1
00:00:11,170 --> 00:00:16,980
welcome back to the lecture for the
2
00:00:14,190 --> 00:00:19,450
introductory chemistry using
3
00:00:16,980 --> 00:00:22,599
Schroedinger and quantum mechanical
4
00:00:19,450 --> 00:00:25,000
methods for the atomic structure so what
5
00:00:22,599 --> 00:00:28,690
we would do in this and in the next
6
00:00:25,000 --> 00:00:33,129
segment is introduced the Schrodinger
7
00:00:28,690 --> 00:00:35,379
equation and also do a model problem
8
00:00:33,129 --> 00:00:37,390
using the particle in a one dimensional
9
00:00:35,379 --> 00:00:41,019
box model this is one of the simplest
10
00:00:37,390 --> 00:00:43,870
models that we have let's take a quick
11
00:00:41,019 --> 00:00:47,050
look at the Schrodinger equation in the
12
00:00:43,870 --> 00:00:50,290
lecture earlier I mentioned that I would
13
00:00:47,050 --> 00:00:52,690
be talking about the time-independent
14
00:00:50,290 --> 00:00:54,879
schrodinger equation in which this
15
00:00:52,690 --> 00:01:02,559
quantity was referred to as the
16
00:00:54,879 --> 00:01:08,200
Hamiltonian and this as a constant but
17
00:01:02,559 --> 00:01:10,060
with dimensions of energy and the
18
00:01:08,200 --> 00:01:12,549
function sigh is the function that we
19
00:01:10,060 --> 00:01:14,409
wanted to find out by solving an
20
00:01:12,549 --> 00:01:16,899
equation of the sort but we don't know
21
00:01:14,409 --> 00:01:20,020
what this is right now we have to
22
00:01:16,899 --> 00:01:22,030
introduce that to understand how this
23
00:01:20,020 --> 00:01:24,969
equation comes about or what is its
24
00:01:22,030 --> 00:01:29,340
origin we can do a very simple example
25
00:01:24,969 --> 00:01:29,340
of a standing wave
26
00:01:32,110 --> 00:01:39,190
and you know that a standing wave is
27
00:01:35,510 --> 00:01:44,450
something that happens between fixed
28
00:01:39,190 --> 00:01:49,729
points and the wave motion of a particle
29
00:01:44,450 --> 00:01:55,549
fixed to the end something of that kind
30
00:01:49,729 --> 00:02:01,159
and let me put it precisely so that the
31
00:01:55,549 --> 00:02:03,979
way when it reflects it still follows
32
00:02:01,159 --> 00:02:05,899
and therefore the standing wave remains
33
00:02:03,979 --> 00:02:08,509
as a wave and the amplitudes don't
34
00:02:05,899 --> 00:02:12,970
cancel each other so if you if you want
35
00:02:08,509 --> 00:02:12,970
to look after the axis this is the
36
00:02:13,540 --> 00:02:21,080
coordinate or the x axis that you might
37
00:02:16,700 --> 00:02:23,630
want to talk about and this is the axis
38
00:02:21,080 --> 00:02:27,550
for the amplitude of the way at any
39
00:02:23,630 --> 00:02:33,170
position X between some pics points
40
00:02:27,550 --> 00:02:35,300
obviously for this way the length of the
41
00:02:33,170 --> 00:02:46,000
repeating unit is obviously called the
42
00:02:35,300 --> 00:02:46,000
wavelength lambda m and here we have 12
43
00:02:46,600 --> 00:02:55,850
yes to this is 1 and this is 2 and then
44
00:02:52,280 --> 00:02:57,739
you have three and three and a half it
45
00:02:55,850 --> 00:02:59,329
has to be either exactly half wavelength
46
00:02:57,739 --> 00:03:02,450
or a full wavelength for this to be a
47
00:02:59,329 --> 00:03:08,510
standing way ok now the equation for the
48
00:03:02,450 --> 00:03:12,200
standing wave for the amplitude a ok or
49
00:03:08,510 --> 00:03:14,450
let us call the amplitude si in relation
50
00:03:12,200 --> 00:03:16,670
to what we have here we will see later
51
00:03:14,450 --> 00:03:19,010
that this I is not necessarily the same
52
00:03:16,670 --> 00:03:21,739
as the side that we talked about but for
53
00:03:19,010 --> 00:03:28,820
that sigh if we have the maximum
54
00:03:21,739 --> 00:03:33,350
amplitude as a this quantity as a then
55
00:03:28,820 --> 00:03:39,739
the wave functions I of X is written as
56
00:03:33,350 --> 00:03:41,629
a sign to pi by lambda of X this is
57
00:03:39,739 --> 00:03:45,470
something that you are familiar with for
58
00:03:41,629 --> 00:03:47,930
a standing wave now this quantity sigh
59
00:03:45,470 --> 00:03:52,010
when you differentiate twice it
60
00:03:47,930 --> 00:03:54,560
satisfies the derivative equation let's
61
00:03:52,010 --> 00:03:59,990
do that for the first derivative dy by
62
00:03:54,560 --> 00:04:06,710
DX as 2 pi by lambda times a sign a
63
00:03:59,990 --> 00:04:11,510
cause 2 pi by lambda X and the second
64
00:04:06,710 --> 00:04:15,500
derivative d square phi by d x square is
65
00:04:11,510 --> 00:04:19,250
equal to minus 4 pie square by lambda
66
00:04:15,500 --> 00:04:22,730
square Phi of X because this will become
67
00:04:19,250 --> 00:04:24,950
sine 2 pi by lambda of X and that's the
68
00:04:22,730 --> 00:04:28,430
same thing as Phi of X therefore you say
69
00:04:24,950 --> 00:04:31,700
that the standing wave satisfies the
70
00:04:28,430 --> 00:04:33,440
differential equation d square Phi by DX
71
00:04:31,700 --> 00:04:38,200
square versailles is the amplitude of
72
00:04:33,440 --> 00:04:41,510
the way with lambda the wave length
73
00:04:38,200 --> 00:04:49,010
associated with that now the nollie if
74
00:04:41,510 --> 00:04:52,220
you remember in the lecture earlier gave
75
00:04:49,010 --> 00:04:56,390
an expression for the matter waves
76
00:04:52,220 --> 00:04:59,000
lambda in terms of the momentum of the
77
00:04:56,390 --> 00:05:01,220
particle in terms of momentum of the
78
00:04:59,000 --> 00:05:04,040
particle you have here and therefore if
79
00:05:01,220 --> 00:05:08,330
I write the wave equation it's d square
80
00:05:04,040 --> 00:05:14,960
Phi by DX square which is equal to minus
81
00:05:08,330 --> 00:05:20,600
4 pie square by H square multiplied by t
82
00:05:14,960 --> 00:05:22,880
square sigh or minus H bar square we
83
00:05:20,600 --> 00:05:25,640
know that H by 2 pi is H bar therefore
84
00:05:22,880 --> 00:05:30,650
they bring that in is minus H bar square
85
00:05:25,640 --> 00:05:34,840
d square y by DX square is equal to T
86
00:05:30,650 --> 00:05:38,419
squared sigh okay this is the equation
87
00:05:34,840 --> 00:05:41,270
for the standing wave using the the
88
00:05:38,419 --> 00:05:42,830
brolly idea and the quantization idea
89
00:05:41,270 --> 00:05:44,870
namely that the energy quantum for
90
00:05:42,830 --> 00:05:47,150
material particles light etcetera given
91
00:05:44,870 --> 00:05:49,640
in terms of the Planck's constant so the
92
00:05:47,150 --> 00:05:52,700
Planck's constant enters naturally here
93
00:05:49,640 --> 00:05:57,169
in describing what happens to the
94
00:05:52,700 --> 00:05:59,120
momentum square on the wave function is
95
00:05:57,169 --> 00:06:02,630
the same thing as the
96
00:05:59,120 --> 00:06:05,990
second derivative on the wave function x
97
00:06:02,630 --> 00:06:10,400
minus h bar square therefore if we write
98
00:06:05,990 --> 00:06:13,669
the kinetic energy p square by 2 m sigh
99
00:06:10,400 --> 00:06:20,360
that turns out to be minus h bar square
100
00:06:13,669 --> 00:06:25,000
by 2 m d square by D X square sign this
101
00:06:20,360 --> 00:06:25,000
being the kinetic energy this is the
102
00:06:25,660 --> 00:06:31,130
difference between if there is a
103
00:06:28,430 --> 00:06:33,919
potential energy V then it's a
104
00:06:31,130 --> 00:06:37,729
difference between the total energy E
105
00:06:33,919 --> 00:06:41,240
and the potential energy V which may be
106
00:06:37,729 --> 00:06:42,740
a function of X for whatever I mean if
107
00:06:41,240 --> 00:06:45,350
there is a potential we have to consider
108
00:06:42,740 --> 00:06:48,350
that therefore what happens this P
109
00:06:45,350 --> 00:06:53,570
Square by 2 m is nothing but e minus v
110
00:06:48,350 --> 00:06:59,210
on sy giving u minus H bar square by 2 m
111
00:06:53,570 --> 00:07:04,010
d square by D X square side now one last
112
00:06:59,210 --> 00:07:07,639
step and then you see the equation H
113
00:07:04,010 --> 00:07:10,700
sigh is equal to e sy making sense to us
114
00:07:07,639 --> 00:07:13,400
because now if you bring the way here
115
00:07:10,700 --> 00:07:16,789
just rewrite the equation you have minus
116
00:07:13,400 --> 00:07:21,560
H bar square by 2 m d square y by DX
117
00:07:16,789 --> 00:07:23,870
square plus ly of side is equal to e of
118
00:07:21,560 --> 00:07:27,430
sign please remember we had already
119
00:07:23,870 --> 00:07:31,490
written this as the kinetic energy and
120
00:07:27,430 --> 00:07:34,940
this is all sigh this is the potential
121
00:07:31,490 --> 00:07:37,220
energy on sy and therefore you say that
122
00:07:34,940 --> 00:07:43,720
this is nothing but kinetic energy plus
123
00:07:37,220 --> 00:07:47,570
potential energy on sighing I'm sorry
124
00:07:43,720 --> 00:07:49,940
giving you yeah constant times e sy and
125
00:07:47,570 --> 00:07:55,039
so you see that this is nothing but the
126
00:07:49,940 --> 00:07:58,160
Hamiltonian on sy giving you a sign this
127
00:07:55,039 --> 00:08:00,110
is a very simple justification I don't
128
00:07:58,160 --> 00:08:01,400
think we can't really say that we have
129
00:08:00,110 --> 00:08:03,349
derived it from any fundamental
130
00:08:01,400 --> 00:08:05,510
principles or whatever is a
131
00:08:03,349 --> 00:08:09,830
justification to see from a simple
132
00:08:05,510 --> 00:08:12,830
standing wave picture and using that the
133
00:08:09,830 --> 00:08:15,200
brolly principle or the proposition
134
00:08:12,830 --> 00:08:18,560
with the Planck's constant it looks like
135
00:08:15,200 --> 00:08:21,320
the particle wave function satisfies the
136
00:08:18,560 --> 00:08:24,340
equation Hamiltonian but the Hamiltonian
137
00:08:21,320 --> 00:08:27,110
looks a lot art it has a derivative
138
00:08:24,340 --> 00:08:29,030
instead of the p square by 2 m that we
139
00:08:27,110 --> 00:08:31,040
have now we have derivative here and
140
00:08:29,030 --> 00:08:33,440
there for the hamiltonian is a
141
00:08:31,040 --> 00:08:36,500
derivative acting on the wave function
142
00:08:33,440 --> 00:08:38,900
and a potential which is of course a
143
00:08:36,500 --> 00:08:40,280
function of the position of whatever
144
00:08:38,900 --> 00:08:43,220
particle or the system that you talk
145
00:08:40,280 --> 00:08:44,720
about the potential generally multiplies
146
00:08:43,220 --> 00:08:51,940
the wave function but the two together
147
00:08:44,720 --> 00:08:56,060
is actually an operator acting on site
148
00:08:51,940 --> 00:08:59,470
the hamiltonian operator acting on site
149
00:08:56,060 --> 00:09:02,560
giving you a constant time society
150
00:08:59,470 --> 00:09:05,030
Schrodinger equation is a very specific
151
00:09:02,560 --> 00:09:08,750
equation for the Hamiltonian operator
152
00:09:05,030 --> 00:09:11,390
and such equations in mathematics are
153
00:09:08,750 --> 00:09:13,780
known as I in value equations for
154
00:09:11,390 --> 00:09:17,450
whatever quantities that appear here
155
00:09:13,780 --> 00:09:20,920
suppose instead of H is any other
156
00:09:17,450 --> 00:09:24,860
operator that we are going to look at a
157
00:09:20,920 --> 00:09:28,040
sigh any operator giving some constant
158
00:09:24,860 --> 00:09:30,110
times sigh please remember this constant
159
00:09:28,040 --> 00:09:32,990
has to have the same dimension as the
160
00:09:30,110 --> 00:09:37,270
cunt as the operator a here in the same
161
00:09:32,990 --> 00:09:39,950
way that this constant has the energy
162
00:09:37,270 --> 00:09:43,070
dimension for the Hamiltonian operator
163
00:09:39,950 --> 00:09:47,030
which is also energy any such equation
164
00:09:43,070 --> 00:09:51,860
in which a can be measured
165
00:09:47,030 --> 00:09:55,240
experimentally such equations are called
166
00:09:51,860 --> 00:09:55,240
I ghen value equations
167
00:09:59,440 --> 00:10:03,680
eigenvalue equations and the Schrodinger
168
00:10:02,360 --> 00:10:05,600
equation the time-independent
169
00:10:03,680 --> 00:10:07,700
schrodinger equation is the eigenvalue
170
00:10:05,600 --> 00:10:10,250
equation for the hamiltonian or the
171
00:10:07,700 --> 00:10:14,570
energy operator this is the picture that
172
00:10:10,250 --> 00:10:17,240
you have to have so let me give you some
173
00:10:14,570 --> 00:10:19,430
small problems associated with whatever
174
00:10:17,240 --> 00:10:22,550
we have done right after this but then
175
00:10:19,430 --> 00:10:24,950
we will go to the next part namely how
176
00:10:22,550 --> 00:10:29,090
do we solve this for the specific case
177
00:10:24,950 --> 00:10:32,470
of a simple model now what's the model
178
00:10:29,090 --> 00:10:35,870
let's look at the modeling all of the
179
00:10:32,470 --> 00:10:41,450
particle in a one dimensional box I have
180
00:10:35,870 --> 00:10:44,920
a small drawing here that tells you that
181
00:10:41,450 --> 00:10:49,940
we have a particle in a finite region
182
00:10:44,920 --> 00:10:54,530
the potentials are in high night at two
183
00:10:49,940 --> 00:10:56,930
points namely points with X equal to 0
184
00:10:54,530 --> 00:10:58,640
and the point x is equal to L meaning
185
00:10:56,930 --> 00:11:02,450
that the particle is confined to a
186
00:10:58,640 --> 00:11:04,760
region of a box of length l and the
187
00:11:02,450 --> 00:11:08,210
particle motion or the particle
188
00:11:04,760 --> 00:11:13,460
coordinate is only one coordinate or one
189
00:11:08,210 --> 00:11:16,310
variable namely X let's assume for the
190
00:11:13,460 --> 00:11:19,610
time being that the potential inside the
191
00:11:16,310 --> 00:11:21,680
box is zero so this is what we call as
192
00:11:19,610 --> 00:11:24,830
the particle in a one dimensional box
193
00:11:21,680 --> 00:11:28,310
with in finite barriers and what does
194
00:11:24,830 --> 00:11:32,300
this particle give you know let's look
195
00:11:28,310 --> 00:11:37,340
at the equations we have minus H bar
196
00:11:32,300 --> 00:11:42,350
square by 2 m d square sigh by D X
197
00:11:37,340 --> 00:11:47,650
square plus we of sci is equal to P of
198
00:11:42,350 --> 00:11:52,070
sci if the potential is infinite then
199
00:11:47,650 --> 00:11:54,980
psy has to be zero in order to satisfy
200
00:11:52,070 --> 00:11:58,700
that therefore at the boundaries x is
201
00:11:54,980 --> 00:12:06,100
equal to 0 X is equal to L the wave
202
00:11:58,700 --> 00:12:06,100
function sigh if x is 0 inside the box
203
00:12:07,150 --> 00:12:15,200
we have V is 0 therefore what we have is
204
00:12:11,870 --> 00:12:23,030
minus H bar square by 2 m d square y by
205
00:12:15,200 --> 00:12:24,440
DX square is equal to e side the total
206
00:12:23,030 --> 00:12:26,750
energy because there is no potential
207
00:12:24,440 --> 00:12:30,710
inside the box if you'll solve this in a
208
00:12:26,750 --> 00:12:35,180
very quick manner namely d square sigh
209
00:12:30,710 --> 00:12:38,540
by the x square plus a constant a
210
00:12:35,180 --> 00:12:45,230
positive constant K squared sy is equal
211
00:12:38,540 --> 00:12:49,010
to 0 where K squared is 2m e by h bar
212
00:12:45,230 --> 00:12:51,830
squared this is the k square is positive
213
00:12:49,010 --> 00:12:54,830
obviously and therefore what you have
214
00:12:51,830 --> 00:12:58,340
here is a simple derivative equation for
215
00:12:54,830 --> 00:13:02,450
second order and you know such functions
216
00:12:58,340 --> 00:13:04,430
can be obtained the solutions can be
217
00:13:02,450 --> 00:13:07,760
obtained from either trigonometric
218
00:13:04,430 --> 00:13:10,340
function or the exponential with
219
00:13:07,760 --> 00:13:14,650
imaginary argument let's use the
220
00:13:10,340 --> 00:13:19,550
trigonometric function namely a sign
221
00:13:14,650 --> 00:13:30,410
let's write that to be consistent we
222
00:13:19,550 --> 00:13:35,920
have a cause k x plus b sine KX where a
223
00:13:30,410 --> 00:13:35,920
and B are arbitrary constants
224
00:13:39,050 --> 00:13:43,199
arbitrary constants now if you look at
225
00:13:41,819 --> 00:13:46,139
that solution with the boundary
226
00:13:43,199 --> 00:13:52,550
condition that you have namely sigh of
227
00:13:46,139 --> 00:13:56,699
zero is zero immediately you have a is
228
00:13:52,550 --> 00:13:59,879
equal to 0 because cos KX is one and
229
00:13:56,699 --> 00:14:03,660
sine KX goes to 0 therefore a is equal
230
00:13:59,879 --> 00:14:07,259
to 0 if you have sigh at L which is the
231
00:14:03,660 --> 00:14:11,970
other extreme of the box please remember
232
00:14:07,259 --> 00:14:18,329
this model at x is equal to L at this
233
00:14:11,970 --> 00:14:21,269
point okay therefore we have sigh of L
234
00:14:18,329 --> 00:14:29,220
is 0 which implies that since a is
235
00:14:21,269 --> 00:14:34,050
already 0 sigh of X is B sine K L and
236
00:14:29,220 --> 00:14:36,720
that's equal to 0 okay we don't want me
237
00:14:34,050 --> 00:14:38,399
to be 0 because if am BR 0 that's anyway
238
00:14:36,720 --> 00:14:40,680
it's a trivial solution for any such a
239
00:14:38,399 --> 00:14:44,370
differential equation doesn't give you
240
00:14:40,680 --> 00:14:45,689
any anything of interest I mean there's
241
00:14:44,370 --> 00:14:47,610
no meaning there's no interpretation
242
00:14:45,689 --> 00:14:49,889
therefore we are going to consider the
243
00:14:47,610 --> 00:14:52,829
case obviously a non-trivial solution
244
00:14:49,889 --> 00:14:59,129
would be not equal to 0 which means sine
245
00:14:52,829 --> 00:15:03,750
KL has to be 0 or que el has to be an
246
00:14:59,129 --> 00:15:08,519
integer times pi n is an integer KL is
247
00:15:03,750 --> 00:15:10,800
equal to n pi and n has to be obviously
248
00:15:08,519 --> 00:15:14,220
we don't want any cual 20 which is also
249
00:15:10,800 --> 00:15:17,540
the case of triviality and so what we
250
00:15:14,220 --> 00:15:21,269
have is n equal to 1 2 3 etcetera
251
00:15:17,540 --> 00:15:25,139
integers or please remember k is equal
252
00:15:21,269 --> 00:15:29,779
to n PI by L look at this k square if
253
00:15:25,139 --> 00:15:32,399
you recall is 2m e by h bar square
254
00:15:29,779 --> 00:15:37,290
therefore this gives you immediately
255
00:15:32,399 --> 00:15:43,410
that M square PI square by L square is
256
00:15:37,290 --> 00:15:46,230
equal to 2 m e by h square times d 4 pie
257
00:15:43,410 --> 00:15:49,529
square that we have canceled things off
258
00:15:46,230 --> 00:15:51,240
and you immediately get the solution
259
00:15:49,529 --> 00:15:57,120
namely e
260
00:15:51,240 --> 00:16:00,330
is equal to H square n square by 8 m l
261
00:15:57,120 --> 00:16:06,839
square and what is the solution for the
262
00:16:00,330 --> 00:16:16,860
wave function sigh of X is be signed k x
263
00:16:06,839 --> 00:16:21,020
which is b sine n pi x by l because k is
264
00:16:16,860 --> 00:16:25,350
n PI by L okay so this is the simplest
265
00:16:21,020 --> 00:16:28,170
solution but two important results one
266
00:16:25,350 --> 00:16:30,600
is that the energy for the particle in
267
00:16:28,170 --> 00:16:32,250
the box which is subject to boundary
268
00:16:30,600 --> 00:16:36,180
conditions that the wave function
269
00:16:32,250 --> 00:16:39,170
vanishes at some boundaries subject to
270
00:16:36,180 --> 00:16:42,630
that the particle energy appears to be
271
00:16:39,170 --> 00:16:46,050
quantized is not arbitrary you recall
272
00:16:42,630 --> 00:16:48,690
the dimension the quantity H square by n
273
00:16:46,050 --> 00:16:59,459
square H square by 8 ml square the
274
00:16:48,690 --> 00:17:04,140
quantity has the dimension of the energy
275
00:16:59,459 --> 00:17:06,630
and it has the only two inputs which is
276
00:17:04,140 --> 00:17:10,640
which are the inputs for this problem
277
00:17:06,630 --> 00:17:14,100
namely the mass of the particle m and
278
00:17:10,640 --> 00:17:16,760
the length of the box l and the other
279
00:17:14,100 --> 00:17:20,189
constant is of course Planck's constant
280
00:17:16,760 --> 00:17:22,439
so now the energy seems to be quantized
281
00:17:20,189 --> 00:17:24,899
in terms of the the two physical
282
00:17:22,439 --> 00:17:26,730
parameters that we introduced which
283
00:17:24,899 --> 00:17:29,159
particle e a larger part a heavier
284
00:17:26,730 --> 00:17:31,679
particle or a lighter particle in a
285
00:17:29,159 --> 00:17:33,390
smaller box or in the larger box but
286
00:17:31,679 --> 00:17:35,880
with all the other conditions being the
287
00:17:33,390 --> 00:17:39,480
same namely potentials being zero inside
288
00:17:35,880 --> 00:17:41,580
the potentials being in finite given
289
00:17:39,480 --> 00:17:44,520
that you see that the energy is
290
00:17:41,580 --> 00:17:48,200
discretized and the energy is in the
291
00:17:44,520 --> 00:17:50,820
units of H square by 8 ml square this is
292
00:17:48,200 --> 00:17:57,929
the fundamental unit for this box and
293
00:17:50,820 --> 00:18:02,669
then it is 14 9 16 25 as the value of n
294
00:17:57,929 --> 00:18:05,669
becomes 1 2 3 4 etc therefore particular
295
00:18:02,669 --> 00:18:09,899
vertical energies are discretized the
296
00:18:05,669 --> 00:18:13,919
second part is the other namely the wave
297
00:18:09,899 --> 00:18:22,340
function is given in terms of B sine n
298
00:18:13,919 --> 00:18:22,340
pi x by l
299
00:18:23,710 --> 00:18:29,540
now what is this wave function from the
300
00:18:27,490 --> 00:18:31,280
beginning of this lecture you might
301
00:18:29,540 --> 00:18:33,290
think that this wave function is
302
00:18:31,280 --> 00:18:35,540
essentially a function telling you how
303
00:18:33,290 --> 00:18:39,200
the particle is oscillating that's not
304
00:18:35,540 --> 00:18:41,750
true okay that picture was a starting
305
00:18:39,200 --> 00:18:43,520
point for us to get an idea that the
306
00:18:41,750 --> 00:18:45,980
Schrodinger equation is like this the
307
00:18:43,520 --> 00:18:48,290
wave function that we have here is not a
308
00:18:45,980 --> 00:18:49,610
function representing how the particle
309
00:18:48,290 --> 00:18:52,730
is moving it is just a function
310
00:18:49,610 --> 00:18:54,770
associated with that particle what's the
311
00:18:52,730 --> 00:19:00,260
meaning of it max bond gave the
312
00:18:54,770 --> 00:19:02,660
interpretation namely that wave function
313
00:19:00,260 --> 00:19:07,160
by itself does not have any meaning but
314
00:19:02,660 --> 00:19:10,760
sy of X square psystar sigh in this case
315
00:19:07,160 --> 00:19:18,610
I is real therefore sy of X sigh of X or
316
00:19:10,760 --> 00:19:25,090
Y squared of X in a small interval DX
317
00:19:18,610 --> 00:19:30,230
gives the probability of the particle
318
00:19:25,090 --> 00:19:35,000
being in the position between X and X
319
00:19:30,230 --> 00:19:38,750
plus DX the probability of locating the
320
00:19:35,000 --> 00:19:44,120
particle between X and X plus DX that's
321
00:19:38,750 --> 00:19:45,830
the number given by the product of the
322
00:19:44,120 --> 00:19:47,960
wave function with itself in this case
323
00:19:45,830 --> 00:19:51,440
because it's real that Max bond
324
00:19:47,960 --> 00:19:53,930
suggested that size squared X DX gives
325
00:19:51,440 --> 00:19:57,170
the probability that the system we found
326
00:19:53,930 --> 00:19:59,390
in the interval X and X plus DX that's
327
00:19:57,170 --> 00:20:03,410
all there is to it therefore let me
328
00:19:59,390 --> 00:20:08,660
conclude immediately what we should be
329
00:20:03,410 --> 00:20:15,590
because if psystar x sigh x which is the
330
00:20:08,660 --> 00:20:18,290
same as shy of x square with a DX is a
331
00:20:15,590 --> 00:20:21,020
probability then if you add all the
332
00:20:18,290 --> 00:20:23,170
probabilities from 0 to l because the
333
00:20:21,020 --> 00:20:26,510
particle can have any position between
334
00:20:23,170 --> 00:20:28,160
the end point but not at the end point
335
00:20:26,510 --> 00:20:30,860
from anywhere as close to the end point
336
00:20:28,160 --> 00:20:32,840
as possible but as close to the other
337
00:20:30,860 --> 00:20:34,430
end point therefore if you integrate the
338
00:20:32,840 --> 00:20:36,880
total probability is this being a
339
00:20:34,430 --> 00:20:43,130
continuous function you have
340
00:20:36,880 --> 00:20:45,860
02 el psy x square DX that probability
341
00:20:43,130 --> 00:20:48,049
has to add to 1 because we have made
342
00:20:45,860 --> 00:20:49,909
sure that the potentials are infinite in
343
00:20:48,049 --> 00:20:51,980
our model therefore the partner cannot
344
00:20:49,909 --> 00:20:53,659
be found outside of the region before
345
00:20:51,980 --> 00:20:57,010
the probability that the particle stays
346
00:20:53,659 --> 00:21:00,289
inside the box is one this gives you
347
00:20:57,010 --> 00:21:06,130
immediately your value for B because you
348
00:21:00,289 --> 00:21:10,490
have B squared sine squared in pi x by l
349
00:21:06,130 --> 00:21:13,460
DX between 0 and y el that's equal to 1
350
00:21:10,490 --> 00:21:17,630
which gives you your value B is equal to
351
00:21:13,460 --> 00:21:19,490
root 2 by L okay therefore you have got
352
00:21:17,630 --> 00:21:24,039
to results for the particle in the box
353
00:21:19,490 --> 00:21:31,159
namely the wave function is root 2 by L
354
00:21:24,039 --> 00:21:35,510
sine n pi x by l and e the particles the
355
00:21:31,159 --> 00:21:40,970
energy is given by h square m square by
356
00:21:35,510 --> 00:21:44,510
8 m l square now because the energy is
357
00:21:40,970 --> 00:21:49,690
given by the quantum number m let me use
358
00:21:44,510 --> 00:21:53,539
a highlighter here because it's given by
359
00:21:49,690 --> 00:21:56,330
yen and the M can take any number of
360
00:21:53,539 --> 00:22:00,080
values and for that in the corresponding
361
00:21:56,330 --> 00:22:01,909
wave function is sine n pi x by l we see
362
00:22:00,080 --> 00:22:03,710
that there are many solutions to the
363
00:22:01,909 --> 00:22:05,510
wave function and many solutions to the
364
00:22:03,710 --> 00:22:07,340
energy this will also turn out to be a
365
00:22:05,510 --> 00:22:09,350
general property when we solve the
366
00:22:07,340 --> 00:22:12,350
Hamiltonian equation the Schrodinger
367
00:22:09,350 --> 00:22:15,710
equation for the systems in all the
368
00:22:12,350 --> 00:22:17,960
other models that in as in one step you
369
00:22:15,710 --> 00:22:19,429
will get all the different types of all
370
00:22:17,960 --> 00:22:21,919
the possible energies and all the
371
00:22:19,429 --> 00:22:25,190
possible wave functions and the best way
372
00:22:21,919 --> 00:22:27,559
to I mean a convenient way I wouldn't
373
00:22:25,190 --> 00:22:29,539
call it mr. a convenient way is to label
374
00:22:27,559 --> 00:22:33,500
the wave function with the quantum
375
00:22:29,539 --> 00:22:35,720
number sy n of X and en for a given
376
00:22:33,500 --> 00:22:39,799
quantum number n so let me summarize and
377
00:22:35,720 --> 00:22:45,639
then stop for this lecture namely the
378
00:22:39,799 --> 00:22:50,320
particle in a 1d box has two results a
379
00:22:45,639 --> 00:22:50,320
quantization of energy or discretization
380
00:22:51,830 --> 00:23:04,140
due to boundary conditions n of energy E
381
00:22:57,540 --> 00:23:07,140
and a probability statement for
382
00:23:04,140 --> 00:23:10,350
determining the position of the particle
383
00:23:07,140 --> 00:23:12,540
in the box at various locations ok let's
384
00:23:10,350 --> 00:23:15,840
continue this in the next part and
385
00:23:12,540 --> 00:23:17,850
complete the remaining that we needed to
386
00:23:15,840 --> 00:23:19,800
do in terms of what are called the
387
00:23:17,850 --> 00:23:21,810
measurables and then how do we interpret
388
00:23:19,800 --> 00:23:23,970
this probability and so on for various
389
00:23:21,810 --> 00:23:26,930
values will do that in the second part
390
00:23:23,970 --> 00:23:26,930
until then thank
391
00:23:43,210 --> 00:23:45,270
you