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