1 00:00:00,012 --> 00:00:04,414 Now, we're going to put to work the main equation of Quantum Mechanics, the 2 00:00:04,414 --> 00:00:09,384 Schrodinger equation, and use it to solve a very interesting problem that will also 3 00:00:09,384 --> 00:00:14,354 illustrate why quantum mechanical effects are not important in our everyday lives 4 00:00:14,354 --> 00:00:18,257 while being essential when we're deal with atomic link scales. 5 00:00:18,257 --> 00:00:23,117 The question we're going to ask is what happens if when enforced a particle like 6 00:00:23,117 --> 00:00:25,971 solution in our Quantum Mechanical problem. 7 00:00:25,971 --> 00:00:30,444 So basically, what it means is that we postulate, let's postulate that the t 8 00:00:30,444 --> 00:00:35,272 equals 0, at initial moment of time, t equals 0, we have the wave-function such 9 00:00:35,272 --> 00:00:38,138 as this, which is called Gaussian wave-packet. 10 00:00:38,138 --> 00:00:44,448 And this wave-function it, it basically describes the particle which is localized 11 00:00:44,448 --> 00:00:49,511 in space around 0 so it's a typical Gaussian around 0 and this spread out of 12 00:00:49,511 --> 00:00:54,657 this Gaussian is a form of g, which is the parameter of the [unknown] in some 13 00:00:54,657 --> 00:00:58,208 coefficient is not very important at this stage. 14 00:00:58,209 --> 00:01:03,393 Now, the calculation that we are going to present is essentially to the evolution of 15 00:01:03,393 --> 00:01:08,357 this initial condition with time, under the action of the Schrodinger equation, 16 00:01:08,357 --> 00:01:11,924 and it's a big technical question. So, for those of you who are not 17 00:01:11,924 --> 00:01:15,857 interested in technical details, I would like just to present the final answers so 18 00:01:15,857 --> 00:01:19,576 that you can understand the bottom line without going into technicalities. 19 00:01:19,576 --> 00:01:25,342 And so, the final answer is here, so here, we have the wave-function as the function 20 00:01:25,342 --> 00:01:29,726 of time models of it squared. So, we describe initial condition 21 00:01:29,726 --> 00:01:35,690 particles and as we will see, this psi squared is, has the same Gaussian form as 22 00:01:35,690 --> 00:01:40,423 the regional wave-function. But the difference here is that the 23 00:01:40,423 --> 00:01:45,252 parameter d, which describes its spreading, it increases with time. 24 00:01:45,252 --> 00:01:50,833 And it increases with time as 1 plus t squared over some tau squared, where tau 25 00:01:50,833 --> 00:01:56,499 squared is this typical time scale at which the spreading occurs. 26 00:01:56,499 --> 00:02:01,746 So for now, look at this dependence of psi squared on x. 27 00:02:01,746 --> 00:02:06,923 And as time goes by, we will see a more and more uniform density. 28 00:02:06,923 --> 00:02:13,414 So eventually, it will be almost flat. Now, this basically implies that the 29 00:02:13,414 --> 00:02:18,662 particle light solution, this type of wave [unknown] is unstable in quantum 30 00:02:18,662 --> 00:02:22,663 mechanics. So but the important thing is the time 31 00:02:22,663 --> 00:02:26,484 scale itself that we, at which this spreading occurs. 32 00:02:26,484 --> 00:02:32,928 And let us estimate this time scale for two sort of diametrically opposite cases. 33 00:02:32,928 --> 00:02:38,238 So, one case will be the case of an electron and microscopic elementary 34 00:02:38,238 --> 00:02:44,088 particle with a mass of about 10 to minus 27 grams and let's assume that it's 35 00:02:44,088 --> 00:02:50,250 localized on atomic length scales [unknown] 10 to minus eight centimeters. 36 00:02:50,250 --> 00:02:54,942 So, if we plug in the numbers, what we're going to, to see is the typical time scale 37 00:02:54,942 --> 00:02:59,614 which the spreading of the localization radius, if you want of this electron 38 00:02:59,614 --> 00:03:05,381 doubles is in the order of 10 to the minus 16th seconds which is an extremely short 39 00:03:05,381 --> 00:03:08,639 time scale hardly observable in any experiment. 40 00:03:09,730 --> 00:03:13,888 Now, if we take on the contrary in a microscopic object, let's say, a typical 41 00:03:13,888 --> 00:03:18,244 human being, which is also described by Laws of Quantum Mechanics so the 42 00:03:18,244 --> 00:03:22,716 difference will be quite noticeable. So here, while the mass, of course, of the 43 00:03:22,716 --> 00:03:26,610 microscopic object here will be around, let's say, 50 kilograms, and let's say 44 00:03:26,610 --> 00:03:29,524 want to be localized on a distance of total 1 centimeter. 45 00:03:29,524 --> 00:03:34,633 Now, if we now plug in the numbers into this equation, what we're going to see is 46 00:03:34,633 --> 00:03:40,708 that the typical time scale at which sort of quantum delocalization of human being 47 00:03:40,708 --> 00:03:44,626 recursed is a [unknown] 10 to, to, to 30 to plus 30 seconds. 48 00:03:44,626 --> 00:03:49,844 So, to put this number in perspective, so this is equal to the 10 to the 23 years or 49 00:03:49,844 --> 00:03:53,842 10 to the 13th, 10 trillion lifetimes of the universe years. 50 00:03:53,843 --> 00:03:56,960 So basically, it's in some sense, meaningless number. 51 00:03:56,960 --> 00:04:01,618 Sp, it's a number which never, which is never relevant in any realistic 52 00:04:01,619 --> 00:04:07,432 circumstances so we should not be worrying too much about being quantum delocalized 53 00:04:07,432 --> 00:04:11,929 due to quantum mechanical effects. And if we now look at different time 54 00:04:11,929 --> 00:04:15,959 scales for various objects that we're dealing in, in our everyday lives, we're 55 00:04:15,959 --> 00:04:20,082 going to see that essentially, for all these objects, even the smallest ones, we 56 00:04:20,082 --> 00:04:24,280 can safely say that quantum mechanical effects are completely irrelevant. 57 00:04:24,280 --> 00:04:29,032 But at the moment, we're going to, we're going to go to a fundamental microscopic 58 00:04:29,032 --> 00:04:31,966 run scales is going to change very significantly. 59 00:04:31,967 --> 00:04:36,282 Now, I'm going to actually derive the main result that they presented in the previous 60 00:04:36,282 --> 00:04:39,240 slide about the revolution of the Gaussian wave predicate. 61 00:04:39,240 --> 00:04:43,504 And as I mentioned to you, this derivation is a bit on the technical side so those of 62 00:04:43,504 --> 00:04:47,584 you who are not really interested in this technicalities may just skip it towards 63 00:04:47,584 --> 00:04:51,825 the end of this lecture. But those who want to actually learn the 64 00:04:51,825 --> 00:04:57,000 technical part of Quantum Mechanics, which is the main part of Quantum Mechanics, 65 00:04:57,000 --> 00:05:00,665 should pay attention perhaps to derive it on your own. 66 00:05:00,665 --> 00:05:05,898 Now, the main element of the solution is the decomposition of the original the 67 00:05:05,898 --> 00:05:09,933 initial condition of this wave-packet Into the plane-waves. 68 00:05:09,933 --> 00:05:14,657 So, the plane-waves are presented here. So, sort of as an intuition as to why we 69 00:05:14,657 --> 00:05:19,640 want to do something like this, we should recall that the Schrodinger's equation was 70 00:05:19,640 --> 00:05:24,029 deduced or derived from almost by construction to describe waves rather than 71 00:05:24,029 --> 00:05:26,604 particles. So, the initial condition that we can 72 00:05:26,604 --> 00:05:29,242 choose is in some sense, an arbitrary function. 73 00:05:29,242 --> 00:05:33,902 So, there's actually no reason why it would be a convenient solution to, to 74 00:05:33,902 --> 00:05:36,802 anything. So, while the plane-waves are indeed the 75 00:05:36,802 --> 00:05:41,716 solutions, and so by the composing this function or any initial condition for this 76 00:05:41,716 --> 00:05:45,779 matrix, this plane-waves, we actually simplify things a lot, as we will see 77 00:05:45,780 --> 00:05:48,837 later. From the mathematical point of view, this 78 00:05:48,837 --> 00:05:53,737 decomposition is nothing but a Fourier transform of this Gaussian wave-function, 79 00:05:53,737 --> 00:05:58,106 into the into the harmonic function, into this cosines and sines. 80 00:05:58,106 --> 00:06:02,867 And we know very well how to do this decomposition but before we before we go 81 00:06:02,867 --> 00:06:07,758 to it, I would like to remind you of the following identity for Gaussian integral 82 00:06:07,758 --> 00:06:11,577 which we, which actually we're going to use at least three times in this 83 00:06:11,577 --> 00:06:15,840 deliberation, in different topics. So, if we have an exponential of a 84 00:06:15,840 --> 00:06:20,455 quadratic function with some coefficients alpha and beta here, so the integral from 85 00:06:20,455 --> 00:06:24,160 minus infinity to plus infinity is well-known and it's given by this 86 00:06:24,160 --> 00:06:26,774 equation. So, we're going to just use it later on. 87 00:06:26,775 --> 00:06:31,571 Now the reasons UNKNOWN] in this, this stage of the derivation is because the 88 00:06:31,571 --> 00:06:36,501 Fourier transform or in other words this, this Fourier harmonics which in the 89 00:06:36,501 --> 00:06:41,125 context of Quantum Mechanics are called wave-function in the momentum space are 90 00:06:41,125 --> 00:06:44,317 determined exactly by this type of Gaussian integral. 91 00:06:44,317 --> 00:06:49,565 Because what we have to do in order to do in [unknown] on this phi of p, we have to 92 00:06:49,565 --> 00:06:55,141 take avoid the integral of this of this exponential e to the power of minus x 93 00:06:55,141 --> 00:07:00,743 squared over 2d squared with, with this e to the power minus i over h bar px. 94 00:07:00,743 --> 00:07:07,623 And if we look at this integral, so, and compare it with this Gaussian expression, 95 00:07:07,623 --> 00:07:12,732 we will, we will see that alpha here is, is equal to 1 over 2 d squared. 96 00:07:12,733 --> 00:07:22,320 And beta is equal to minus i h bar p. So if we use now this formula, so beta 97 00:07:22,320 --> 00:07:29,255 squared is equal to minus p squared over h squared and divided by 4 alpha will give 98 00:07:29,255 --> 00:07:33,964 us the Fourier expression. So, if I completely ignore the overall 99 00:07:33,964 --> 00:07:37,334 coefficient, it's not really important for the following. 100 00:07:37,334 --> 00:07:43,527 So, now let me just erase this and get rid of this Gaussian integral. 101 00:07:43,528 --> 00:07:51,028 So now we're going to interpret, actually, this Fourier transform in an interesting 102 00:07:51,028 --> 00:07:54,342 way. So the reason we have constructed this 103 00:07:54,342 --> 00:07:59,886 initial condition in this form, was because we wanted to localize our particle 104 00:07:59,886 --> 00:08:03,010 sort of in the certain length scales of order d. 105 00:08:03,010 --> 00:08:07,660 So, the uncertainty of our initial condition in, in, in the real space is of 106 00:08:07,660 --> 00:08:12,535 order d, but by looking at this sort of wave-function in the momentum space, we 107 00:08:12,535 --> 00:08:17,510 see that the uncertainty in momentum delta p is a [unknown] h bar over d. 108 00:08:17,510 --> 00:08:21,699 So, it's inverse proportional to the, a localization distance in real space. 109 00:08:21,699 --> 00:08:26,802 And so, this sort of dual relation, so the more it's localized in real space, the 110 00:08:26,802 --> 00:08:30,950 less it's localized in the momentum space and vice-versa is a particular 111 00:08:30,950 --> 00:08:35,098 manifestation of what is known as the general Heisenberg uncertainty of 112 00:08:35,098 --> 00:08:40,582 principle, which is which is written here. So, that in, in general, whatever wave 113 00:08:40,582 --> 00:08:45,808 function you have, whatever quantum state you possibly can construct, there is 114 00:08:45,808 --> 00:08:51,307 always a constraint that the uncertainty in position, then the uncertainty in 115 00:08:51,307 --> 00:08:56,493 momentum is larger or equal than h bar, h bar being the plane constant. 116 00:08:56,493 --> 00:09:02,193 So now, we're in the position to solve the main technical problem that we formulated 117 00:09:02,193 --> 00:09:08,080 in the beginning called this, this segment that is to solve the Schrodinger equation 118 00:09:08,081 --> 00:09:12,860 the free Schrodinger equation here. So, the [unknown] here is just p squared 119 00:09:12,860 --> 00:09:15,560 over 2m with initial condition written as so. 120 00:09:15,560 --> 00:09:20,180 So, instead of writing it as a Gaussian real space, we write it as a linear 121 00:09:20,180 --> 00:09:25,108 combination of plane -waves that was basically the Fourier transform in the 122 00:09:25,108 --> 00:09:29,001 previous slide. Now, in order to write the time-dependent 123 00:09:29,001 --> 00:09:34,092 wave-function we need to take into account two simple circumstances. 124 00:09:34,092 --> 00:09:38,370 The first circumstance is the fact that the Schrodinger equation, the fact that 125 00:09:38,370 --> 00:09:40,867 the Schrodinger equation is a linear equation. 126 00:09:40,868 --> 00:09:46,649 Which implies that if we have several solutions to this equation, psi 1, psi 2, 127 00:09:46,649 --> 00:09:49,532 psi 3, etc. Their sum is also a solution, with the 128 00:09:49,532 --> 00:09:53,962 initial condition being simply the sum of the corresponding initial conditions. 129 00:09:53,962 --> 00:10:00,650 Now, the second circumstance is we have to recall that a plane wave written as so is, 130 00:10:00,650 --> 00:10:04,774 is an, is an eta solution to the Schrodinger equation. 131 00:10:04,774 --> 00:10:09,373 In some sense, we have constructed Schrodinger equation such that it would 132 00:10:09,373 --> 00:10:14,828 give us this guy as a solution. And here, the epsilon of p is the energy 133 00:10:14,828 --> 00:10:20,257 of the electron, the energy of free electron and simply equal to p squared 134 00:10:20,257 --> 00:10:22,882 over 2m. Now if we take a look at the initial 135 00:10:22,882 --> 00:10:27,098 condition again written in this form, we see that it's an integral over momentum, 136 00:10:27,098 --> 00:10:31,339 but integral is a sum, in some sense which just happens to be in infinite psi. 137 00:10:31,340 --> 00:10:38,237 So, we can use the fact that a Schrodinger equation is a linear equation and write 138 00:10:38,238 --> 00:10:43,936 psi of x and t, the full solution as so. So basically, all we have to do is to 139 00:10:43,936 --> 00:10:51,034 replace this plane-wave at, at t equals 0 with the plane-wave at a finite time with 140 00:10:51,034 --> 00:10:56,569 epsilon of p being p squared over 2m. And so, to write now the wave-function in 141 00:10:56,569 --> 00:11:02,482 a more convenient way so what remains now is to calculate the Gaussian integral over 142 00:11:02,482 --> 00:11:06,737 momentum. And it can be done again using the same 143 00:11:06,737 --> 00:11:13,577 identity that we used in the previous slide but now, the parameter alpha here is 144 00:11:13,577 --> 00:11:19,467 going to be d squared over 2h squared plus i, it divided by 2m h bar and the 145 00:11:19,467 --> 00:11:26,354 parameter beta now is ix over h bar. So again, if we use the same identity, 146 00:11:26,354 --> 00:11:33,642 what we get is the, the following expression for the wave-function. 147 00:11:33,642 --> 00:11:38,825 We see that this expression is an intrinsically complex function but if 148 00:11:38,825 --> 00:11:44,000 we're interested just in determining the density of our particles or where the 149 00:11:44,000 --> 00:11:49,025 particle is located, so the only thing which matters, as I mentioned already and 150 00:11:49,025 --> 00:11:53,750 we'll discuss it in more details a little later, is the absolute value of the 151 00:11:53,750 --> 00:11:57,756 wave-function squared. So this absolute value squared can be 152 00:11:57,756 --> 00:12:00,318 calculated. Again, we don't, we don't worry too much 153 00:12:00,318 --> 00:12:03,902 about the overall coefficients when we look at what is what appears in the 154 00:12:03,902 --> 00:12:08,083 exponential. So, we can write this we can write this 155 00:12:08,084 --> 00:12:14,863 density as an exponential, basically from here, minus x squared over d squared, but 156 00:12:14,863 --> 00:12:20,960 there is also another term, which is 1 over 1 plus t squared or tau squared, and 157 00:12:20,960 --> 00:12:26,875 tau here is exactly the time scale we discussed in the beginning, which is mass 158 00:12:26,875 --> 00:12:32,883 times the delocalization length, in some sense, d squared divided by h bar. 159 00:12:32,884 --> 00:12:38,355 So this completes the derivation of the main result that we discussed earlier and 160 00:12:38,355 --> 00:12:43,511 also completes the first lecture. Thank you very much and I will see you in 161 00:12:43,511 --> 00:12:45,425 class later in the week.