1
00:00:02,860 --> 00:00:12,630
Shall we? Shall we begin then? So yesterday we looked at this this pair of wells that was separated by a barrier
2
00:00:12,640 --> 00:00:15,459
so that classically the particle couldn't get from one well to the other.
3
00:00:15,460 --> 00:00:21,310
And we found that the particle got from one well to the other and used that to make a model of
4
00:00:21,310 --> 00:00:26,140
an ammonia maser with a nitrogen atom passing through the barrier formed by the hydrogen atoms.
5
00:00:28,570 --> 00:00:36,250
So let's look at this phenomenon from another another perspective, from the perspective of scattering it, a scattering experiment.
6
00:00:38,020 --> 00:00:42,160
And we'll we'll come on to what this has to do with radioactivity.
7
00:00:43,000 --> 00:00:45,960
I hope at the end. So consider this set up.
8
00:00:45,970 --> 00:00:52,780
We have a we have a potential barrier here of height v nought usual square form for computational convenience.
9
00:00:53,680 --> 00:01:00,549
And we have some incoming stream of particles. We have a beam of particles here represented by the wave function plus each of the ICS.
10
00:01:00,550 --> 00:01:05,140
So this these are of course particles with well-defined momentum approaching the barrier.
11
00:01:05,800 --> 00:01:09,820
We expect to see some of these particles reflected classically.
12
00:01:10,270 --> 00:01:14,580
If the energy of these incoming particles were less than V nought, they would all be reflected.
13
00:01:14,590 --> 00:01:22,990
So we put in a reflected wave here which goes like e to the minus I kicks remember the time dependence of everything in quantum mechanics.
14
00:01:23,620 --> 00:01:34,269
These are in for a state of well-defined energy is e to the minus i.e. upon bar t so these things are
15
00:01:34,270 --> 00:01:40,900
going have sort of e to the minus omega t type dependence and therefore we have a minus sign here.
16
00:01:41,080 --> 00:01:43,420
You're looking at a wave which is travelling to the left.
17
00:01:43,420 --> 00:01:51,489
If you have a plus sign here, you're looking at wave which is travelling to the right and then we expect some of the particles to get through.
18
00:01:51,490 --> 00:01:52,660
So we put in a wave.
19
00:01:53,170 --> 00:02:02,920
In this portion we say the trial solution should look like C with some unknown constant C each of the six and then within the barrier.
20
00:02:02,920 --> 00:02:12,430
If because we can look for solutions at energies lower than V zero, we're going to have B plus each, the k x plus.
21
00:02:12,430 --> 00:02:20,440
The trial trial solution will be a combination of a of a of a of an exponential growth and an exponential decay.
22
00:02:21,280 --> 00:02:28,270
So this is a more complicated so so this is a bit different from what we've done before in two ways.
23
00:02:29,290 --> 00:02:42,500
One is that. Well, the main thing is that we are initial a problem inherently has a lack of left right symmetry right.
24
00:02:42,530 --> 00:02:50,359
We the the potential that we're discussing here has left right symmetry symmetrical around the origin, which I've got to say.
25
00:02:50,360 --> 00:02:55,670
But the origin is here in the middle of the well. This is minus A and this is a over here.
26
00:02:56,000 --> 00:03:01,340
So the potential is going to be an even function of X, same as ever.
27
00:03:01,730 --> 00:03:06,740
But our problem, our initial conditions, the physical situation we wish to discuss,
28
00:03:06,740 --> 00:03:11,080
has a built in asymmetry because the particles have to come in from one side or the other.
29
00:03:11,090 --> 00:03:17,240
Now we could. So that's. And that is computationally very inconvenient.
30
00:03:17,420 --> 00:03:25,370
It stops us using this nice trick, which doesn't it makes it difficult for users for us to use this nice trick of looking for solutions to the
31
00:03:25,370 --> 00:03:30,860
problem which have well-defined parity and thus discussing only what happens that this boundary condition.
32
00:03:31,190 --> 00:03:34,849
With this setup, we're going to have to discuss this boundary condition and this boundary condition,
33
00:03:34,850 --> 00:03:39,020
because you can see there are fundamental differences between what's happening on those two sides.
34
00:03:39,020 --> 00:03:48,349
So you can handle this problem using looking for solutions of well-defined parity, but it's slightly unnatural.
35
00:03:48,350 --> 00:03:52,339
And I think well, it's actually a very well, it's actually a very good way to go.
36
00:03:52,340 --> 00:03:57,070
But it's it's not such an obvious and intuitive way to go, even though it's computationally simpler.
37
00:03:57,090 --> 00:04:01,280
I think it's worthwhile just seeing what happens when you when you play the
38
00:04:01,280 --> 00:04:04,690
game straightforwardly and you'll see the algebra becomes quite unpleasant,
39
00:04:07,850 --> 00:04:11,540
which illustrates the benefits that we had before by assuming well-defined parity.
40
00:04:12,680 --> 00:04:19,940
All right. The other thing that's different here is that because we are considering particles which are free,
41
00:04:20,150 --> 00:04:26,150
you know that because the potential goes to zero outside this interval here, the particles.
42
00:04:26,150 --> 00:04:32,090
And we're going to consider particles with with positive energy, the particles are going to be able to push off to infinity.
43
00:04:32,600 --> 00:04:36,980
So we are not going to find discrete energy levels. We're going to be able to find solutions for any energy.
44
00:04:37,580 --> 00:04:45,890
Right? That's whereas previously we were we had a potential which forbade going to infinity and that made the energy levels discrete.
45
00:04:46,700 --> 00:04:53,390
So those are the those are the differences because we're we're dealing with a different physical situation and it has implications for the mass.
46
00:04:53,870 --> 00:04:59,300
Right. So what do we have to do? Well, we it's very boring. We have to impose continuity of the wave function.
47
00:04:59,690 --> 00:05:03,160
So the wave function here is this wave, plus this wave.
48
00:05:03,860 --> 00:05:11,400
And then it has to be continuous at this boundary X equals A, so it has to give you the same numerical value as the sum of these two things here.
49
00:05:11,420 --> 00:05:22,190
So let's just quickly write that down. So we have a plus e to the minus i k which is the incoming wave evaluated at that barrier x equals minus AA,
50
00:05:22,430 --> 00:05:34,579
plus a minus e to them plus i k and that would better equal b plus E to the minus k minus sorry.
51
00:05:34,580 --> 00:05:37,730
Plus B minus E to the minus.
52
00:05:38,000 --> 00:05:42,590
No that one's got plus k. Right. Many double negatives here unfortunately.
53
00:05:44,330 --> 00:05:46,370
Oh, we forgot I forgot to say, of course,
54
00:05:46,370 --> 00:05:58,699
that we will have as ever that K is equal to the square root of two m times the energy of our squared because p squared over two m is the energy and
55
00:05:58,700 --> 00:06:11,960
p is h bar squared k squared three p is h mark and we will have the big k is equal to the squares of two and v zero minus the overage bar squared.
56
00:06:17,400 --> 00:06:23,520
So this is the continent, this is the condition for the wave function to be continuous at x equals minus say we
57
00:06:23,520 --> 00:06:27,840
require as yesterday that the gradient of the wave function is also continuous there.
58
00:06:28,140 --> 00:06:32,580
So we have to take the gradient of that function on the left and evaluate it x equals
59
00:06:32,580 --> 00:06:42,630
minus j and we find that i k common factor a plus e to the minus i k a minus a minus.
60
00:06:42,990 --> 00:06:59,460
Each of the i k close brackets is equal to big k common factor b plus e to the minus k minus b minus each of the k close brackets.
61
00:07:01,470 --> 00:07:07,560
Then we have sets two equations. Now we have two more equations because we have to get everything hunky dory on the right hand boundary,
62
00:07:07,830 --> 00:07:11,580
which is not now dealt with by symmetry as it was yesterday.
63
00:07:11,580 --> 00:07:16,710
So this is the because this is where life becomes, everything becomes difficult.
64
00:07:17,040 --> 00:07:30,660
So we have c e to the i k is equal to B plus e to the k plus b minus E to the minus k
65
00:07:30,990 --> 00:07:38,729
and we have that i k over k I'll write it us of c e to the well maybe I should do that
66
00:07:38,730 --> 00:07:45,809
one like c e to the I k that's the gradient on the right side is equal to big k common
67
00:07:45,810 --> 00:07:59,160
factor open brackets B plus E to the k a minus B minus E to the minus k close bracket.
68
00:07:59,160 --> 00:08:05,520
And I live in hope. And so my anxiety that that that has been those equations have incorrectly stated.
69
00:08:07,410 --> 00:08:12,660
So what do we have to do? We now have four equations and five unknowns, I think.
70
00:08:12,660 --> 00:08:19,110
Right there are two ways to Bayes and a C, so we will not be able to get rid of all of them.
71
00:08:19,110 --> 00:08:25,919
We will be able to express in principle any one of a B of the APIs and C's in terms of the other one.
72
00:08:25,920 --> 00:08:34,829
And that physically corresponds to the point that the flux of incoming particles is controlled by a plus, and that's in your control.
73
00:08:34,830 --> 00:08:37,140
You can put in more particles or fewer particles,
74
00:08:37,350 --> 00:08:42,840
and that will obviously lead to more particles coming out or fewer particles coming out depending on the coming flux.
75
00:08:43,110 --> 00:08:52,470
So the general idea is you expect to be able the goal is to express any one of these things as a function of a plus,
76
00:08:52,650 --> 00:08:58,740
as a multiple of plus, and we expect them to be linear and A-plus. So that's why we've got two few equations.
77
00:08:58,740 --> 00:09:08,100
We don't physically expect to be able to determine everything. So what we should do is what we should do is engage in an elimination exercise.
78
00:09:08,340 --> 00:09:11,970
A reasonable way to go is to take these two equations here.
79
00:09:12,300 --> 00:09:21,510
Divide this equation by this equation, say, and that will get rid of C and we'll give you a relationship between B plus and B minus.
80
00:09:23,550 --> 00:09:28,560
And then you can take that relationship between B plus and B minus and use it in these
81
00:09:28,560 --> 00:09:34,020
two equations to to express the right to get rid of B minus from these equations,
82
00:09:34,650 --> 00:09:39,780
these two right hand sides. So they both become simple multiples of B plus.
83
00:09:40,440 --> 00:09:46,589
And then you could divide these two equations one by another, the B plus, which will be a common factor on the right hand side.
84
00:09:46,590 --> 00:09:52,920
It will go away and you will be left with a relationship between a plus and a minus, a single relationship, an A-plus and a minus.
85
00:09:53,280 --> 00:09:59,430
So that will be the promised relationship that expresses the number of reflected particles is a multiple of the number of incident particles.
86
00:10:01,440 --> 00:10:09,180
So once you found what a minus is in in terms of a plus, you can go back to your original expression here,
87
00:10:09,300 --> 00:10:16,970
which had only B plus on the right hand side, A minus can be expressed as a function of a plus, the well-defined what B pluses and minuses.
88
00:10:17,110 --> 00:10:22,650
And they can all be they can all be determined. So let me not do all that algebra.
89
00:10:23,730 --> 00:10:31,560
That's the strategy. The execution, of course, is quite tedious and the scope for making errors is quite large.
90
00:10:31,860 --> 00:10:38,909
And in fact I find that there's a typo right there in Equation 540 in the book because when you when you
91
00:10:38,910 --> 00:10:45,000
do eliminate between these two equations here to find out the relationship between B minus and B plus,
92
00:10:45,360 --> 00:11:03,750
it should be that B minus is one minus I K over k over one plus i k over k e to the two big k a b plus.
93
00:11:05,010 --> 00:11:13,130
So that differs from what's in the book, partly by arrangement of this, but more importantly by this having been left out, that's got slipped out.
94
00:11:13,250 --> 00:11:17,780
Now in in the doing the typesetting. Okay.
95
00:11:18,560 --> 00:11:29,210
So we have that relationship there. We stuff this back into the other places and we find that a minus is equal to AA plus.
96
00:11:29,540 --> 00:11:29,810
So.
97
00:11:29,830 --> 00:11:41,270
So I've described how we what we do, we take we take this B minus, use it to get rid of B minus from here and replace that with B plus in some factor.
98
00:11:41,510 --> 00:11:46,190
Then we divide these two equations and then we get this relationship I'm about to write down between a minus and D
99
00:11:46,190 --> 00:12:03,860
plus and it is a minus is a plus E to the minus to i k a q minus one over Q plus one where q is itself pretty yucky.
100
00:12:04,010 --> 00:12:09,710
It's kosh to k minus i.
101
00:12:10,250 --> 00:12:15,290
K on k. Hyperbolic shine of two.
102
00:12:15,320 --> 00:12:20,360
K all over.
103
00:12:23,090 --> 00:12:28,250
Kosh two k minus.
104
00:12:28,520 --> 00:12:33,170
Big K over eight k of shine.
105
00:12:35,750 --> 00:12:40,729
So the algebra is, as promised, all altogether more yet more messy than it was yesterday.
106
00:12:40,730 --> 00:12:47,070
Because we're not exploiting parity. We're not dealing with finding a right.
107
00:12:47,330 --> 00:12:49,110
So what do we want to know about this physically?
108
00:12:49,130 --> 00:12:54,260
What we want to know about this physically, I think, is what is the chance that the particle is reflected?
109
00:12:54,270 --> 00:12:56,550
What is the chance that the particle gets through?
110
00:12:56,570 --> 00:13:03,920
So classically everything will be reflected and the modulus of a minus would be the same as the modulus of a plus.
111
00:13:03,920 --> 00:13:10,760
Right. And you can see that that isn't looking very promising, because that would require that.
112
00:13:12,320 --> 00:13:14,960
Well, basically, the cube is simply enormous, right.
113
00:13:15,350 --> 00:13:21,980
If Cube were very large, then Q minus one would be the same as Q plus one and and everything would be reflected.
114
00:13:22,550 --> 00:13:25,400
But in reality, it's not all going to be reflected.
115
00:13:25,400 --> 00:13:35,450
Something is going to get through how to find see what we could you could take take this a minus expression from it,
116
00:13:36,170 --> 00:13:44,600
as I've described, obtain B plus from B plus, obtain B minus to put these back into the into this equation here, say and find.
117
00:13:44,600 --> 00:13:47,840
See, that's too much like hard work.
118
00:13:48,410 --> 00:13:51,950
It's easier to say that, look, there's going to be conservation of particles.
119
00:13:57,130 --> 00:14:04,300
We've got a well-defined theoretical apparatus, apparatus here, which is not going to which which conserves probability.
120
00:14:04,840 --> 00:14:13,420
So the incoming particles, the A plus or either are all going to go out at the end of the day, either to the left or to the right.
121
00:14:14,170 --> 00:14:28,510
So we can argue that a plus mod squared, which is well, that is the spatial density of incoming particles, if you like.
122
00:14:29,110 --> 00:14:35,500
If you multiply that by the speed of the incoming particles, which is P over M, so Bach over M,
123
00:14:36,280 --> 00:14:42,069
you will get the flux of incoming particles and the flux of incoming particles has to equal to flux of the outgoing particles,
124
00:14:42,070 --> 00:14:47,110
which is a minus square, the square of a minus, the density of outgoing particles.
125
00:14:47,110 --> 00:14:57,160
Again times times h bach over M for the speed plus c mod squared.
126
00:14:57,610 --> 00:15:03,640
Right. So conservation of particles implies this relationship between these amplitudes.
127
00:15:05,150 --> 00:15:13,480
And of course, you can in principle check whether this relation algebraic relationship is satisfied by these equations,
128
00:15:14,080 --> 00:15:19,569
by hard slog, because I've described how you can in fact find C we've already find out found a minus.
129
00:15:19,570 --> 00:15:23,139
You could in fact in principle find C and check that it satisfied this equation.
130
00:15:23,140 --> 00:15:35,860
But we don't want to do all that algebra. So so the point is that what we want to say is the, the, the, the flux of well,
131
00:15:36,130 --> 00:15:44,860
what we want to say is the following actually is the fraction the fraction of particles that get through.
132
00:15:57,500 --> 00:16:05,660
Is obviously the ratio of the incoming flux and the LB Well, the ratio of the outgoing flux to the incoming flux.
133
00:16:06,290 --> 00:16:07,819
So it's going to be this, this,
134
00:16:07,820 --> 00:16:18,110
this fraction that we want to call it F is going to be mod C squared over a plus squared because the contents of proportionality,
135
00:16:18,110 --> 00:16:27,860
namely H back over M between this quantity and the outgoing flux on the right is this is the
136
00:16:27,860 --> 00:16:32,210
same as the console proportionality between this constant and the incoming flux on the left.
137
00:16:32,420 --> 00:16:40,579
So the fraction of particles that get through will be given by this ratio here, which given that relationship.
138
00:16:40,580 --> 00:16:53,059
So in other words, see, let's, let's we can write that now is as a plus mod squared minus a minus mod squared of a plus mod squared,
139
00:16:53,060 --> 00:17:01,340
but we've got a minus mod squared from that expression at the top and it has a multiple of a plus mod squared.
140
00:17:01,700 --> 00:17:11,210
So we can write this as one minus Q minus one over Q plus one mod squared.
141
00:17:19,070 --> 00:17:25,700
Well the mod square of this ratio is the mod square, the ratio of the mod squares of the top of the bottom.
142
00:17:25,700 --> 00:17:37,610
So this can be written as one minus one Q minus one mod square over Q plus one mod square.
143
00:17:38,090 --> 00:17:45,200
So let's address ourselves to what these mod squares are. So what's Q minus one?
144
00:17:47,060 --> 00:17:58,370
Well, Q minus one is going to be well, it's obviously on the top, it will have the existing top minus the bottom.
145
00:17:58,700 --> 00:18:00,890
So when we take away the bottom from the top,
146
00:18:01,070 --> 00:18:19,459
the causes go away and we are left with I think over i k minus ik over k times shine to k and that will be over.
147
00:18:19,460 --> 00:18:24,500
I'll just call it the bottom because it's the we're not really going to take much interest in what this bottom is.
148
00:18:24,950 --> 00:18:34,340
It is the bottom that you see up there, cost to K minus K over K, etcetera, and Q plus one.
149
00:18:36,200 --> 00:18:39,800
The reason we won't care about the bottom is of course it will cancel when we take this ratio.
150
00:18:41,390 --> 00:18:49,730
So for Q plus one, we unfortunately find that the code, the Cochise ad and the shine's irritating refused to cancel.
151
00:18:50,210 --> 00:19:12,860
So this becomes two cosh, two K we're adding so we have minus I k over k plus k of ik only in a bracket shine to k and again that's over the bottom.
152
00:19:18,190 --> 00:19:26,140
So what we need to do now is take the mod square of these two numbers, ratio them and and take it from one.
153
00:19:26,710 --> 00:19:33,820
So the fraction that gets through is going to be one minus that.
154
00:19:34,090 --> 00:19:37,660
So the top of that is completely imaginary rights, pure imaginary.
155
00:19:37,870 --> 00:19:54,540
We should take out an AI from that bracket and then we will find we are staring at K over k plus k over k squared times shine squared to k.
156
00:19:54,550 --> 00:19:59,530
So that's, that's that's Q minus one mod squared.
157
00:19:59,530 --> 00:20:03,450
As regards the top, the bottom we're not interested in because we're going to cancel with the other bottom.
158
00:20:04,420 --> 00:20:14,889
And now we have to put underneath the mod square of this which will be for kosh squared to k, right.
159
00:20:14,890 --> 00:20:17,530
Because this is the real part of it, this is the imaginary part of it.
160
00:20:18,100 --> 00:20:35,770
We take out a factor awry and now we're staring at plus k over k minus k over k squared shine squared took nearly that.
161
00:20:38,290 --> 00:20:43,810
So now we put this all these two bits. It'll simplify if we put these two bits on a common denominator.
162
00:20:44,200 --> 00:20:47,409
So the top one is on.
163
00:20:47,410 --> 00:20:51,790
The common denominator will be this this bottom plus that stuff there.
164
00:20:52,270 --> 00:21:05,110
So this will be a four course squared to K and now we're going to have shine squared.
165
00:21:05,230 --> 00:21:14,860
Let's write it in plus. Yeah, shine squared to K brackets now brackets.
166
00:21:14,860 --> 00:21:19,600
What. We will have this bracket squared.
167
00:21:21,400 --> 00:21:25,270
Well we'll have this bracket squared sorry. Minus this bracket squared.
168
00:21:26,770 --> 00:21:32,440
And when we square these brackets we're going to get K squids of a K squids which will cancel because of that minus sign.
169
00:21:32,440 --> 00:21:36,639
And what will not go away is the mixed term, the product of multiplying this on this,
170
00:21:36,640 --> 00:21:41,320
which generates two and the product of multiplying this on this which generates another two.
171
00:21:41,320 --> 00:21:47,950
So we will get four and will be with the minus sign because this minus sign will is there, you know,
172
00:21:48,520 --> 00:21:55,210
when this comes up here, that minus sign will stick out and this minus sign will make the mixed term minus there.
173
00:21:55,540 --> 00:22:00,520
So this is going to be times minus four and it's over the bottom as you see it,
174
00:22:01,180 --> 00:22:21,820
over four squared to K plus K over K minus K over k squared shine squared to k and the top simplifies most beautifully
175
00:22:22,150 --> 00:22:32,410
because kosh squared minus nine squared is one so the fours can be cancelled and this actually is nothing but one over.
176
00:22:32,980 --> 00:22:51,340
So the fraction is one over cost squared to k plus a quarter of k of a k minus k of a k squared shines squared to care.
177
00:22:56,940 --> 00:23:01,770
How much fun? So what do we learn from this?
178
00:23:02,970 --> 00:23:07,530
What we learn from this is is most interestingly,
179
00:23:07,530 --> 00:23:13,290
is what happens if we have a rather high barrier in the particles of very short of energy to get through.
180
00:23:13,360 --> 00:23:19,829
All right. So so K is a measure of the deficits in energy that the particles.
181
00:23:19,830 --> 00:23:24,480
Right. That they have by how much they don't have enough energy classically to get through the barrier.
182
00:23:26,310 --> 00:23:28,620
If the barrier is very high and they don't have much energy,
183
00:23:28,890 --> 00:23:38,040
then we're looking at the cost of a of a largish number and the shine of a largish number.
184
00:23:39,420 --> 00:24:01,590
And so what we can say is that for large K, we can say that kosh two K behaves pretty much like shine to K behaves like E to the two K.
185
00:24:02,520 --> 00:24:07,110
All right. But we are interested in fact, in cost squared and shine squared.
186
00:24:07,530 --> 00:24:11,830
So F is looking like one over E to the four.
187
00:24:12,840 --> 00:24:21,600
Okay. So if K is an appreciable number, this probability of penetration is becoming small.
188
00:24:21,750 --> 00:24:27,990
Crucial result is that the probability of getting through there is decreasing exponentially fast in the height of the barrier.
189
00:24:31,130 --> 00:24:34,370
So you don't need a very high barrier to make this quite a small effect.
190
00:24:40,490 --> 00:24:49,600
And somewhere here we have. So this is who? This machine goes to sleep as well as the trouble shouldn't go to sleep.
191
00:24:58,730 --> 00:25:02,090
Give up. Is there anything there? I'll just draw it.
192
00:25:02,780 --> 00:25:09,280
Is it. Is it sort of. Yeah. I know that we're saving the planet by having the machine turn itself off, but.
193
00:25:10,100 --> 00:25:20,239
I can't see it. So what, you want to do some steps that sort of become typically what happens when.
194
00:25:20,240 --> 00:25:23,959
KS Very large in detail.
195
00:25:23,960 --> 00:25:37,110
You might want to know. The smaller the smaller k a.
196
00:25:39,360 --> 00:25:45,490
Sorry. Right.
197
00:25:45,500 --> 00:25:52,630
So these results offer a barrier which is so in these in these results, the barrier is not terribly high.
198
00:25:52,640 --> 00:26:03,950
So. So we have V0. Sorry, we have an e is equal to 0.750.
199
00:26:13,970 --> 00:26:24,690
No, no, no. Sorry. What if I don't? What I don't want to over.
200
00:26:27,080 --> 00:26:32,070
Yeah, that is correct. Sorry.
201
00:26:34,710 --> 00:26:40,920
Yeah. The height of the height of the barrier is sorry there's this parameter w it wasn't that which we talked about yesterday,
202
00:26:40,920 --> 00:26:45,660
which is a measure of the width and the height of the parameter of the barrier.
203
00:26:45,870 --> 00:26:53,610
So it's two v0 a squared over each bar.
204
00:26:54,690 --> 00:27:00,899
This animal, right. That's your dimensionless measure of the height and the width of the barrier in
205
00:27:00,900 --> 00:27:04,080
terms of the mass of the particle with no reference to the energy of the particle.
206
00:27:05,460 --> 00:27:09,420
Sorry, that's not the case then. What's being plotted here is.
207
00:27:09,660 --> 00:27:17,400
Is the probability of getting through as a function of your energy of av0.
208
00:27:17,640 --> 00:27:23,100
For barriers of different WS. So I think it's a 0.5 at the top there.
209
00:27:23,370 --> 00:27:29,970
Yeah. So here's a relatively weak barrier which gives you a fairly small energy is a chance of getting through.
210
00:27:30,510 --> 00:27:34,350
It's not a very fat barrier as the crucial thing. This is a fatter barrier.
211
00:27:34,380 --> 00:27:42,960
This is a fatter barrier. And so you can see how, as a function of the energy, your chance of getting through rises in detail.
212
00:27:42,990 --> 00:27:46,410
Okay. So if we can get these things to stay alive for later,
213
00:27:47,460 --> 00:27:54,300
what's what's physically interesting about this or an interesting application of this is to radioactive decay.
214
00:28:00,510 --> 00:28:04,830
So this is obviously a very simple minded, very simple minded model that we have so far.
215
00:28:05,760 --> 00:28:08,610
But the general idea, for example, is this.
216
00:28:12,210 --> 00:28:22,980
So what we should say is that inside two, three, eight uranium, which is the non fissile source of uranium, you have a number of alpha particles.
217
00:28:27,340 --> 00:28:32,440
It is a simple minded picture. So what does the potential energy of an alpha particle.
218
00:28:32,440 --> 00:28:35,770
So. So we kind of consider this to be so.
219
00:28:36,040 --> 00:28:43,420
Two, three, eight. Uranium which decays to 2 to 3, four.
220
00:28:44,530 --> 00:28:53,800
Thorium and an alpha particle with a half life of I think it's 6.4 giga years.
221
00:28:55,270 --> 00:29:04,660
So it takes the age of the universe, typically for a uranium two, three eight made in some supernova to eject an alpha particle.
222
00:29:05,230 --> 00:29:08,380
So what's happening here from this perspective? What's happening?
223
00:29:08,470 --> 00:29:15,340
So what we should do is we should think about this alpha particle in this uranium two, three, four nucleus as a kind of dynamical system.
224
00:29:17,020 --> 00:29:21,610
So the alpha particle, when it's a long way from from when it's a decent distance,
225
00:29:21,940 --> 00:29:28,150
more than ten to the -15 metres or so away from the thorium nucleus is repelled by the electrostatic repulsion.
226
00:29:28,450 --> 00:29:32,680
So the potential energy curve has a sort of one overall type behaviour here.
227
00:29:34,090 --> 00:29:39,250
If you get when it gets close enough to the thorium nucleus, the strong interaction,
228
00:29:39,430 --> 00:29:45,700
it's able to exchange gluons and stuff with, with the, with the alpha particles.
229
00:29:46,270 --> 00:29:50,560
Well with the nucleons inside there and it and it feels an attraction.
230
00:29:50,950 --> 00:29:59,200
So there is a, well it looks a bit like this, except this is extremely narrow.
231
00:29:59,200 --> 00:30:03,670
So the width of this right is in say ten to the -15 metres.
232
00:30:03,990 --> 00:30:15,790
So typical nucleus size. So inside that uranium 238, did you mine in Australia or something?
233
00:30:15,790 --> 00:30:21,489
There's some alpha particle moving around in here with a large velocity,
234
00:30:21,490 --> 00:30:30,819
a sort of relativistic velocity motion inside nuclei is kind of relativistic, so it bangs to and fro across here, right?
235
00:30:30,820 --> 00:30:37,240
If you're moving, if you got ten to the -15 metres to cover and you're travelling at some speed comparable to the speed of light,
236
00:30:37,690 --> 00:30:40,960
uh, that means that you, you cross this thing.
237
00:30:41,430 --> 00:30:44,980
What, what does this give me ten to the -23. Sorry.
238
00:30:46,360 --> 00:30:52,150
Yeah. You need about ten to the -20 3 seconds to cross.
239
00:30:52,330 --> 00:30:58,690
So roughly ten to the 23 times a second. This alpha particle bangs to and fro, to and fro, to and fro.
240
00:30:59,170 --> 00:31:03,250
This will be the classical picture and it needs to do this.
241
00:31:04,120 --> 00:31:10,660
So it does this for on the order of 6.4 giga is so far for many giga years.
242
00:31:11,020 --> 00:31:16,630
So for on the order of, shall we say, ten to the 17 seconds, which is a third of the age of the universe.
243
00:31:17,050 --> 00:31:26,620
So it makes so it makes about ten to the 40 impacts on the barrier.
244
00:31:29,760 --> 00:31:35,000
And then wonderful moment. It gets out on the 10th to the 40th attack, whatever.
245
00:31:35,790 --> 00:31:38,100
It slips through here. It goes off to infinity.
246
00:31:39,270 --> 00:31:46,860
So this astonishing phenomenon of of a systems with incredibly small dynamical times, the smallest dynamical times,
247
00:31:48,240 --> 00:31:58,150
you know, in the in the typical physical world, doing something on a time scale, which is the age of the universe.
248
00:31:58,260 --> 00:32:03,890
It is the most astonishing phenomenon. But how does it happen? It happens through this exponential decay.
249
00:32:03,900 --> 00:32:08,700
The height and width of this barrier are substantial,
250
00:32:09,360 --> 00:32:14,310
but that each of the four is that each of the four times the height and width of the barrier
251
00:32:14,610 --> 00:32:19,860
amplifies this so much that your chance of getting out turns out to be only one in ten to the 40.
252
00:32:21,960 --> 00:32:30,750
So that of a neutron that got trapped in there in a supernova before the sun was born pops out today.
253
00:32:33,450 --> 00:32:36,569
So we should now. So. So that's.
254
00:32:36,570 --> 00:32:39,330
That's the end of games with square potential. Well,
255
00:32:39,330 --> 00:32:47,250
so I hope you get the idea that it's a it's a rather artificial it's a it's a scheme for finding solutions
256
00:32:47,250 --> 00:32:56,250
to the to the time independent Schrodinger equation which can illustrate interesting physical phenomena.
257
00:32:56,250 --> 00:32:58,740
Although it's the potentials themselves are very artificial.
258
00:32:59,070 --> 00:33:07,590
And we should now just ask ourselves what of the results that we've obtained would be spoilt?
259
00:33:07,890 --> 00:33:13,110
What would change if the potential if the changes in potential weren't abrupt?
260
00:33:13,260 --> 00:33:16,319
Right. And in the real world they're not going to be just step potentials.
261
00:33:16,320 --> 00:33:22,380
We've used step potentials as a computational convenience in the real well, they're going to have to extend over some distance.
262
00:33:22,710 --> 00:33:27,360
And one wants to understand it's important to understand which of these results would
263
00:33:27,360 --> 00:33:32,400
survive and which would would be spoilt by by taking a more realistic potential.
264
00:33:32,820 --> 00:33:37,830
And I focussed on problems where stuff would survive and,
265
00:33:37,880 --> 00:33:44,100
and tried to neglect problems or haven't spoken about problems which would be seriously damaged, but you can be misled.
266
00:33:44,790 --> 00:33:54,419
So in particular, if you if we would do an a calculation precisely analogous to this for particles encountering a square potential.
267
00:33:54,420 --> 00:34:05,760
Well we could all this calculation could be pushed through with the minor modification that in here we would have B plus E to the I,
268
00:34:06,030 --> 00:34:12,749
k, x and B minus E to the minus like big k x.
269
00:34:12,750 --> 00:34:20,610
Right. We would have to. So if we had particles moving in here from infinity with an energy greater than zero,
270
00:34:21,540 --> 00:34:26,429
they the particles when they got here would speed up and slow down when they got here and stuff.
271
00:34:26,430 --> 00:34:28,410
And classically, all the particles would pass through.
272
00:34:28,830 --> 00:34:36,000
If you solve this problem using this apparatus here, what you're going to find is that some of the particles are reflected from this barrier.
273
00:34:36,810 --> 00:34:41,490
Well, some of the particles reflected, sorry, from the whole set up. I don't want to say which barrier that are reflected from,
274
00:34:41,490 --> 00:34:48,300
because there are two barriers they can be reflect from and the results are a superposition of those and some particles get through.
275
00:34:51,870 --> 00:34:58,290
And if you do this calculation, you were learning something which will be profoundly changed.
276
00:34:58,500 --> 00:35:03,959
If you are more realistic and say, well, my real potential well of course,
277
00:35:03,960 --> 00:35:08,190
has he's going to have somewhat sloppy, you know, somewhat sloppy boundaries.
278
00:35:08,820 --> 00:35:16,140
And the issue is how steep does something have to be for this to be a decent guide?
279
00:35:16,890 --> 00:35:26,000
The good news is that the the results to that kind of calculation are not going to be profoundly affected if by the by the statements,
280
00:35:26,010 --> 00:35:28,110
they'll be somewhat affected, but not enormously affected.
281
00:35:28,110 --> 00:35:35,549
So long as we stick, we would be misled if we put particles in it, sufficient energy that they were classically able to get over the top.
282
00:35:35,550 --> 00:35:42,930
But if we stick to particles which are classically forbidden in here, we're not going to be enormously deceived by taking sharp boundaries.
283
00:35:44,040 --> 00:35:49,770
How do we do this? Well, what you need to do is numerically solve the time.
284
00:35:50,610 --> 00:35:55,860
It solved the wave equation to solve the time independent Schrodinger equation for some kind
285
00:35:55,860 --> 00:36:03,749
of a for some kind of a potential change which can be made either steep or less steep.
286
00:36:03,750 --> 00:36:16,230
So if you take that, the potential as a function of X is equal to some constant brackets, times nought if model if X is less than minus say.
287
00:36:16,950 --> 00:36:28,220
And in this zone here is something like one minus, uh, sorry, one plus sign pi x over a that.
288
00:36:28,380 --> 00:36:34,580
For Model X less than A and you take it to a one down here, if not x,
289
00:36:34,580 --> 00:36:40,250
if x is greater than I hope I've done that the way I should have done that, then you will.
290
00:36:40,400 --> 00:36:44,930
So this is this is just a simple functional form that describes a curve that looks like this.
291
00:36:44,930 --> 00:36:56,670
Right? It goes from the note here. It's precisely v nought when you're more than a way and it's precisely zero if you're to the left of minus
292
00:36:56,690 --> 00:37:02,030
A and it moves smoothly and continuously with a continuous gradient from one thing to the other thing.
293
00:37:02,030 --> 00:37:05,450
And by changing a, you can make this steeper or less steep.
294
00:37:06,740 --> 00:37:13,760
And it's very straightforward. I urge you to, to try it on your laptop to solve the to solve the time independent Schrodinger equation.
295
00:37:13,760 --> 00:37:18,800
Numerically, there's a problem describing how to do it. I think it's certainly in the book, possibly in a problem set.
296
00:37:19,040 --> 00:37:22,219
And what do you find when you do it? You get this kind of curve here.
297
00:37:22,220 --> 00:37:28,860
So this is the reflection probability as a function of K.
298
00:37:29,360 --> 00:37:39,919
So so that's right. And this is for this is was what I did for an energy E which was equal to 0.7 V zero.
299
00:37:39,920 --> 00:37:45,440
So all of these solutions are for energy equals point is 0.7 V zero,
300
00:37:45,440 --> 00:37:52,640
which in the square with the if we have an abrupt you know sudden change in the
301
00:37:53,810 --> 00:38:00,080
in the potential gives us this probability of roughly 0.1 of being reflected.
302
00:38:01,010 --> 00:38:05,389
So this is the property of reflection. I say something different, this is the probability of reflection.
303
00:38:05,390 --> 00:38:08,930
And the square one gives you the shot, one gives you this.
304
00:38:09,230 --> 00:38:16,219
The numerics reproduce this if you take a and a is now this not the width of a well,
305
00:38:16,220 --> 00:38:19,430
but the width of the transition well to a really is the width of the transition.
306
00:38:21,020 --> 00:38:26,840
If is less than one, then the numerics reproduce the analytic solution.
307
00:38:27,110 --> 00:38:31,280
But if K is bigger than one, you see there's a very look at this is a logarithmic scale, right?
308
00:38:31,280 --> 00:38:35,450
This is this is a probability of 0.1.01.001.
309
00:38:35,450 --> 00:38:41,270
So the probability of reflection drops like a stone as K becomes bigger than one.
310
00:38:42,530 --> 00:38:49,700
So the the abrupt transition is going to be profoundly misleading when unless the transition.
311
00:38:51,960 --> 00:38:59,310
So the step in this case where we have what's crucial here is that we have a transition
312
00:38:59,700 --> 00:39:06,810
from between between two zones within within which the particle is classically allowed.
313
00:39:07,030 --> 00:39:14,100
Right. So the step between classically allowed regions.
314
00:39:23,760 --> 00:39:38,520
Is misleading. It exaggerates reflection if K is greater than on the order of one that is to.
315
00:39:38,520 --> 00:39:41,729
So what does that tell me? K is two pi over lambda.
316
00:39:41,730 --> 00:39:51,150
So that tells me that a if a the transmission width is greater than two pi over the debris wavelength.
317
00:39:51,720 --> 00:40:02,450
Right. So the transition really has to be quite abrupt in terms of this natural, natural sense of scale.
318
00:40:04,220 --> 00:40:08,030
If you ask, so what's the debris wavelength for an electron?
319
00:40:09,860 --> 00:40:26,280
The answer is that it is on the order of 1.2 times ten to the minus nine energy over one EVA to the half metres.
320
00:40:26,810 --> 00:40:28,400
So the de broglie wavelength,
321
00:40:28,400 --> 00:40:35,750
this quantity for an electron and I mentioned electrons obviously because they're things that we do far around laboratories.
322
00:40:36,230 --> 00:40:39,620
People used to follow them around their homes even when they had cathode ray tubes.
323
00:40:42,020 --> 00:40:45,770
So it's a it's a it's a typical kind of particle you want to understand about.
324
00:40:47,690 --> 00:40:55,340
Then the debris wavelength is is a nanometre or so times the energy in electron volts.
325
00:40:55,760 --> 00:41:02,180
Look, that's a minus a half, isn't it? Because the the higher the energy, the shorter the debris wavelength.
326
00:41:04,760 --> 00:41:10,310
So if you're if you're constructing a a step potential,
327
00:41:11,030 --> 00:41:18,200
typically you you are going to be doing it by having some kind of doing some kind of solid state physics so
328
00:41:18,200 --> 00:41:24,200
that those sheets of glass provide pretty much a step change in they provide a change in the refractive index,
329
00:41:24,200 --> 00:41:28,259
which affects photons. Right. So photons hitting the window have a chance of being reflected,
330
00:41:28,260 --> 00:41:32,980
the chance of being transmitted basically as if it were being bounced off a step potential.
331
00:41:32,990 --> 00:41:36,020
Why? Because the photons have wavelengths.
332
00:41:36,230 --> 00:41:45,770
Those photons that we're we're bouncing off the windows have wavelengths of 500 nanometres or something and atoms.
333
00:41:46,790 --> 00:41:53,300
So the size of an atom is, of course, on the order of 5.1 nanometres.
334
00:41:54,620 --> 00:41:59,959
So it's easy using atoms to make, to make changes that occur over a few atoms.
335
00:41:59,960 --> 00:42:08,300
Therefore over on the order of a nanometre, you so you can make if you if you are using atoms to make the barrier,
336
00:42:08,710 --> 00:42:12,320
you know you're propagating an electron through some kind of solid state material.
337
00:42:12,650 --> 00:42:19,430
You can probably you can probably make a step change which has a you can change the
338
00:42:19,430 --> 00:42:23,299
effective potential of the electron experiences within on the order of a nanometre.
339
00:42:23,300 --> 00:42:28,710
So you may be able to get useful results out of this provided your energies are lower than one EV.
340
00:42:29,150 --> 00:42:33,410
But that's extremely challenging. In practice, your energies will typically be higher than one.
341
00:42:34,130 --> 00:42:43,490
So these results are going to be basically misleading. What you see here is, is return of common sense and rationality.
342
00:42:43,760 --> 00:42:50,239
If you if you roll a piece of chalk off the edge of this table, it will, of course, fall.
343
00:42:50,240 --> 00:42:56,480
It won't be reflected. It's not gonna be reflected by the lower potential onset of lower potential.
344
00:42:57,440 --> 00:43:10,669
And that's what's what the numerics are saying here, that unless you have a that in practice when something encounters a drop in potential,
345
00:43:10,670 --> 00:43:16,340
for example, the reflection of chance is going to be in fact, very small because this is not going to be abrupt.
346
00:43:16,340 --> 00:43:22,340
It's going to be like this tiny bit easy, and then everything is basically going to get through.
347
00:43:22,790 --> 00:43:31,130
So what happens? What actually happens is that when you when you have a slow change, a gradual change in the potential,
348
00:43:31,370 --> 00:43:39,230
is that the wavelength as the as the as the electron or the particle comes along, it comes this region of lower potential energy.
349
00:43:39,710 --> 00:43:48,020
We would say it speeds up. The numerics will show you is that the wavelength of the wave is getting shorter.
350
00:43:48,320 --> 00:43:52,040
So the momentum is getting larger because as P is H mark.
351
00:43:52,190 --> 00:43:57,860
Yes, it's speeding up and it's just it the there's no reflected wave.
352
00:43:58,070 --> 00:44:05,960
So the whole thing just just moves into a new regime with a short wavelength of everything changing continuously.
353
00:44:08,180 --> 00:44:12,649
Well, I think that's pretty much all I want to say. So we'll finish there. And that's the end of step potentials.
354
00:44:12,650 --> 00:44:14,060
And on Monday we can start on.