1 00:00:01,710 --> 00:00:05,934 In this video I'm going to talk about the, perhaps the simplest, kind of, 2 00:00:05,934 --> 00:00:09,830 problem in time-dependent quantum mechanics. 3 00:00:09,830 --> 00:00:13,484 That is the problem of a sudden perturbation, so, which happens when a 4 00:00:13,484 --> 00:00:17,390 potential and a quantum problem changes very fast as compared to all other 5 00:00:17,390 --> 00:00:24,410 relevant time scales and the problem. So in another term for such a change, 6 00:00:24,410 --> 00:00:29,880 sudden change in the potential is a Quantum Quench. 7 00:00:29,880 --> 00:00:35,910 So, and this may happen, for instance if we have a potential which changes. 8 00:00:35,910 --> 00:00:38,560 Very rapidly from one form to a different form. 9 00:00:38,560 --> 00:00:42,655 So, let's say here for example I'm showing a harmonic oscillator, but don't 10 00:00:42,655 --> 00:00:47,335 show and for a negative time, we have a particle, let's say, just sitting here on 11 00:00:47,335 --> 00:00:51,755 a stationary state described by the usual Gaussian wave function with is the, 12 00:00:51,755 --> 00:00:55,850 Eigenfunction of the grounds, the grounds that wave function the harmonic 13 00:00:55,850 --> 00:01:02,814 oscillator. And let's imagine that at t equals zero 14 00:01:02,814 --> 00:01:08,840 the frequency of the harmonic oscillator potential changes rapidly. 15 00:01:08,840 --> 00:01:13,390 So it increases rapidly, such that it becomes a completely different potential 16 00:01:13,390 --> 00:01:16,680 now. So what kind of questions can we ask? 17 00:01:16,680 --> 00:01:21,450 So well the main before asking questions let's see what's going on here. 18 00:01:21,450 --> 00:01:26,931 So the mean sort of inside in this problem is that while the potential 19 00:01:26,931 --> 00:01:35,070 changes very rapidly, the wave function does not have the time to change. 20 00:01:35,070 --> 00:01:39,570 So the wave function right before the quench is exactly equal to the wave 21 00:01:39,570 --> 00:01:45,598 function right after the quench. In the approximation of an infinitely 22 00:01:45,598 --> 00:01:51,210 fast or an infinitely sudden you know, quench of the potential. 23 00:01:51,210 --> 00:01:54,410 Let's say in this case from one harmonic oscillator frequency to a different 24 00:01:54,410 --> 00:01:58,847 harmonic oscillator frequency. So but they another important insight is 25 00:01:58,847 --> 00:02:04,820 that the quantum state that used to be a good stable stationary ground state. 26 00:02:04,820 --> 00:02:09,370 It's not longer, an eignestate of our potential, but what is it? 27 00:02:09,370 --> 00:02:14,170 So this wave function, the old wave function, is just some wave function for 28 00:02:14,170 --> 00:02:18,620 the, from the point of view of the new Hamiltonian. 29 00:02:18,620 --> 00:02:23,176 So for instance let me call this guy here psi naught of 1, which refers to the 30 00:02:23,176 --> 00:02:27,531 ground state of the old potential that we had for negative for t is smaller than 31 00:02:27,531 --> 00:02:32,488 zero. And now after we quench it, so this sign 32 00:02:32,488 --> 00:02:37,980 naught 1 is no longer an Eigenstate of the of the new Hamiltonian. 33 00:02:37,980 --> 00:02:43,210 Let's say we have this Hamiltonian h 2 when it acts on sine naught 1. 34 00:02:43,210 --> 00:02:48,260 It's no longer equal to an energy of anything times sine naught 1. 35 00:02:48,260 --> 00:02:51,546 It's no longer proportional to sine naught one so this state is no longer an 36 00:02:51,546 --> 00:02:55,492 eigen state. What it is though, it's, as always we can 37 00:02:55,492 --> 00:02:59,858 use the eigen states of the new Hamiltonian so we can use these eigen 38 00:02:59,858 --> 00:03:05,279 states as a base. So we can, what we can do, we can expand 39 00:03:05,279 --> 00:03:09,626 our wave function which was the eigenstate of the old Hamiltonian in 40 00:03:09,626 --> 00:03:14,318 terms of the well some coefficients here, let me call it c sub n, and n in this 41 00:03:14,318 --> 00:03:20,840 case goes from zero to infinity. In this particular case of Harmonic 42 00:03:20,840 --> 00:03:24,214 oscillator. So and this can be expanded in terms of 43 00:03:24,214 --> 00:03:28,940 the wave functions of the new harmonic oscillator Hamiltonian. 44 00:03:28,940 --> 00:03:31,500 But in general, this coefficients are all nonzero. 45 00:03:31,500 --> 00:03:36,180 So what basically what happens after we do this quench, is that the wave 46 00:03:36,180 --> 00:03:42,810 function, which used to be nice, pure. State with a well defined energy, well 47 00:03:42,810 --> 00:03:46,350 defined frequency, now becomes essentially a bunch of de, incoherent 48 00:03:46,350 --> 00:03:51,556 harmonics with different frequencies. It's something if you, if I were trying 49 00:03:51,556 --> 00:03:55,710 to suggest a possible classical analogue of what happens here. 50 00:03:55,710 --> 00:03:59,630 Imagine you have a guitar string and you, you generated a pure frequency of you 51 00:03:59,630 --> 00:04:02,934 know, pure sound with the initial frequency of the string, and then you 52 00:04:02,934 --> 00:04:06,854 suddenly change the length of the string, well just you know pinch it at a certain 53 00:04:06,854 --> 00:04:11,300 distance. So then you're going to generate all 54 00:04:11,300 --> 00:04:13,500 kinds of new sounds all kinds of harmonic, it's no longer going to be a 55 00:04:13,500 --> 00:04:16,563 pure sound. A similar thing is happening here but 56 00:04:16,563 --> 00:04:19,770 instead of a sound, instead of the strings we're having. 57 00:04:19,770 --> 00:04:25,026 You know, quantum states which become now a linear combination of the new 58 00:04:25,026 --> 00:04:28,932 eigenstates. So an, an interesting illustration 59 00:04:28,932 --> 00:04:32,442 perhaps of a physically relevant phenomenon that can be discussed in this 60 00:04:32,442 --> 00:04:37,430 context here. So let's say again let me go back to, 61 00:04:37,430 --> 00:04:43,003 this original potential. So let's say we have many particles 62 00:04:43,003 --> 00:04:46,490 sitting on the ground state, in the ground state. 63 00:04:46,490 --> 00:04:50,333 So this basically means that there is a [UNKNOWN] of sorts, which we gain a 64 00:04:50,333 --> 00:04:55,150 produced [UNKNOWN]. In many example labratories world wide, 65 00:04:55,150 --> 00:04:59,620 so this is now. done very frequently and the recumulous. 66 00:04:59,620 --> 00:05:06,070 So, and let me say I change the frequency of the trap a, a as I just showed you. 67 00:05:06,070 --> 00:05:09,782 So and what happens is that this particles that used to be in the ground 68 00:05:09,782 --> 00:05:14,198 state of the old potential did now get redistributed in essential all possible 69 00:05:14,198 --> 00:05:20,312 states of the new potential. And after time goes by, so there is some 70 00:05:20,312 --> 00:05:23,168 relaxation going on and the system relaxes to new state, but now the 71 00:05:23,168 --> 00:05:26,534 temperature of the state is going to be large because the temperature is actually 72 00:05:26,534 --> 00:05:29,696 essentially the measure of the typical energy of a particle emitted by the 73 00:05:29,696 --> 00:05:34,080 system. So essentially what I'm saying here is 74 00:05:34,080 --> 00:05:37,072 that any change in the potential in this case it's a harmonic oscillator, but it's 75 00:05:37,072 --> 00:05:42,120 a completely general statement. Any sudden change of the potential always 76 00:05:42,120 --> 00:05:44,475 hits it up. So, it doesn't matter whether we increase 77 00:05:44,475 --> 00:05:46,810 the frequency, if we decrease the frequency. 78 00:05:46,810 --> 00:05:49,960 If we, let's say had the lowest energy state in the beginning with the lowest 79 00:05:49,960 --> 00:05:53,160 possible temperature, let's say even zero, absolute zero in this model, you 80 00:05:53,160 --> 00:05:57,312 could do the quench. So the particle are going to be 81 00:05:57,312 --> 00:06:01,300 redistributed among all possible energies. 82 00:06:01,300 --> 00:06:04,080 And the adventure that I'm going to relax, to find a temperature. 83 00:06:04,080 --> 00:06:06,590 Which is always higher than the original one. 84 00:06:06,590 --> 00:06:10,167 So this is an interesting phenomena so we can understand without any calculations, 85 00:06:10,167 --> 00:06:14,374 just by this discussion. Now another thing that we just mentioned 86 00:06:14,374 --> 00:06:19,086 in the end so here in the context of this of this expansion of the regional wave 87 00:06:19,086 --> 00:06:22,806 function in terms of the new eigenfunctions of the harmonic oscillator 88 00:06:22,806 --> 00:06:28,210 or well any other potential for this matter. 89 00:06:28,210 --> 00:06:31,794 So this coefficient determine the probability of us finding the particle in 90 00:06:31,794 --> 00:06:35,210 the single particle case or the density of particles let's say in the mini 91 00:06:35,210 --> 00:06:39,870 particle case in this specific new eigenstate. 92 00:06:39,870 --> 00:06:51,562 So cn squared is exactly the probability of finding our particle in the state psi 93 00:06:51,562 --> 00:07:02,880 sub n2, 2 again refers to the potential after the quench. 94 00:07:02,880 --> 00:07:09,250 Well at, oops, at T greater than zero. Okay. 95 00:07:09,250 --> 00:07:14,530 It's not always working this writing on a tablet. 96 00:07:14,530 --> 00:07:18,625 Now, so now let me move actually to a similar but physically quite different 97 00:07:18,625 --> 00:07:22,850 problem which also involves some perturbation, but it's a perturbation of 98 00:07:22,850 --> 00:07:28,996 a completely different time. So the toy problem that we're going to 99 00:07:28,996 --> 00:07:34,676 solve was actually a serious research problem back you know 70 years ago or so, 100 00:07:34,676 --> 00:07:40,640 in the good old days of theoretical physics. 101 00:07:40,640 --> 00:07:45,740 And it was, actually considered by. A famous Russian theorist mostly working 102 00:07:45,740 --> 00:07:51,616 in nuclear physics Arkady Migdal in 1939. So this problem I'm going to discuss 103 00:07:51,616 --> 00:07:55,336 actually appears also in the uh, [UNKNOWN] course on theoretical physics 104 00:07:55,336 --> 00:08:00,250 for those of you who have access to this material. 105 00:08:00,250 --> 00:08:03,970 and, Well, the problem, itself, is, quite, 106 00:08:03,970 --> 00:08:06,980 straightforward. So they formulated the problem. 107 00:08:06,980 --> 00:08:10,620 So let's say we have an atom. So, for the sake of simplicity, I'll be 108 00:08:10,620 --> 00:08:14,880 talking about, the, a hydrogen atom. But, in principle. 109 00:08:14,880 --> 00:08:19,577 We can consider any atom including heavy atoms for which it's the most 110 00:08:19,577 --> 00:08:23,990 interesting. due to various applications in nuclear 111 00:08:23,990 --> 00:08:29,540 physics. And lets imagine that a nucleus of this 112 00:08:29,540 --> 00:08:37,250 atom which has isotomic electron or electrons in it's ground state. 113 00:08:37,250 --> 00:08:41,070 Receives a sharp impulse, which gives it a velocity, v. 114 00:08:41,070 --> 00:08:44,283 So we kick it with a with a radiation, or with a hammer, or we drop it from an 115 00:08:44,283 --> 00:08:47,650 airplane, or whatever you want to think about. 116 00:08:47,650 --> 00:08:51,410 So, and what we want to know is what's going to happen with the electron. 117 00:08:51,410 --> 00:08:55,840 So basically here, the sharp, sudden perturbation. 118 00:08:55,840 --> 00:08:59,561 Is that initially we have this atom at rest, and at certain moment of time, 119 00:08:59,561 --> 00:09:04,980 let's sat t equals 0, this atom starts moving with a certain velocity, v. 120 00:09:04,980 --> 00:09:07,500 And if it does so we want to see when the [INAUDIBLE] we want to see if the 121 00:09:07,500 --> 00:09:11,550 electron gets excited. But how do we calculate the probability 122 00:09:11,550 --> 00:09:16,030 for the electron to get excited you know that is to say that it would go to an 123 00:09:16,030 --> 00:09:20,580 excited state with the higher quantum number or the atom would even get 124 00:09:20,580 --> 00:09:25,724 ionized. That is, to say, that the electron just 125 00:09:25,724 --> 00:09:28,820 leaves the atom and goes into the continuum. 126 00:09:28,820 --> 00:09:34,280 And they just go two separate ways, the atom of the nucleus, and the electron. 127 00:09:34,280 --> 00:09:37,680 So here naively it seems that the potential doesn't really change. 128 00:09:37,680 --> 00:09:42,442 We don't quench the potential. but if we look that this problem a little 129 00:09:42,442 --> 00:09:48,030 closer we see that well the atom when it receives this impulse or this kick. 130 00:09:48,030 --> 00:09:51,294 So the atom starts moving so the, and the atom is the potential which binds the 131 00:09:51,294 --> 00:09:56,776 electron to it in the first place. So therefore, the potential in which the 132 00:09:56,776 --> 00:10:00,350 electron is supposed to be bound starts moving. 133 00:10:00,350 --> 00:10:03,742 And so this is a sharp change. And so the original, but the wave 134 00:10:03,742 --> 00:10:06,590 function of the electron does not have time to change. 135 00:10:06,590 --> 00:10:10,302 So if let's say the electron was here, in the vicinity of the lowest orbit so it 136 00:10:10,302 --> 00:10:13,782 was localized in the vicinity of the lowest orbit it will just try to stay 137 00:10:13,782 --> 00:10:18,222 there. And the question we're asking is whether 138 00:10:18,222 --> 00:10:23,016 the atom when it starts moving. with a certain velocity whether it will 139 00:10:23,016 --> 00:10:26,433 carry the electron with it or whether it will leave it behind or maybe whether the 140 00:10:26,433 --> 00:10:30,670 electron will sort of dissipate into higher energy states. 141 00:10:30,670 --> 00:10:35,770 So and for this to solve this problem what turns out to be useful is to go into 142 00:10:35,770 --> 00:10:42,770 another frame of reference. Which is moving together with the atom. 143 00:10:42,770 --> 00:10:46,034 So essentially, we want just to move together with the atom and write the wave 144 00:10:46,034 --> 00:10:50,090 function of the electron in this new frame of reference. 145 00:10:50,090 --> 00:10:54,177 So this is really the key to solving the problem because in the frame of reference 146 00:10:54,177 --> 00:11:00,270 moving together with atom, the the potential is exactly the same. 147 00:11:00,270 --> 00:11:04,010 It was the [UNKNOWN] potential to begin with is the [UNKNOWN] potential here, and 148 00:11:04,010 --> 00:11:07,365 in this frame of reference the atom, the nucleus itself is static, but the 149 00:11:07,365 --> 00:11:10,995 electron now is moving and so essentially what we have to do, we have to write the 150 00:11:10,995 --> 00:11:14,515 electron wave function after the quench, after, well, in this case it's not a 151 00:11:14,515 --> 00:11:17,980 quench a, I'm sorry I shouldn't have said that, after this sort of jolt of the 152 00:11:17,980 --> 00:11:24,697 atom. and this wave function will have two 153 00:11:24,697 --> 00:11:28,945 pieces, so the first piece is going to be just, the standard, ground-state wave 154 00:11:28,945 --> 00:11:34,070 function of the hydrogen atom, in the case of hydrogen atom. 155 00:11:34,070 --> 00:11:37,850 And there will be a piece, a new piece, which is related to the velocity of the 156 00:11:37,850 --> 00:11:43,475 electron moving, with velocity v, so here essentially it's a plane wave. 157 00:11:43,475 --> 00:11:47,255 So a way to write the way function in the frame of reference moving with the 158 00:11:47,255 --> 00:11:52,138 nucleus. is right as a product of a localized way 159 00:11:52,138 --> 00:11:57,140 function in the lowest orbit times the plane wave. 160 00:11:57,140 --> 00:12:00,353 And here again this mv is simply the momentum of the electron in this frame of 161 00:12:00,353 --> 00:12:05,145 reference. And this is just the standard plain wave 162 00:12:05,145 --> 00:12:08,810 expression. So now the final step is to calculate the 163 00:12:08,810 --> 00:12:13,562 excitation probability of the electron. That is, to find the total probability of 164 00:12:13,562 --> 00:12:17,166 an electron, of the electron going to, to some higher energy state, including the 165 00:12:17,166 --> 00:12:21,404 continuum states. And well, the good thing is that we know 166 00:12:21,404 --> 00:12:25,970 the wave function now. So in this moving frame of reference, let 167 00:12:25,970 --> 00:12:30,695 me just write it again, so psi-naught of r is equal to 1 over the square root of 168 00:12:30,695 --> 00:12:35,795 pi a-cubed, so this is Bohr's radius, e to the power minus r over the Bohr radius 169 00:12:35,795 --> 00:12:44,612 and times this plain wave e to the minus i over h-bar mv dot r. 170 00:12:44,612 --> 00:12:49,168 So this our wave function we just derived, sort of, which is by logical 171 00:12:49,168 --> 00:12:54,028 reasoning. And if we want to know, let's say what 172 00:12:54,028 --> 00:12:58,708 the probability of a transition into a higher energy state is, let's say, some 173 00:12:58,708 --> 00:13:03,782 other state. I'm not going to specify what it is, but 174 00:13:03,782 --> 00:13:08,760 some, let's say psi. Size of k, whatever k is. 175 00:13:08,760 --> 00:13:13,375 So k could be a collection of indices involving the radial quantum numbers and 176 00:13:13,375 --> 00:13:19,380 the angular quantum numbers or it can be a label of a continuum state. 177 00:13:19,380 --> 00:13:25,006 Whatever this label, quantum label is. So the amplitude, the transition 178 00:13:25,006 --> 00:13:31,630 amplitude is going to be, let me write it, a of zero to k will be equal to the 179 00:13:31,630 --> 00:13:40,297 overlap integral of this synod. Let me write it star although it doesn't 180 00:13:40,297 --> 00:13:46,645 really matter psi sub key of r integrated over the volume. 181 00:13:46,645 --> 00:13:52,257 And the probability to go from this ground state to an excited state, w sub 182 00:13:52,257 --> 00:13:57,961 k, is going to be equal to the absolute value squared of this transition 183 00:13:57,961 --> 00:14:02,740 amplitude. So this is what it means. 184 00:14:02,740 --> 00:14:08,175 So another way to connect it to what I. Talked about previously is to notice that 185 00:14:08,175 --> 00:14:14,370 this wave function can expanded in the basis of all eigen states of our problem. 186 00:14:14,370 --> 00:14:18,402 In this case actually it's both discreet state we describe atoms, I'm sorry 187 00:14:18,402 --> 00:14:23,335 electron localized near the atom and also the continuum states. 188 00:14:23,335 --> 00:14:27,170 Okay, and it seems that, well the problem is very complicated, so we have to sum 189 00:14:27,170 --> 00:14:32,529 over all this probability. But, well fortunately we don't have to do 190 00:14:32,529 --> 00:14:36,094 so, really. So this was just a, an unnecessary 191 00:14:36,094 --> 00:14:39,533 comment. As a matter of fact, if we're interested 192 00:14:39,533 --> 00:14:43,925 in calculating the total excitation probability, we, may want to notice that, 193 00:14:43,925 --> 00:14:47,829 the total probability to find the electron in some quantum state is equal 194 00:14:47,829 --> 00:14:53,040 to 1, so the electron will stay somewhere, right? 195 00:14:53,040 --> 00:14:57,631 So it will be in some state. So therefore the probability to remain in 196 00:14:57,631 --> 00:15:02,814 the ground state plus the excitation probability, which is essentially the sum 197 00:15:02,814 --> 00:15:10,270 in this ungeneralized sense it also involves an integral of this w sub k. 198 00:15:10,270 --> 00:15:14,130 This this must be equal to 1. And this excitation probability is 199 00:15:14,130 --> 00:15:18,070 exactly what we're trying to find. So, from this equation we simply can 200 00:15:18,070 --> 00:15:22,286 write that this excitation probability is equal to 1 minus the probability not to 201 00:15:22,286 --> 00:15:25,900 get excited. The probability to remain. 202 00:15:25,900 --> 00:15:29,250 In the grounds state now of the moving atom, of the moving nucleus. 203 00:15:30,390 --> 00:15:33,426 Well I just realized actually that there is a slight problem with the notations 204 00:15:33,426 --> 00:15:36,632 here. Because what we're actually looking for 205 00:15:36,632 --> 00:15:40,780 is the transition amplitudes and the transition probability of, from an 206 00:15:40,780 --> 00:15:44,732 electron. That is a moving electron relative to the 207 00:15:44,732 --> 00:15:48,551 stationary nucleus. But we want to find transition's 208 00:15:48,551 --> 00:15:53,450 amplitude into, stationary states, localized near this nucleus. 209 00:15:53,450 --> 00:15:57,450 So therefore, you know? This, us, this case, for instance. 210 00:15:57,450 --> 00:16:01,105 they do not have this, plane wave piece. So let me just. 211 00:16:01,105 --> 00:16:06,145 sort of introduce tilde here which would refer to the fact that these guys are 212 00:16:06,145 --> 00:16:10,897 moving together with that while this state psi-naught that we get in the 213 00:16:10,897 --> 00:16:18,890 previous slide is actually moving with velocity v relative to that. 214 00:16:18,890 --> 00:16:22,322 Now with this correction to notations and with this comment about the total 215 00:16:22,322 --> 00:16:26,340 probability of. Staying in the new ground state, plus the 216 00:16:26,340 --> 00:16:30,025 probability of excitation being equal to one, we can write down this excitation 217 00:16:30,025 --> 00:16:33,410 probability. Let me just do that. 218 00:16:33,410 --> 00:16:38,805 psi star is going to be equal to 1 minus the probability to stay in the ground 219 00:16:38,805 --> 00:16:43,951 state, which means that I have to take this integral of psi-naught with 220 00:16:43,951 --> 00:16:50,129 psi-naught-tilde. This is a ground state of the stationary 221 00:16:50,129 --> 00:16:55,030 atom integrated over the volume and square the absolute value of it. 222 00:16:55,030 --> 00:17:01,250 And so here I know everything there is to know, so for example, this part. 223 00:17:01,250 --> 00:17:05,764 Here is, exactly, this psi naught tilde in my new inconvenient, perhaps, 224 00:17:05,764 --> 00:17:10,100 notations. So we can just plug in this these 225 00:17:10,100 --> 00:17:17,760 functions, and integrate over volume. And I will leave this calculation to you. 226 00:17:17,760 --> 00:17:21,960 So let me just maybe write an intermediate step, and the final result. 227 00:17:21,960 --> 00:17:26,030 So the intermediate step would be simply. Would simply involve writing down 228 00:17:26,030 --> 00:17:30,640 explicitly this integral, so it's going to be 1 minus 1 over pi. 229 00:17:30,640 --> 00:17:37,540 A Bohr radius cube from the normalization factor in the in the ground state wave 230 00:17:37,540 --> 00:17:46,480 function times the exponential, well intregal of the exponential. 231 00:17:46,480 --> 00:17:55,760 Minus 2r over the Bohr radius, minus this plane wave, i over h bar, p with r. 232 00:17:55,760 --> 00:17:58,550 And so integrate it over the volume, and everything is squared. 233 00:17:58,550 --> 00:18:01,790 And so this is really the integral that needs to be calculated. 234 00:18:01,790 --> 00:18:04,738 Well I, I could, I could take another five minutes to, go over with you through 235 00:18:04,738 --> 00:18:07,554 the calculation of the integral, but I think it's probably not the best use of 236 00:18:07,554 --> 00:18:09,974 your time, and my time too, so let me just leave it, for those who are 237 00:18:09,974 --> 00:18:16,110 interested, to do it on your own. And let me just write the final result. 238 00:18:16,110 --> 00:18:22,318 So the final result will be this, 1 minus 1 divided by this ratio here, 1 over 4, 239 00:18:22,318 --> 00:18:28,149 mv squared, n squared v squared a squared. 240 00:18:28,149 --> 00:18:32,501 This is the Bohr radius divided by h bar squared, and everything to the fourth 241 00:18:32,501 --> 00:18:36,001 power. So, I'm not necessarily saying that this 242 00:18:36,001 --> 00:18:39,777 is the most important and illuminating result in physics, but it's a result, 243 00:18:39,777 --> 00:18:43,671 something that you can actually measure and check, so what happens if we kick an 244 00:18:43,671 --> 00:18:47,944 atom. So what is the probability of it being 245 00:18:47,944 --> 00:18:51,463 excited and therefore it would mean protons and there all kinds of different 246 00:18:51,463 --> 00:18:56,526 phenomenon associated with it. So, and well one can round the very 247 00:18:56,526 --> 00:19:00,775 safety checks on this result. So let's say if the velocity is zero, so 248 00:19:00,775 --> 00:19:06,110 we don't really keep the atom so it was stationary have one minus one. 249 00:19:06,110 --> 00:19:08,966 So, the probability to excite the atom is exactly zero, as it should be, because we 250 00:19:08,966 --> 00:19:13,447 don't really do anything. On the other hand, if the velocity here, 251 00:19:13,447 --> 00:19:17,840 well, now in our altruistic case, goes to infinity. 252 00:19:17,840 --> 00:19:21,116 So the, one or infinity is zero, so the probability to excite that atom is 253 00:19:21,116 --> 00:19:24,260 exactly one. So we just leave the electron behind 254 00:19:24,260 --> 00:19:27,450 that's what going to happen. The nucleus will just go away. 255 00:19:27,450 --> 00:19:31,481 That's what's going to happen. In any case, so this is just an example 256 00:19:31,481 --> 00:19:36,236 of that I wanted to show you. So it's actually two examples in this 257 00:19:36,236 --> 00:19:40,400 part of, time dependent quantum mechanical problems. 258 00:19:40,400 --> 00:19:46,736 And in both of these cases we dealt with very sharp, very sudden perturbation's. 259 00:19:46,736 --> 00:19:52,440 To our Hamiltonian and they resulted in redistribution essentially of energies. 260 00:19:52,440 --> 00:19:56,527 So the energy of the original state or is the, well is the original state of the 261 00:19:56,527 --> 00:20:00,560 old Hamiltonian became a linear combination. 262 00:20:00,560 --> 00:20:05,740 A rather complicated linear combination of Eigenstates of the new Hamiltonian. 263 00:20:05,740 --> 00:20:08,872 And this is very typical for sharp perturbations or non adiabatic 264 00:20:08,872 --> 00:20:11,896 perturbations. Now in the next video we're going to 265 00:20:11,896 --> 00:20:15,260 study actually the opposite case of what's very slow perturbation or 266 00:20:15,260 --> 00:20:18,856 adiabatic perturbation which in some sense is more interesting in that it 267 00:20:18,856 --> 00:20:25,178 involves new mathematical structures. A new phenomena, so-called topological 268 00:20:25,178 --> 00:20:28,769 Berry phase, which is currently actually a subject, the subject of intense 269 00:20:28,769 --> 00:20:32,200 research in, in condensed matter physics.