
very intense,
purple is weak,

505
00:37:55,000 --> 00:38:00,000
green is rather week,
and the red is very intense.

506
00:38:00,000 --> 00:38:02,000
What is happening,
here?

507
00:38:02,000 --> 00:38:06,000
Well, what is happening is that
in this discharge,

508
00:38:06,000 --> 00:38:12,000
there is enough energy to put
some of the hydrogen atoms in a

509
00:38:12,000 --> 00:38:17,000
high energy state.
And we will call that state E

510
00:38:17,000 --> 00:38:21,000
sub i,
the energy of the initial

511
00:38:21,000 --> 00:38:24,000
state.
Actually, that is an unstable

512
00:38:24,000 --> 00:38:28,000
situation.
That hydrogen atom wants to

513
00:38:28,000 --> 00:38:32,000
relax.
It wants to be in the lower

514
00:38:32,000 --> 00:38:36,000
energy state.
And what happens is that it

515
00:38:36,000 --> 00:38:38,000
does relax.
It relaxes.

516
00:38:38,000 --> 00:38:42,000
The electron falls into the
lower energy state,

517
00:38:42,000 --> 00:38:47,000
but, because it is lower
energy, it has to give up a

518
00:38:47,000 --> 00:38:50,000
photon.
And so the photon that is

519
00:38:50,000 --> 00:38:56,000
emitted, the energy of that
photon has to be exactly the

520
00:38:56,000 --> 00:39:02,000
difference in energy between the
energy of the initial state and

521
00:39:02,000 --> 00:39:07,000
the final state.
That is the quantum nature of

522
00:39:07,000 --> 00:39:12,000
the hydrogen atom.
The photon that comes out has

523
00:39:12,000 --> 00:39:15,000
to have that energy difference
exactly.

524
00:39:15,000 --> 00:39:18,000
And, therefore,
the frequency,

525
00:39:18,000 --> 00:39:24,000
here, of the radiation coming
out is going to correspond to

526
00:39:24,000 --> 00:39:29,000
that exact energy difference.
And, for the different

527
00:39:29,000 --> 00:39:33,000
energies, for the different
transitions, you are going to

528
00:39:33,000 --> 00:39:38,000
have very specific values of the
frequency of the radiation or

529
00:39:38,000 --> 00:39:41,000
the wavelength of the radiation.
For example,

530
00:39:41,000 --> 00:39:46,000
some of those hydrogen atoms in
this discharge have been excited

531
00:39:46,000 --> 00:39:49,000
to this excited state,
which we will call B.

532
00:39:49,000 --> 00:39:54,000
The energy difference between B
and this ground state here is

533
00:39:54,000 --> 00:39:58,000
small.
Therefore, we are going to have

534
00:39:58,000 --> 00:40:01,000
some radiation that is low
frequency because delta E,

535
00:40:01,000 --> 00:40:05,000
the difference in the energy
between the two states is small.

536
00:40:05,000 --> 00:40:09,000
And then there are going to be
some hydrogen atoms that are

537
00:40:09,000 --> 00:40:12,000
going to be excited to this
state, the A state.

538
00:40:12,000 --> 00:40:15,000
And, relatively speaking,
that energy difference is

539
00:40:15,000 --> 00:40:17,000
large.
There are going to be some

540
00:40:17,000 --> 00:40:21,000
hydrogen atoms that are going to
be relaxing to this ground

541
00:40:21,000 --> 00:40:23,000
state.
And when they do,

542
00:40:23,000 --> 00:40:27,000
since that energy difference is
large, the frequency of the

543
00:40:27,000 --> 00:40:31,000
radiation coming off is going to
be high.

544
00:40:31,000 --> 00:40:35,000
Or, correspondingly,
the wavelength of the radiation

545
00:40:35,000 --> 00:40:39,000
coming off is going to be low
because the wavelength is

546
00:40:39,000 --> 00:40:42,000
inversely proportional to the
frequency.

547
00:40:42,000 --> 00:40:45,000
And, likewise,
for this transition we are

548
00:40:45,000 --> 00:40:50,000
going to have some long
wavelength radiation emitted.

549
00:40:50,000 --> 00:40:54,000
Well, let's see if we can
understand specifically the

550
00:40:54,000 --> 00:40:57,000
spectrum, here,
in the visible range for the

551
00:40:57,000 --> 00:41:03,000
hydrogen atom.
What I have done is to draw an

552
00:41:03,000 --> 00:41:09,000
energy level diagram again.
Here is the energy of n equals

553
00:41:09,000 --> 00:41:12,000
1, n equals 2,
n equals 3, etc.

554
00:41:12,000 --> 00:41:16,000
Here is the n equals infinity,
up here.

555
00:41:16,000 --> 00:41:22,000
It turns out that this purple
line is a transition from a

556
00:41:22,000 --> 00:41:30,000
hydrogen atom in the n equals 6
state to the n equals 2 state.

557
00:41:30,000 --> 00:41:36,000
That blue line is a transition
from n equals 5 to n equals 2,

558
00:41:36,000 --> 00:41:43,000
the green line is a transition
from n equals 4 to n equals 2

559
00:41:43,000 --> 00:41:48,000
and the red line from n equals 3
to n equals 2.

560
00:41:48,000 --> 00:41:54,000
Notice that since this energy
difference here is small,

561
00:41:54,000 --> 00:41:59,000
this line has a long
wavelength.

562
00:41:59,000 --> 00:42:04,000
Since this energy difference is
larger, this line has a shorter

563
00:42:04,000 --> 00:42:08,000
wavelength.
All of these transitions that

564
00:42:08,000 --> 00:42:13,000
you are seeing in the visible
range have the final state of n

565
00:42:13,000 --> 00:42:16,000
equals 2.
Now, how do we know that the

566
00:42:16,000 --> 00:42:22,000
Schrödinger equation is making
predictions that are consistent

567
00:42:22,000 --> 00:42:27,000
with the frequencies or the
wavelengths of the radiation

568
00:42:27,000 --> 00:42:32,000
that we observe here?
Well, we have got to do a plug

569
00:42:32,000 --> 00:42:34,000
here.
This is the frequency that we

570
00:42:34,000 --> 00:42:37,000
would expect.
It is the energy difference

571
00:42:37,000 --> 00:42:40,000
between two states over H.
Here is the initial energy.

572
00:42:40,000 --> 00:42:44,000
Here is the final energy.
We said that the Schrödinger

573
00:42:44,000 --> 00:42:48,000
equation tells us that the
energies of the state are minus

574
00:42:48,000 --> 00:42:50,000
R sub H over n.

575
00:42:50,000 --> 00:42:54,000
For the initial energy level,
it is minus R sub H over n sub

576
00:42:54,000 --> 00:42:57,000
i squared. We plug

577
00:42:57,000 --> 00:43:01,000
that in.
For the final energy level,

578
00:43:01,000 --> 00:43:05,000
well, it is minus R sub H over
n sub f,

579
00:43:05,000 --> 00:43:08,000
the final quantum number
squared.

580
00:43:08,000 --> 00:43:11,000
We plug that in.
We rearrange some things.

581
00:43:11,000 --> 00:43:15,000
And then this is the prediction
for the frequency.

582
00:43:15,000 --> 00:43:20,000
I said that this level right
here is the n equals 2 level,

583
00:43:20,000 --> 00:43:24,000
so we will plug in the two for
the final quantum state.

584
00:43:24,000 --> 00:43:28,000
And so this is then the
prediction for the frequencies

585
00:43:28,000 --> 00:43:33,000
of the radiation to the n equals
2 level.

586
00:43:33,000 --> 00:43:37,000
You just plug in n equals 6,
n equals 5, n equals 4,

587
00:43:37,000 --> 00:43:39,000
n equals 3.
And you know what?

588
00:43:39,000 --> 00:43:43,000
The frequencies that are
predicted match what we observe

589
00:43:43,000 --> 00:43:46,000
to one part in 10^8.
There really is precise

590
00:43:46,000 --> 00:43:51,000
agreement between the results of
the Schrödinger equation and

591
00:43:51,000 --> 00:43:55,000
what we actually observe in
nature, therefore,

592
00:43:55,000 --> 00:44:01,000
we think it is correct.
And there are no experiments

593
00:44:01,000 --> 00:44:06,000
that cast any doubt,
so far, on the Schrödinger

594
00:44:06,000 --> 00:44:09,000
equation.
Now, the set of transitions

595
00:44:09,000 --> 00:44:14,000
that we just looked at are these
transitions.

596
00:44:14,000 --> 00:44:20,000
Plotted here is another
energy-level diagram for the

597
00:44:20,000 --> 00:44:24,000
hydrogen atom.
Here is the n equals 1 state,

598
00:44:24,000 --> 00:44:30,000
n equals 2, n equals 3,
n equals 4.

599
00:44:30,000 --> 00:44:34,000
And we looked at all the
transitions that end up in the n

600
00:44:34,000 --> 00:44:36,000
equals 2 state.
This set of transitions.

601
00:44:36,000 --> 00:44:41,000
It is called the Balmer series.
Now, the n equals 2 state is

602
00:44:41,000 --> 00:44:44,000
not the ground state.
There is a transition from n

603
00:44:44,000 --> 00:44:47,000
equals 6 to n equals 1,
the ground state.

604
00:44:47,000 --> 00:44:50,000
It is this one right here.
But, you see,

605
00:44:50,000 --> 00:44:53,000
that is a very high energy
transition.

606
00:44:53,000 --> 00:44:57,000
Actually, this transition,
from n equals 6 to n equals 1,

607
00:44:57,000 --> 00:45:03,000
is in the ultraviolet range of
the electromagnetic spectrum.

608
00:45:03,000 --> 00:45:06,000
And so you cannot see it with
the experiment that we did,

609
00:45:06,000 --> 00:45:09,000
but it is there.
And then, of course,

610
00:45:09,000 --> 00:45:12,000
the hydrogen atoms that relaxed
to n equals 2,

611
00:45:12,000 --> 00:45:16,000
well, they actually eventually
relax to n equals 1.

612
00:45:16,000 --> 00:45:19,000
And there is a transition
there, it is over here,

613
00:45:19,000 --> 00:45:23,000
n equals 2 to n equals 1.
But, again, that is a high

614
00:45:23,000 --> 00:45:26,000
energy transition.
It is also in the ultraviolet

615
00:45:26,000 --> 00:45:30,000
range of the electromagnetic
spectrum.

616
00:45:30,000 --> 00:45:34,000
And so all of these transitions
that end up in n equals 1 are in

617
00:45:34,000 --> 00:45:36,000
the UV range.
They are called the Lyman

618
00:45:36,000 --> 00:45:38,000
series.
All the transitions that end up

619
00:45:38,000 --> 00:45:40,000
in n equals 2 are in the
visible.

620
00:45:40,000 --> 00:45:44,000
They are called the Balmer.
n equals 3, the Paschen,

621
00:45:44,000 --> 00:45:47,000
is in the infrared.
Brackett is in the infrared.

622
00:45:47,000 --> 00:45:50,000
Pfund is in the infrared.
And so we cannot see these

623
00:45:50,000 --> 00:45:52,000
easily.
Now, these are named for the

624
00:45:52,000 --> 00:45:55,000
different discoverers.
The reason there are so many

625
00:45:55,000 --> 00:46:01,000
discovers is because these are
all different frequency ranges.

626
00:46:01,000 --> 00:46:04,000
Different frequencies of
radiation require different

627
00:46:04,000 --> 00:46:08,000
kinds of detectors.
And so, depending on exactly

628
00:46:08,000 --> 00:46:11,000
what kind of detector the
experimentalist had,

629
00:46:11,000 --> 00:46:16,000
that will dictate then which
one of these sets of transitions

630
00:46:16,000 --> 00:46:20,000
he was able to discover.
But not only does this work for

631
00:46:20,000 --> 00:46:24,000
emission, but this also works
for absorption.

632
00:46:24,000 --> 00:46:28,000
That is, we can have a hydrogen
atom in the ground state,

633
00:46:28,000 --> 00:46:33,000
a low energy state.
And if there is a photon around

634
00:46:33,000 --> 00:46:37,000
that is exactly the energy
difference between these two

635
00:46:37,000 --> 00:46:40,000
states, well,
that photon can be absorbed.

636
00:46:40,000 --> 00:46:44,000
If that photon is a little bit
higher in energy,

637
00:46:44,000 --> 00:46:47,000
it won't be absorbed.
If it is a little lower,

638
00:46:47,000 --> 00:46:50,000
it won't be absorbed.
It has to be exactly the

639
00:46:50,000 --> 00:46:54,000
difference in the energies
between these two states.

640
00:46:54,000 --> 00:47:00,000
Again, that is the quantum
nature of the hydrogen atom.

641
00:47:00,000 --> 00:47:04,000
And now, if you are calculating
the frequency for absorption,

642
00:47:04,000 --> 00:47:09,000
what I have done here is I have
reversed n sub i and n

643
00:47:09,000 --> 00:47:12,000
sub f.
This is 1 over n sub i squared

644
00:47:12,000 --> 00:47:17,000
instead of 1 over
n sub f squared,

645
00:47:17,000 --> 00:47:21,000
which was the case in emission.
And I have reversed this so

646
00:47:21,000 --> 00:47:25,000
that you get a positive value
for the frequency of the

647
00:47:25,000 --> 00:47:28,000
radiation absorbed.
We will have two different

648
00:47:28,000 --> 00:47:33,000
equations for absorption and for
emission.

649
00:47:33,000 --> 00:47:36,000
So, those are the Schrödinger
equation results.

650
00:47:36,000 --> 00:47:40,000
I forgot to announce that there
is a forum this evening from

651
00:47:40,000 --> 00:47:44,000
5:00 to 6:00.
You should have signed up in

652
00:47:44,000 --> 00:47:45,000
recitation.
If you didn't,

653
00:47:45,000 --> 00:47:49,000
and still want to come,
send us an email and come.

654
00:47:49,000 --> 00:47:54,000
And we need the diffraction
glasses back as you are exiting.

655
00:47:54,000 --> 00:47:57,000
See you Friday.