1
00:00:11,180 --> 00:00:18,790
so we shall continue the particle in a
2
00:00:14,690 --> 00:00:24,279
2d box but for the moment let us
3
00:00:18,790 --> 00:00:26,600
consider a little bit on this famous
4
00:00:24,279 --> 00:00:29,030
principle called the uncertainty
5
00:00:26,600 --> 00:00:33,100
principle which was first four forward
6
00:00:29,030 --> 00:00:33,100
by Werner Heisenberg
7
00:00:38,750 --> 00:00:45,680
now there is a very beautiful lecture on
8
00:00:43,310 --> 00:00:52,710
the Heisenberg's uncertainty principle
9
00:00:45,680 --> 00:00:57,180
by Professor we balakrishnan and it's
10
00:00:52,710 --> 00:01:04,110
there in the NPTEL website under basic
11
00:00:57,180 --> 00:01:06,390
courses or in physics this is on quantum
12
00:01:04,110 --> 00:01:08,790
mechanics the very first lecture is on
13
00:01:06,390 --> 00:01:10,920
the Heisenberg's uncertainty principle I
14
00:01:08,790 --> 00:01:13,530
would like everyone I would like to
15
00:01:10,920 --> 00:01:15,990
recommend that to every one of you to go
16
00:01:13,530 --> 00:01:17,760
through that lecture but this is very
17
00:01:15,990 --> 00:01:20,160
very preliminary it's not anything like
18
00:01:17,760 --> 00:01:22,590
what was there but you would appreciate
19
00:01:20,160 --> 00:01:25,140
that lecture far more when you listen to
20
00:01:22,590 --> 00:01:27,810
versal balakrishnan account of how the
21
00:01:25,140 --> 00:01:30,030
Heisenberg's uncertainty principle is to
22
00:01:27,810 --> 00:01:33,780
be understood we will do a much simpler
23
00:01:30,030 --> 00:01:36,420
exercise since you are beginning this is
24
00:01:33,780 --> 00:01:44,010
meant for the introductory very first
25
00:01:36,420 --> 00:01:48,210
year students now uncertainty Delta X in
26
00:01:44,010 --> 00:01:56,760
any measurement measurement quantity X
27
00:01:48,210 --> 00:02:03,420
is given by this simple statement that
28
00:01:56,760 --> 00:02:10,280
it's the difference between the average
29
00:02:03,420 --> 00:02:15,600
of the square of that variable minus the
30
00:02:10,280 --> 00:02:19,320
square of the average square of the
31
00:02:15,600 --> 00:02:23,310
average of the variable and this whole
32
00:02:19,320 --> 00:02:27,060
thing is under a square root ok this is
33
00:02:23,310 --> 00:02:29,720
the angular brackets tell you that the
34
00:02:27,060 --> 00:02:29,720
average month
35
00:02:30,319 --> 00:02:37,519
what is inside is the one for which the
36
00:02:33,980 --> 00:02:40,299
average is taken therefore the average
37
00:02:37,519 --> 00:02:43,790
is taken for the square of the value X
38
00:02:40,299 --> 00:02:45,920
here the average is taken for the value
39
00:02:43,790 --> 00:02:47,840
itself and then it is squared the
40
00:02:45,920 --> 00:02:52,359
difference between the two the square
41
00:02:47,840 --> 00:02:52,359
root of this is called the uncertainty
42
00:02:53,980 --> 00:03:11,090
average of the square minus square of
43
00:03:01,669 --> 00:03:12,560
the average let's see this I don't know
44
00:03:11,090 --> 00:03:15,500
how to say it in English it's the square
45
00:03:12,560 --> 00:03:18,400
root okay or you can write with in
46
00:03:15,500 --> 00:03:20,989
bright red square likewise the
47
00:03:18,400 --> 00:03:23,359
uncertainty this is for the position
48
00:03:20,989 --> 00:03:25,129
variable and this is for the momentum
49
00:03:23,359 --> 00:03:28,459
variable I have introduced this in a
50
00:03:25,129 --> 00:03:31,280
separate account I might tell you how
51
00:03:28,459 --> 00:03:33,609
this formula comes about and so on but
52
00:03:31,280 --> 00:03:36,829
let us just introduce these things as
53
00:03:33,609 --> 00:03:42,590
defined in textbooks the Delta P is
54
00:03:36,829 --> 00:03:49,239
again the average of the square of the
55
00:03:42,590 --> 00:03:49,239
momentum minus the woman 2 squared
56
00:03:49,850 --> 00:04:01,160
Delta X Delta P the product of the two
57
00:03:52,880 --> 00:04:06,620
is greater than or equal to H bar my two
58
00:04:01,160 --> 00:04:09,530
ok this is the Heisenberg's statement
59
00:04:06,620 --> 00:04:13,990
about the uncertainty between X and P
60
00:04:09,530 --> 00:04:16,970
what it means is that if for some
61
00:04:13,990 --> 00:04:20,690
preparation of the states we are able to
62
00:04:16,970 --> 00:04:24,710
minimize this by making sure that this
63
00:04:20,690 --> 00:04:27,110
average and this squared average or very
64
00:04:24,710 --> 00:04:28,790
close to each other therefore we are
65
00:04:27,110 --> 00:04:32,390
able to measure the position very very
66
00:04:28,790 --> 00:04:37,300
very accurately if you do that what
67
00:04:32,390 --> 00:04:41,420
uncertainty principle tells you that is
68
00:04:37,300 --> 00:04:46,010
in the denominator not for the
69
00:04:41,420 --> 00:04:48,920
uncertainty in Delta P is very large it
70
00:04:46,010 --> 00:04:51,320
is not possible for us to control the
71
00:04:48,920 --> 00:04:57,770
uncertainties to both of them to
72
00:04:51,320 --> 00:05:01,220
absolute minimum except not violate this
73
00:04:57,770 --> 00:05:04,160
particular relation therefore this is
74
00:05:01,220 --> 00:05:07,480
one of the statements that you might see
75
00:05:04,160 --> 00:05:09,680
in textbooks very often regarding the
76
00:05:07,480 --> 00:05:11,660
uncertainty in the position measurement
77
00:05:09,680 --> 00:05:13,780
and uncertainty in the momentum
78
00:05:11,660 --> 00:05:17,740
measurement what it also means is that
79
00:05:13,780 --> 00:05:22,820
position and momentum cannot be
80
00:05:17,740 --> 00:05:26,050
simultaneously used as variables for
81
00:05:22,820 --> 00:05:28,100
describing the state of a particle as as
82
00:05:26,050 --> 00:05:30,020
independent quantities for describing
83
00:05:28,100 --> 00:05:32,120
the state of the particle the state of
84
00:05:30,020 --> 00:05:34,460
the particle can either be very
85
00:05:32,120 --> 00:05:37,100
precisely stated using the position or
86
00:05:34,460 --> 00:05:41,270
very precisely stated using its momentum
87
00:05:37,100 --> 00:05:43,640
but not both and therefore this brings
88
00:05:41,270 --> 00:05:46,700
down the whole structure of classical
89
00:05:43,640 --> 00:05:48,950
mechanics where one would imagine in the
90
00:05:46,700 --> 00:05:52,880
solution of the Newton's equation the
91
00:05:48,950 --> 00:05:55,370
precise statement for the position and
92
00:05:52,880 --> 00:05:58,280
velocity of a particle at one instant of
93
00:05:55,370 --> 00:06:00,260
time and be able to solve therefore if
94
00:05:58,280 --> 00:06:02,390
you can specify the velocity obviously
95
00:06:00,260 --> 00:06:03,380
you can also specify the momentum of the
96
00:06:02,390 --> 00:06:05,150
particle
97
00:06:03,380 --> 00:06:09,760
therefore position and momentum can be
98
00:06:05,150 --> 00:06:11,870
simultaneously used as descriptors for
99
00:06:09,760 --> 00:06:14,330
defining the state of a classical
100
00:06:11,870 --> 00:06:16,820
particle but they can't be used as
101
00:06:14,330 --> 00:06:19,820
descriptors for the state of a quantum
102
00:06:16,820 --> 00:06:22,550
particles in the relation between the
103
00:06:19,820 --> 00:06:24,470
two is given by this famous Heisenberg's
104
00:06:22,550 --> 00:06:26,720
uncertainty principle and professor
105
00:06:24,470 --> 00:06:28,610
balakrishna's lecture tells you how to
106
00:06:26,720 --> 00:06:32,860
generalize the Heisenberg's uncertainty
107
00:06:28,610 --> 00:06:35,180
states in using other classical
108
00:06:32,860 --> 00:06:40,160
formulations and eventually what is
109
00:06:35,180 --> 00:06:45,080
known as the commutator now let's use
110
00:06:40,160 --> 00:06:50,840
the wave functions I of X for the one
111
00:06:45,080 --> 00:06:54,250
dimensional box root 2 by L sine pi x by
112
00:06:50,840 --> 00:06:57,560
l we will take the n equal to 1 case
113
00:06:54,250 --> 00:07:01,460
quantum number and if we try to
114
00:06:57,560 --> 00:07:06,260
calculate the average value X for the
115
00:07:01,460 --> 00:07:13,780
particle in this state whose wave
116
00:07:06,260 --> 00:07:16,990
function and the probability of the
117
00:07:13,780 --> 00:07:23,150
particle at various points is
118
00:07:16,990 --> 00:07:28,250
symmetrically the same on either side of
119
00:07:23,150 --> 00:07:32,110
l by 2 ok it should be immediately clear
120
00:07:28,250 --> 00:07:35,720
that the average value for the particle
121
00:07:32,110 --> 00:07:37,730
position given that these are the
122
00:07:35,720 --> 00:07:41,930
probabilities for the particles position
123
00:07:37,730 --> 00:07:44,240
being here or here or here or here by
124
00:07:41,930 --> 00:07:47,060
looking at this being a symmetrical
125
00:07:44,240 --> 00:07:52,870
graph you can immediately see it should
126
00:07:47,060 --> 00:07:55,400
be L by 2 but that's also the
127
00:07:52,870 --> 00:07:59,600
expectation value or the average value
128
00:07:55,400 --> 00:08:05,240
this is cool the average value in
129
00:07:59,600 --> 00:08:11,540
quantum mechanics for any variable a in
130
00:08:05,240 --> 00:08:15,540
the state's I is given by psystar a on
131
00:08:11,540 --> 00:08:17,880
sign D tau
132
00:08:15,540 --> 00:08:20,040
which is the volume element or the area
133
00:08:17,880 --> 00:08:22,560
element or the length element similar to
134
00:08:20,040 --> 00:08:26,220
whether it's a one dimensional box or a
135
00:08:22,560 --> 00:08:30,300
two or three dimensional / the integral
136
00:08:26,220 --> 00:08:37,950
sliced off sigh dito okay this is a
137
00:08:30,300 --> 00:08:42,390
postulate I don't want to tell you how
138
00:08:37,950 --> 00:08:44,190
this can be arrived at using arguments
139
00:08:42,390 --> 00:08:46,710
you will find such things in physics
140
00:08:44,190 --> 00:08:50,690
books but for the particular course that
141
00:08:46,710 --> 00:08:53,150
you have started taking this is the
142
00:08:50,690 --> 00:09:00,500
postulated introduction for the
143
00:08:53,150 --> 00:09:02,910
expectation value of any variable a host
144
00:09:00,500 --> 00:09:06,480
corresponding representation as an
145
00:09:02,910 --> 00:09:09,210
operator is given by this a hat and the
146
00:09:06,480 --> 00:09:12,270
a hat is between the wave function sine
147
00:09:09,210 --> 00:09:16,200
and the complex conjugate size taught
148
00:09:12,270 --> 00:09:20,460
ifs is a complex function otherwise both
149
00:09:16,200 --> 00:09:22,110
of them are sign this prescription must
150
00:09:20,460 --> 00:09:27,210
be kept in mind this is introduced as a
151
00:09:22,110 --> 00:09:33,240
postulate a form and let me calculate DX
152
00:09:27,210 --> 00:09:35,670
of the particle it's very easy now now
153
00:09:33,240 --> 00:09:40,530
for the average value X is given by the
154
00:09:35,670 --> 00:09:44,670
integral root 2 x l root 2 by l because
155
00:09:40,530 --> 00:09:53,490
it's i star psy and you have sine pi x
156
00:09:44,670 --> 00:09:57,210
by l x sine pi x by l d x between 0 and
157
00:09:53,490 --> 00:10:00,690
yell for the particle in the quantum
158
00:09:57,210 --> 00:10:05,340
state with the quantum number one which
159
00:10:00,690 --> 00:10:07,680
is what we call as sy13 and x of course
160
00:10:05,340 --> 00:10:09,540
doesn't change anything i mean it simply
161
00:10:07,680 --> 00:10:15,950
multiplies to this therefore this
162
00:10:09,540 --> 00:10:22,080
integral is to my l 0 to l sine square I
163
00:10:15,950 --> 00:10:24,270
X by L multiplied by X DX calculate this
164
00:10:22,080 --> 00:10:24,640
integral and show that the answer is l
165
00:10:24,270 --> 00:10:29,650
by
166
00:10:24,640 --> 00:10:32,860
okay that's where you to do the exercise
167
00:10:29,650 --> 00:10:36,190
what about the momentum you have to be
168
00:10:32,860 --> 00:10:38,920
careful in ensuring that the momentum
169
00:10:36,190 --> 00:10:46,150
operator which is a derivative operator
170
00:10:38,920 --> 00:10:49,000
is placed as written here namely to by L
171
00:10:46,150 --> 00:10:55,140
that comes from the two constants I
172
00:10:49,000 --> 00:10:59,440
starts I then you have sine pi x by l
173
00:10:55,140 --> 00:11:06,100
between 0 to l and the momentum operator
174
00:10:59,440 --> 00:11:11,760
is minus IH bar d by DX acting on sine
175
00:11:06,100 --> 00:11:14,830
pi x by l DX see that the operator is
176
00:11:11,760 --> 00:11:16,510
sandwiched between the wave function and
177
00:11:14,830 --> 00:11:18,310
the complex conjugate of the wave
178
00:11:16,510 --> 00:11:20,110
function but here the wave function is
179
00:11:18,310 --> 00:11:22,690
real therefore you don't see the
180
00:11:20,110 --> 00:11:25,600
difference between the two what is this
181
00:11:22,690 --> 00:11:27,670
it's very easy to see that this will
182
00:11:25,600 --> 00:11:30,280
give you the derivative will give you a
183
00:11:27,670 --> 00:11:35,400
cost and the sign costs will give you a
184
00:11:30,280 --> 00:11:39,220
sign 2 pi x by l and not in this
185
00:11:35,400 --> 00:11:43,840
interval is actually 0 what about the
186
00:11:39,220 --> 00:11:46,750
average value p squared the average
187
00:11:43,840 --> 00:11:53,350
value of P squared is given by 2 by L
188
00:11:46,750 --> 00:11:56,110
again sine square sine pi x by l and now
189
00:11:53,350 --> 00:11:59,230
you remember it's minus h bar square d
190
00:11:56,110 --> 00:12:05,980
square by DX square for the operator p
191
00:11:59,230 --> 00:12:08,440
square sine pi x by l DX and it's
192
00:12:05,980 --> 00:12:10,240
between 0 and yell i didn't write the
193
00:12:08,440 --> 00:12:13,900
denominator because we have chosen the
194
00:12:10,240 --> 00:12:16,420
wave function by ensuring that the wave
195
00:12:13,900 --> 00:12:18,580
function is the integral of the square
196
00:12:16,420 --> 00:12:20,440
of the wave function is actually one in
197
00:12:18,580 --> 00:12:24,160
the entire region therefore I didn't
198
00:12:20,440 --> 00:12:29,380
write the denominator that's one this of
199
00:12:24,160 --> 00:12:33,490
course you know is nothing but 2 m.e the
200
00:12:29,380 --> 00:12:35,290
total energy this is P squared on the
201
00:12:33,490 --> 00:12:37,329
wave function you remember P Square by 2
202
00:12:35,290 --> 00:12:41,199
m on the wave function give you the
203
00:12:37,329 --> 00:12:45,489
II therefore this is 2 m.e therefore you
204
00:12:41,199 --> 00:12:49,540
see that p square is immediately given
205
00:12:45,489 --> 00:12:53,679
by the energy that we know you can write
206
00:12:49,540 --> 00:12:55,869
that okay what about DX square if I have
207
00:12:53,679 --> 00:13:02,589
to do X square all I need to do the same
208
00:12:55,869 --> 00:13:05,790
thing right x square on sci 1 and i have
209
00:13:02,589 --> 00:13:13,360
the integral that needs to be evaluated
210
00:13:05,790 --> 00:13:18,790
is integral to by l sine square pi x by
211
00:13:13,360 --> 00:13:21,069
l and x square DX therefore you know the
212
00:13:18,790 --> 00:13:25,089
value of x square you know the value of
213
00:13:21,069 --> 00:13:28,179
x you know the value of P is 0 you know
214
00:13:25,089 --> 00:13:32,319
the value of P Square has nothing but 2
215
00:13:28,179 --> 00:13:35,410
m.e this is the only integral that I
216
00:13:32,319 --> 00:13:38,949
have not calculated once you have done
217
00:13:35,410 --> 00:13:44,769
that you can calculate Delta X Delta P
218
00:13:38,949 --> 00:13:48,249
has nothing but the square root of P
219
00:13:44,769 --> 00:13:56,319
squared minus p of course you know
220
00:13:48,249 --> 00:14:02,230
that's 0 times x squared minus x whole
221
00:13:56,319 --> 00:14:04,959
square okay and you should be able to
222
00:14:02,230 --> 00:14:09,519
verify that this answer is greater than
223
00:14:04,959 --> 00:14:12,489
or equal to H bar by 2 okay so this is
224
00:14:09,519 --> 00:14:15,339
the statement of the Heisenberg's
225
00:14:12,489 --> 00:14:18,579
uncertainty principle for the particle
226
00:14:15,339 --> 00:14:22,959
in a one dimensional box now exactly the
227
00:14:18,579 --> 00:14:24,220
same statement can be extent it can be
228
00:14:22,959 --> 00:14:25,779
extended to a particle in a two
229
00:14:24,220 --> 00:14:30,879
dimensional box except that now you have
230
00:14:25,779 --> 00:14:34,290
x and y s two independent coordinates px
231
00:14:30,879 --> 00:14:38,369
and py as two independent coordinates
232
00:14:34,290 --> 00:14:41,290
therefore you have a corresponding
233
00:14:38,369 --> 00:14:45,660
uncertainty relation in two dimensions
234
00:14:41,290 --> 00:14:51,180
with one exception namely x and y are
235
00:14:45,660 --> 00:14:56,320
independent coordinates therefore x + py
236
00:14:51,180 --> 00:15:00,300
can be simultaneously measured or can be
237
00:14:56,320 --> 00:15:04,690
ascribed as a property to the system y +
238
00:15:00,300 --> 00:15:08,890
P X can be simultaneously specified for
239
00:15:04,690 --> 00:15:13,170
the particle x and y can be specified px
240
00:15:08,890 --> 00:15:17,650
and py can be specified but not x + bx +
241
00:15:13,170 --> 00:15:21,730
y npy that's the only thing you have to
242
00:15:17,650 --> 00:15:26,190
remember the independence of the degrees
243
00:15:21,730 --> 00:15:28,120
of freedom ensures that the operators
244
00:15:26,190 --> 00:15:30,730
corresponding to those degrees of
245
00:15:28,120 --> 00:15:32,110
freedom commute with each other and if I
246
00:15:30,730 --> 00:15:34,690
have not spoken to you much about
247
00:15:32,110 --> 00:15:37,120
commutation that would be in the next
248
00:15:34,690 --> 00:15:39,400
lecture but in this part I would simply
249
00:15:37,120 --> 00:15:42,880
want you to calculate the Heisenberg
250
00:15:39,400 --> 00:15:45,880
uncertainty principle as given this is
251
00:15:42,880 --> 00:15:48,700
one simple way of doing it you can find
252
00:15:45,880 --> 00:15:52,570
similar treatments for the uncertainty
253
00:15:48,700 --> 00:15:54,640
when you go to study the other systems
254
00:15:52,570 --> 00:15:57,130
like the harmonic oscillator the
255
00:15:54,640 --> 00:16:00,640
hydrogen atom so on what is key to
256
00:15:57,130 --> 00:16:03,010
remember is the definition for the Delta
257
00:16:00,640 --> 00:16:06,400
X I gave you and the definition for the
258
00:16:03,010 --> 00:16:08,700
Delta P I gave you those are fundamental
259
00:16:06,400 --> 00:16:11,560
I have not told you where they come from
260
00:16:08,700 --> 00:16:13,270
maybe in a separate lecture or in the
261
00:16:11,560 --> 00:16:15,850
class when we discuss these things
262
00:16:13,270 --> 00:16:18,070
through elaborations I will tell you
263
00:16:15,850 --> 00:16:21,820
what the origin of the Delta X and Delta
264
00:16:18,070 --> 00:16:25,570
P but these are definitions which you
265
00:16:21,820 --> 00:16:28,450
have to start with working and then feel
266
00:16:25,570 --> 00:16:30,760
more comfortable go back and look at the
267
00:16:28,450 --> 00:16:32,980
whole process of the derivation we will
268
00:16:30,760 --> 00:16:34,960
continue this exercise to complete what
269
00:16:32,980 --> 00:16:37,320
is known as the introductory but
270
00:16:34,960 --> 00:16:40,300
postulated basis of quantum mechanics
271
00:16:37,320 --> 00:16:42,910
for this course in the next part of this
272
00:16:40,300 --> 00:16:44,290
lecture which is the the third part for
273
00:16:42,910 --> 00:16:46,750
the particle in a two dimensional box
274
00:16:44,290 --> 00:16:49,600
with that we will complete the two
275
00:16:46,750 --> 00:16:52,840
simple models particle in a 1b and a 2d
276
00:16:49,600 --> 00:16:54,970
box will meet again for the last portion
277
00:16:52,840 --> 00:16:58,560
of the particle in the thule box lecture
278
00:16:54,970 --> 00:16:58,560
the next thing thank you
279
00:17:14,319 --> 00:17:16,380
you