1 00:00:11,260 --> 00:00:20,880 so let us continue from where we left 2 00:00:15,000 --> 00:00:28,509 namely the wavefunction sy n of X as 3 00:00:20,880 --> 00:00:35,710 root 2 by L sine n PI X by L and the 4 00:00:28,509 --> 00:00:41,469 energy en s H square M square by 8 M L 5 00:00:35,710 --> 00:00:45,580 square since psystar sy is the 6 00:00:41,469 --> 00:00:50,320 probability density at any point X and 7 00:00:45,580 --> 00:00:53,620 sy starts I at the small interval DX 8 00:00:50,320 --> 00:00:58,210 around X gives you the probability that 9 00:00:53,620 --> 00:01:01,870 the particle is in that small region if 10 00:00:58,210 --> 00:01:08,350 you plot size squared of X as a function 11 00:01:01,870 --> 00:01:11,439 of X you get some ideas about what these 12 00:01:08,350 --> 00:01:20,259 probabilities mean so let me show this 13 00:01:11,439 --> 00:01:27,249 graph for some value of the Box length L 14 00:01:20,259 --> 00:01:30,189 and we have a root 2 by L which is a pre 15 00:01:27,249 --> 00:01:32,259 factor for the the wave function so the 16 00:01:30,189 --> 00:01:35,590 wave function has a dimension it has a 17 00:01:32,259 --> 00:01:38,259 dimension of 1 by square root of the 18 00:01:35,590 --> 00:01:41,590 length okay and then you see that psy 19 00:01:38,259 --> 00:01:45,520 star psy DL takes care of the 20 00:01:41,590 --> 00:01:50,200 probability being a number now if you 21 00:01:45,520 --> 00:01:52,689 plot the sy star sign for the first wave 22 00:01:50,200 --> 00:01:54,219 function namely n equal to 1 you see 23 00:01:52,689 --> 00:01:56,709 that this is nothing but the half sine 24 00:01:54,219 --> 00:02:01,359 wave with a little bit of tapering on 25 00:01:56,709 --> 00:02:05,529 the edges then it is n equal to 2 please 26 00:02:01,359 --> 00:02:08,560 remember that the expression if you look 27 00:02:05,529 --> 00:02:11,319 at the wave function here is the wave 28 00:02:08,560 --> 00:02:14,110 function itself if you look at the wave 29 00:02:11,319 --> 00:02:16,360 function n equal to 1 is 1/2 sine wave n 30 00:02:14,110 --> 00:02:21,159 equal to 2 is a full sine wave this is a 31 00:02:16,360 --> 00:02:22,450 3 1/2 sine wave and so on therefore if 32 00:02:21,159 --> 00:02:24,670 you take the square of this way 33 00:02:22,450 --> 00:02:27,010 obviously 34 00:02:24,670 --> 00:02:29,830 the picture that you get is the picture 35 00:02:27,010 --> 00:02:31,960 that I showed you know so this is the 36 00:02:29,830 --> 00:02:34,120 square of the N equal to two way and 37 00:02:31,960 --> 00:02:39,130 this is the square of the N equal to 3 38 00:02:34,120 --> 00:02:44,380 sine three PI X by M since it is zero to 39 00:02:39,130 --> 00:02:46,510 L you have that shape and also the 40 00:02:44,380 --> 00:02:49,590 shapes for n equal to four and as you 41 00:02:46,510 --> 00:02:51,820 see that as n increases these 42 00:02:49,590 --> 00:02:54,940 oscillations become so close to each 43 00:02:51,820 --> 00:02:57,250 other that for very very large n it 44 00:02:54,940 --> 00:03:02,830 looks like that the probability density 45 00:02:57,250 --> 00:03:05,350 is uniformed okay what does that mean if 46 00:03:02,830 --> 00:03:08,230 we have to do this exercise for a 47 00:03:05,350 --> 00:03:10,480 classical system let's assume that the 48 00:03:08,230 --> 00:03:12,550 particle is moving in the Box in some 49 00:03:10,480 --> 00:03:14,260 way and let's not ask the question how 50 00:03:12,550 --> 00:03:15,520 does it get deflected from one end to 51 00:03:14,260 --> 00:03:17,110 the other and there are questions for 52 00:03:15,520 --> 00:03:18,730 which we don't have any answers and 53 00:03:17,110 --> 00:03:20,770 questions which are also meaningless 54 00:03:18,730 --> 00:03:24,280 let's assume that we are looking at the 55 00:03:20,770 --> 00:03:28,300 particle at a given instant of time and 56 00:03:24,280 --> 00:03:30,430 using a small window if the particle is 57 00:03:28,300 --> 00:03:33,310 moving at constant speed because it's 58 00:03:30,430 --> 00:03:37,090 kinetic energy is a constant energy E 59 00:03:33,310 --> 00:03:39,130 and if it is a constant then the 60 00:03:37,090 --> 00:03:43,390 probability of locating the particle in 61 00:03:39,130 --> 00:03:45,640 every small region of the same size is 62 00:03:43,390 --> 00:03:49,209 uniform it's the same value that's what 63 00:03:45,640 --> 00:03:52,480 is meant by this particular flash movie 64 00:03:49,209 --> 00:03:54,970 telling you that the probability p1 for 65 00:03:52,480 --> 00:03:56,799 an interval of Delta X if you are 66 00:03:54,970 --> 00:04:00,340 locating the particle in that interval 67 00:03:56,799 --> 00:04:02,760 across this its uniform now if the Delta 68 00:04:00,340 --> 00:04:07,450 X is different it's going to be 69 00:04:02,760 --> 00:04:11,200 different by that corresponding ratio 70 00:04:07,450 --> 00:04:14,019 namely Delta x2 by M this is what we 71 00:04:11,200 --> 00:04:16,750 expect the particles position and its 72 00:04:14,019 --> 00:04:18,729 probability to be associated with but in 73 00:04:16,750 --> 00:04:20,620 quantum mechanics we don't see that in 74 00:04:18,729 --> 00:04:22,960 this particular case we see is like a 75 00:04:20,620 --> 00:04:25,510 different result you see that the 76 00:04:22,960 --> 00:04:28,240 probability is not uniform and the 77 00:04:25,510 --> 00:04:30,510 probability is also not the same for 78 00:04:28,240 --> 00:04:33,190 different energies at any given location 79 00:04:30,510 --> 00:04:36,610 so let's look at this particular graph 80 00:04:33,190 --> 00:04:37,710 here let's look at this region right in 81 00:04:36,610 --> 00:04:41,850 the middle 82 00:04:37,710 --> 00:04:44,340 for n equal to one the red line you see 83 00:04:41,850 --> 00:04:48,780 that near the middle if you are looking 84 00:04:44,340 --> 00:04:51,870 at a probability of this kind the way to 85 00:04:48,780 --> 00:04:56,220 represent this is using this simple 86 00:04:51,870 --> 00:05:05,520 picture and if we have we function in 87 00:04:56,220 --> 00:05:07,289 this finite region the probability in 88 00:05:05,520 --> 00:05:10,530 the middle region for a small interval 89 00:05:07,289 --> 00:05:21,590 DX if you are looking at it it's the 90 00:05:10,530 --> 00:05:24,300 area here DX that small this size square 91 00:05:21,590 --> 00:05:30,539 calculate it right at the middle value 92 00:05:24,300 --> 00:05:34,800 of the interval and likewise if you have 93 00:05:30,539 --> 00:05:39,389 a particle with a slightly different 94 00:05:34,800 --> 00:05:45,389 energy this energy is e 1 and that's H 95 00:05:39,389 --> 00:05:50,789 square by 8 M L square if the particle 96 00:05:45,389 --> 00:05:55,370 has energy e 2 now you see that the sine 97 00:05:50,789 --> 00:06:00,090 wave gives you for the same thing yeah 98 00:05:55,370 --> 00:06:05,550 two components a maximum at this point 99 00:06:00,090 --> 00:06:11,009 and again a maximum this is L by 4 this 100 00:06:05,550 --> 00:06:14,039 is L by 2 this is 3 L by 4 and this is L 101 00:06:11,009 --> 00:06:16,440 and this is 0 okay so you see that the 102 00:06:14,039 --> 00:06:19,800 probability maximizes in the 103 00:06:16,440 --> 00:06:23,220 intermediate region see L by 4 and 3 L 104 00:06:19,800 --> 00:06:25,409 by 4 but its minimum in the middle and 105 00:06:23,220 --> 00:06:27,810 also the probability is not uniformed 106 00:06:25,409 --> 00:06:30,659 the probability of locating the particle 107 00:06:27,810 --> 00:06:34,320 in a certain region is dependent on what 108 00:06:30,659 --> 00:06:37,380 that region is and also what the energy 109 00:06:34,320 --> 00:06:40,699 associated with that particular wave 110 00:06:37,380 --> 00:06:42,979 function is for that particle so this is 111 00:06:40,699 --> 00:06:45,599 something very unique and very special 112 00:06:42,979 --> 00:06:48,419 with quantum mechanics and it's not 113 00:06:45,599 --> 00:06:50,790 something that we can immediately accept 114 00:06:48,419 --> 00:06:53,040 as a 115 00:06:50,790 --> 00:06:55,680 I mean something that makes sense that's 116 00:06:53,040 --> 00:06:58,410 all the 80s if you write the Schrodinger 117 00:06:55,680 --> 00:07:00,840 equation and if the interpretation of 118 00:06:58,410 --> 00:07:04,560 Max Monde as the most meaningful 119 00:07:00,840 --> 00:07:06,420 interpretation for the wave function is 120 00:07:04,560 --> 00:07:08,880 to be accepted then these are 121 00:07:06,420 --> 00:07:12,300 consequences with that accepting 122 00:07:08,880 --> 00:07:16,290 accepting those what are called the 123 00:07:12,300 --> 00:07:20,010 cornerstones and the probability varies 124 00:07:16,290 --> 00:07:23,010 depending on what the energy is as you 125 00:07:20,010 --> 00:07:25,740 see in this particular graph you see 126 00:07:23,010 --> 00:07:27,600 that the probabilities according to the 127 00:07:25,740 --> 00:07:32,250 square of the wave function plot that 128 00:07:27,600 --> 00:07:35,940 you have here for example in the 1/4 129 00:07:32,250 --> 00:07:39,870 length of the box or the 3/4 length of 130 00:07:35,940 --> 00:07:42,000 the box is different and you have a 131 00:07:39,870 --> 00:07:44,420 little bit of a different energy then 132 00:07:42,000 --> 00:07:48,780 you see that at that point it's actually 133 00:07:44,420 --> 00:07:50,250 very nearly zero and then it varies but 134 00:07:48,780 --> 00:07:53,730 what happens when the yen becomes 135 00:07:50,250 --> 00:07:56,460 extremely large you see that all these 136 00:07:53,730 --> 00:07:58,920 things become more or less uniform I 137 00:07:56,460 --> 00:08:01,140 mean there's so many oscillations that 138 00:07:58,920 --> 00:08:03,630 if you take any particular small region 139 00:08:01,140 --> 00:08:05,340 if the end is extremely large the 140 00:08:03,630 --> 00:08:08,480 probability of locating the particle in 141 00:08:05,340 --> 00:08:11,910 that region is nearly the same as 142 00:08:08,480 --> 00:08:14,700 locating the particle in any other but 143 00:08:11,910 --> 00:08:17,850 the same extent of that region therefore 144 00:08:14,700 --> 00:08:20,850 the classical idea gets closer and 145 00:08:17,850 --> 00:08:23,430 closer to the reality of a quantum 146 00:08:20,850 --> 00:08:25,410 equation when the energies of the 147 00:08:23,430 --> 00:08:31,040 particles become extremely large no 148 00:08:25,410 --> 00:08:35,220 there are two conditions namely in the 149 00:08:31,040 --> 00:08:38,370 energy expression that you have here you 150 00:08:35,220 --> 00:08:40,680 remember H Square and the N square the 151 00:08:38,370 --> 00:08:41,310 mass of the particle is m and the length 152 00:08:40,680 --> 00:08:43,950 of the box 153 00:08:41,310 --> 00:08:47,010 l both of them determine what is the 154 00:08:43,950 --> 00:08:54,330 spread between different energies so if 155 00:08:47,010 --> 00:09:01,950 you plot a 1 it's 1 times H square by 8 156 00:08:54,330 --> 00:09:04,380 ml square e 2 is 4 frames that unit e 3 157 00:09:01,950 --> 00:09:10,140 is 9 times 158 00:09:04,380 --> 00:09:17,010 that unit e4 is 16 times that unit and 159 00:09:10,140 --> 00:09:19,830 as you increase this energy you see that 160 00:09:17,010 --> 00:09:21,870 the gap between the energies increased 161 00:09:19,830 --> 00:09:24,060 but you can also see the following 162 00:09:21,870 --> 00:09:26,040 namely as you increase these energies 163 00:09:24,060 --> 00:09:28,920 the probabilities of finding the 164 00:09:26,040 --> 00:09:32,700 particle in any given region more or 165 00:09:28,920 --> 00:09:35,160 less approaches a constant value which 166 00:09:32,700 --> 00:09:37,530 is proportional to the extent of that 167 00:09:35,160 --> 00:09:40,500 region divided by the total length of 168 00:09:37,530 --> 00:09:44,490 that region for very very large values 169 00:09:40,500 --> 00:09:47,130 of n that's the classical limit and that 170 00:09:44,490 --> 00:09:49,140 classical limit can be obtained by doing 171 00:09:47,130 --> 00:09:51,990 some manipulations here by choosing a 172 00:09:49,140 --> 00:09:54,800 heavy particle so that the gaps are 173 00:09:51,990 --> 00:09:59,280 smaller and therefore the larger 174 00:09:54,800 --> 00:10:01,790 energies are reached very quickly or the 175 00:09:59,280 --> 00:10:04,950 length of the box is very large 176 00:10:01,790 --> 00:10:07,080 macroscopic dimension macroscopic 177 00:10:04,950 --> 00:10:09,060 particle you see that the quantum 178 00:10:07,080 --> 00:10:12,600 conditions are becoming less and less 179 00:10:09,060 --> 00:10:15,030 important but smaller particles narrow 180 00:10:12,600 --> 00:10:18,270 region the quantization results are 181 00:10:15,030 --> 00:10:20,040 somewhat unique and that's where we have 182 00:10:18,270 --> 00:10:22,080 to spend a lot of time trying to 183 00:10:20,040 --> 00:10:25,290 understand why quantum mechanics is 184 00:10:22,080 --> 00:10:28,260 important for atomic particles in atomic 185 00:10:25,290 --> 00:10:30,000 dimensions when y n is very large when y 186 00:10:28,260 --> 00:10:32,790 m is very large you see that these 187 00:10:30,000 --> 00:10:36,150 quantization energies the gaps between 188 00:10:32,790 --> 00:10:38,730 them are not very important and a very 189 00:10:36,150 --> 00:10:41,520 large amount of energy that is very high 190 00:10:38,730 --> 00:10:43,890 values of the quantum number or easily 191 00:10:41,520 --> 00:10:45,840 reached and the particle behaves more or 192 00:10:43,890 --> 00:10:48,690 less like classically so this is an 193 00:10:45,840 --> 00:10:51,840 important are what is called the 194 00:10:48,690 --> 00:10:54,020 correspondence limit as one would call 195 00:10:51,840 --> 00:10:54,020 it 196 00:10:57,820 --> 00:11:07,340 and this was first considered by needs 197 00:11:01,040 --> 00:11:09,160 more it's more the correspondence 198 00:11:07,340 --> 00:11:12,890 principle is that quantum mechanics 199 00:11:09,160 --> 00:11:14,540 approaches classical predictions 200 00:11:12,890 --> 00:11:16,760 classical mechanical predictions for 201 00:11:14,540 --> 00:11:21,430 very very large values of the quantum 202 00:11:16,760 --> 00:11:24,550 number as in this case very large 203 00:11:21,430 --> 00:11:28,610 particles of macroscopic dimensions and 204 00:11:24,550 --> 00:11:31,580 particles of microscopic size then 205 00:11:28,610 --> 00:11:33,800 quantum is less than less important and 206 00:11:31,580 --> 00:11:35,450 that's how we were never able to 207 00:11:33,800 --> 00:11:39,050 discover quantum mechanics until we 208 00:11:35,450 --> 00:11:43,060 started looking at the atomic details 209 00:11:39,050 --> 00:11:45,020 more and more closely so this is the 210 00:11:43,060 --> 00:11:47,480 association that I would like you to 211 00:11:45,020 --> 00:11:49,640 have with the particle in a box model we 212 00:11:47,480 --> 00:11:53,330 will look at the particle in a 2d box 213 00:11:49,640 --> 00:11:56,060 and some other simpler models before we 214 00:11:53,330 --> 00:12:00,050 discuss what are called the expectation 215 00:11:56,060 --> 00:12:03,290 values or the average values to be seen 216 00:12:00,050 --> 00:12:04,760 or to be calculated using quantum 217 00:12:03,290 --> 00:12:07,640 mechanics as we will see in the 218 00:12:04,760 --> 00:12:10,310 experiments we will put up for a later 219 00:12:07,640 --> 00:12:12,310 lecture but remember the message that we 220 00:12:10,310 --> 00:12:14,690 have here quantization of energies 221 00:12:12,310 --> 00:12:16,220 probability descriptions and the 222 00:12:14,690 --> 00:12:19,220 probability descriptions being very 223 00:12:16,220 --> 00:12:22,190 different from classical expectations 224 00:12:19,220 --> 00:12:24,530 and when the system becomes more and 225 00:12:22,190 --> 00:12:27,230 more classical with larger and larger 226 00:12:24,530 --> 00:12:29,450 energies these are the pointers that you 227 00:12:27,230 --> 00:12:31,910 have to have in order to understand 228 00:12:29,450 --> 00:12:33,800 things better in the next lecture we 229 00:12:31,910 --> 00:12:36,740 will start looking at it not one 230 00:12:33,800 --> 00:12:38,770 dimension but motion in a plane that is 231 00:12:36,740 --> 00:12:42,470 there are two coordinates X and y or 232 00:12:38,770 --> 00:12:45,290 even three XY and Z and look at the 233 00:12:42,470 --> 00:12:48,950 particles and will see one more unique 234 00:12:45,290 --> 00:12:53,950 result namely the degeneracy we will 235 00:12:48,950 --> 00:12:53,950 look at that in the next lecture 236 00:12:55,400 --> 00:12:58,520 thank you 237 00:13:14,410 --> 00:13:16,470 you