1
00:00:13.504 --> 00:00:15.302
Ladies and gentlemen.
2
00:00:15.615 --> 00:00:24.378
This morning you will have heard lectures in 3 dialects of English, one with a slight Swedish accent,
3
00:00:24.591 --> 00:00:30.961
one with a slight Japanese accent and one with a strong American accent.
4
00:00:32.297 --> 00:00:44.213
Sometimes it helps for people for whom English isn’t a native language to hear the material spoken with a slight accent.
5
00:00:44.213 --> 00:00:47.383
I hope mine can be understood.
6
00:00:48.201 --> 00:00:57.508
Another thing, I believe that I was scheduled to stop this lecture promptly at 11.30 or as some people say half 12,
7
00:00:57.710 --> 00:01:08.022
I will try to stop sooner than that because I wish to go on the trip to Mainau.
8
00:01:08.654 --> 00:01:13.381
(applause).
9
00:01:17.573 --> 00:01:23.875
However I also wish to eat lunch today so that, anyway but I do have a lot of material to cover
10
00:01:24.077 --> 00:01:32.151
and therefore I am going to read the text reasonably quickly.
11
00:01:32.525 --> 00:01:40.628
Quantum mechanics is a very wonderful tool for dealing with problems in atomic molecular and condensed matter physics,
12
00:01:40.871 --> 00:01:44.414
as well as with many parts of chemistry.
13
00:01:44.663 --> 00:01:54.110
It can be based on a small number of axiomatic statements which are easy to apply but hard to understand.
14
00:01:54.110 --> 00:01:58.513
This same quantum mechanics can be extended in very plausible ways
15
00:01:58.513 --> 00:02:04.201
to apply to electromagnetic radiation in interaction with matter.
16
00:02:04.201 --> 00:02:08.183
Many problems in nuclear physics have been treated with quantum mechanics
17
00:02:08.183 --> 00:02:18.212
and the reconciliation of quantum mechanics and special relativity theory has had considerable success.
18
00:02:18.212 --> 00:02:23.107
With general relativity things are rather more difficult still.
19
00:02:23.107 --> 00:02:29.569
Many problems of sub nuclear physics and high energy phenomena have also been treated with quantum mechanics
20
00:02:29.569 --> 00:02:36.425
but the theory is pushed rather far from its roots when dealing with such problems.
21
00:02:36.684 --> 00:02:42.993
My main object in this lecture is to deal with the interpretation of quantum mechanics on the atomic level.
22
00:02:42.993 --> 00:02:48.442
And with the application of that theory to more macroscopic systems
23
00:02:48.764 --> 00:02:54.456
which for example among other things might include measuring instruments.
24
00:02:54.703 --> 00:03:02.688
It is tempting to think that quantum mechanics might be applicable as well to much larger parts of the universe
25
00:03:02.688 --> 00:03:06.637
such as with life and beyond.
26
00:03:06.854 --> 00:03:12.771
But I will try to give you some reasons why such a temptation should be resisted.
27
00:03:13.065 --> 00:03:21.894
Experience and common sense teaches that to learn anything about a sub microscopic system is a difficult task.
28
00:03:21.894 --> 00:03:26.107
Our intuition is a most unreliable guide in this domain.
29
00:03:26.399 --> 00:03:37.696
We have ingrained concepts about the meaning of reality which no doubt will be defined by professor Wigner in the next lecture.
30
00:03:37.696 --> 00:03:42.187
And about causality, the relation between cause and effect.
31
00:03:42.187 --> 00:03:47.855
And about our human role as observers of natural phenomena.
32
00:03:47.855 --> 00:03:55.982
All of these things, concepts are most inappropriate for dealing with the sub microscopic world.
33
00:03:55.982 --> 00:03:59.948
The development of the mathematical structure of quantum mechanics proved easier
34
00:03:59.948 --> 00:04:07.701
than the formulation of a satisfactory interpretation of the theory and description of the process of measurement.
35
00:04:07.914 --> 00:04:13.642
Quantum mechanics can do a lot for us if we regard it simply as a set of computational rules
36
00:04:13.642 --> 00:04:16.304
for dealing with simple dynamical systems.
37
00:04:16.901 --> 00:04:21.403
For instance we can calculate the energies of stationary states of atoms
38
00:04:21.403 --> 00:04:28.006
and make an immediate connection with high precision spectroscopic observations of the last century.
39
00:04:28.508 --> 00:04:34.078
For this one does not need to worry very much about the interpretation of the wave function
40
00:04:34.078 --> 00:04:36.765
which describes the state of the system.
41
00:04:37.059 --> 00:04:41.919
But only to calculate the energies of stationary states.
42
00:04:42.182 --> 00:04:48.195
When we come to discuss scattering processes or radiative transitions between stationary states,
43
00:04:48.195 --> 00:04:51.478
or the theory of measurement of dynamical variables
44
00:04:51.725 --> 00:04:57.704
however we have greater need for a better understanding of the meaning of the wave function.
45
00:04:57.704 --> 00:05:01.796
In the last few years I have seen a number of papers
46
00:05:01.796 --> 00:05:07.006
dealing with interesting applications of quantum mechanics to large scale phenomena.
47
00:05:07.725 --> 00:05:14.998
These range from problems in physics, optical communication theory, molecular biology,
48
00:05:14.998 --> 00:05:18.210
all the way up to the theory of the whole universe.
49
00:05:18.210 --> 00:05:23.454
I have been quite suspicious about the validity of most of this research.
50
00:05:23.454 --> 00:05:33.555
One example of this is found in the theory expounded by Kip Thorne and Carton Caves of Caltech
51
00:05:33.555 --> 00:05:38.184
on quantum demolition measurements of gravity waves.
52
00:05:38.184 --> 00:05:44.233
Gravity waves probably have not been detected yet, perhaps they have but not everybody thinks so.
53
00:05:44.233 --> 00:05:52.096
But many people are working on much more sensitive detectors and probably some day gravity waves will be detected.
54
00:05:53.179 --> 00:06:04.479
However exceeding the limitations imposed by quantum mechanics in such measurements
55
00:06:04.479 --> 00:06:15.457
is the subject of the discipline called quantum non demolition measurements.
56
00:06:15.457 --> 00:06:25.077
This work represents part of a major program at Caltech to build the most sensitive detector possible for gravity waves.
57
00:06:25.339 --> 00:06:32.846
Estimates based on an intuitive application of the uncertainty principle to the experimental configuration
58
00:06:32.846 --> 00:06:38.750
set a quantum limit on the smallest signal which could be received.
59
00:06:38.750 --> 00:06:45.361
Thorne, Caves and others have searched for methods to “beat the quantum limit”.
60
00:06:45.361 --> 00:06:50.575
Their papers are very impressive and persuasive.
61
00:06:50.575 --> 00:07:01.447
Unfortunately for them or more likely for me, they make use of a form of assumption about quantum mechanics
62
00:07:01.447 --> 00:07:04.392
with which I do not agree.
63
00:07:04.641 --> 00:07:12.764
To put it briefly I could say that they postulate the reduction of the wave packet hypothesis of von Neumann.
64
00:07:12.764 --> 00:07:22.414
There are very similar ideas expressed in the first chapter of Professor Dirac’s book on quantum mechanics.
65
00:07:22.614 --> 00:07:31.863
I partly disagree with the statements expressed by those authors
66
00:07:31.863 --> 00:07:44.327
but I disagree more strongly with the use of those hypothesis by people involved in gravity wave measurements.
67
00:07:44.327 --> 00:07:48.863
A gravity wave detector typically consists of a very massive cylinder.
68
00:07:48.863 --> 00:07:58.777
One of its very high Q normal modes is put into vibration by a passing pulse of gravitation radiation.
69
00:07:58.777 --> 00:08:04.591
In simple terms the system consists of a forced simple harmonic oscillator
70
00:08:04.591 --> 00:08:09.828
and should be very familiar to all students of text book quantum mechanics.
71
00:08:09.828 --> 00:08:16.673
At issue is the question how one can measure coordinate and momentum of such a large system.
72
00:08:16.673 --> 00:08:23.968
The method is to trace out the time dependent shape of a gravitational wave pulse
73
00:08:23.968 --> 00:08:29.272
by following the motion of a reference surface on the massive oscillator.
74
00:08:29.272 --> 00:08:32.100
There is general agreement that the gravitational fields
75
00:08:32.100 --> 00:08:37.241
which are to be studied in this research can be treated completely classically.
76
00:08:37.241 --> 00:08:43.837
So there is no need to worry about the quantum theory of gravitation in this study.
77
00:08:43.837 --> 00:08:49.220
No need to talk of gravitons or other quantal aspects of gravitation.
78
00:08:49.220 --> 00:08:54.460
There will be plenty of non quantum mechanical disturbances of the detector to cope with
79
00:08:54.460 --> 00:08:58.590
but Thorne and Caves wish to go further than the quantum limits
80
00:08:58.590 --> 00:09:05.171
since this is required for the detection of some types of expected gravitational radiation.
81
00:09:05.171 --> 00:09:14.706
My feeling is that they may probably succeed if the von Neumann hypothesis is correct and will fail if it is not.
82
00:09:14.981 --> 00:09:26.312
Perhaps in 3 or 4 years one of those gentlemen will be here to tell you what they did.
83
00:09:26.513 --> 00:09:33.095
Another field where quantum mechanics has been applied to macroscopic phenomena is in the theory of optical communication,
84
00:09:33.095 --> 00:09:42.474
here we might consider a signal generator such as a laser, a transmission medium, perhaps between the earth and the moon,
85
00:09:42.474 --> 00:09:49.644
and a detector which might consist of a photoelectric device and associated electronic circuits.
86
00:09:49.644 --> 00:09:55.867
Since the von Neumann reduction hypothesis plays an essential role in this theory,
87
00:09:55.867 --> 00:10:01.540
the foundations of which were laid at Bell telephone labs with many other people working on the subject.
88
00:10:01.540 --> 00:10:05.091
I believe that this theory is fatally flawed.
89
00:10:05.405 --> 00:10:11.599
Quantum mechanics is now over 50 years old, 55 years old.
90
00:10:11.599 --> 00:10:23.080
I taught graduate courses in that subject for over 35 years at Columbia, Stanford, Oxford, Yale and the University of Arizona.
91
00:10:24.144 --> 00:10:30.462
My lectures always began with an explanation that one must first learn the rules of calculation in quantum mechanics
92
00:10:30.462 --> 00:10:33.939
before one can understand the physical meaning of the subject.
93
00:10:34.250 --> 00:10:43.556
Somehow the time always ran out before I could give a proper discussion of the interpretation of quantum mechanics.
94
00:10:43.861 --> 00:10:49.741
I did give an hour’s lecture on this subject here in Lindau in 1968.
95
00:10:49.988 --> 00:10:55.254
This was subsequently published in Physics Today.
96
00:10:55.254 --> 00:11:03.852
2 months ago I gave a long series of Leigh Page’s lectures at Yale university
97
00:11:03.852 --> 00:11:12.703
on the theory of measurement in quantum mechanics and that will be eventually published in that university’s press.
98
00:11:13.155 --> 00:11:22.358
In the little time available to me today I will have to confine my discussion to a very simple form of quantum mechanics
99
00:11:22.358 --> 00:11:27.200
and I will have to keep hidden many elegant features of the more general theory.
100
00:11:27.200 --> 00:11:34.223
I will mostly be considering a dynamical system in which one particle is moving along a straight line.
101
00:11:34.223 --> 00:11:40.987
The dynamical variables to be measured will be limited to a coordinate such as X and an energy
102
00:11:40.987 --> 00:11:46.157
such as the Hamiltonian denoted by a symbol H for Hamiltonian.
103
00:11:46.480 --> 00:11:50.258
I will mostly use the wave mechanical formulation of quantum mechanics
104
00:11:50.258 --> 00:11:59.718
in which the state of a system is described by the shorting of wave function which is a function of X and T, psi of X and T.
105
00:12:01.534 --> 00:12:04.889
Suppose that we have a simple problem in classical mechanics,
106
00:12:04.889 --> 00:12:13.094
a particle of mass M moves a long a line under the action of some specified conservative potential energy field, V of X.
107
00:12:13.094 --> 00:12:21.104
The system is described by the mass M and the form of the potential function V of X.
108
00:12:21.104 --> 00:12:28.421
The state of the system can be specified by giving the particles coordinate X and velocity V.
109
00:12:28.421 --> 00:12:35.198
The object of the exercise is usually to predict the future state of the system at a time greater than zero.
110
00:12:35.879 --> 00:12:43.198
And that’s done with the help of the Newtonian equations of motion, given the initial state at time T equals zero.
111
00:12:43.198 --> 00:12:50.326
It should be obvious that if at a certain time we want to change the mass of the particle or the force,
112
00:12:50.326 --> 00:12:56.817
F of X which is acting on the particle we will subsequently have a different dynamical system
113
00:12:56.817 --> 00:13:03.082
and a different set of differential equation or equations of motion to solve.
114
00:13:03.082 --> 00:13:09.034
It is a good idea to know at all times what problem we are trying to solve.
115
00:13:09.301 --> 00:13:18.114
There’s a great deal of profundity in that last sentence and it is inadequately observed I would say.
116
00:13:18.114 --> 00:13:24.832
With a little more sophistication we can introduce concepts like momentum which is the product of mass times velocity
117
00:13:24.832 --> 00:13:30.825
and we can introduce a Hamiltonian function of X and P.
118
00:13:30.825 --> 00:13:40.822
The Newtonian equations of motion are replaced by Hamilton’s equations which I believe are on the new graph near the bottom.
119
00:13:44.622 --> 00:13:48.840
I now want to deal with the corresponding problem in quantum mechanics.
120
00:13:48.840 --> 00:13:56.116
Starting simply I look at the Schrödinger equation for a completely isolated system.
121
00:14:15.395 --> 00:14:22.349
Now that equation may not be carved in marble, but I’m going to take it very literally.
122
00:14:22.349 --> 00:14:27.814
I have seen students who wear this equation inscribed on T-shirts.
123
00:14:28.648 --> 00:14:30.362
(laughter).
124
00:14:31.246 --> 00:14:37.166
As a bit of fancy I would like to pretend that Moses found this equation on a tablet at Mount Sinai.
125
00:14:37.473 --> 00:14:43.546
While he understood thou shalt not kill and other commandments,
126
00:14:43.546 --> 00:14:48.189
he did not know the meaning of the strange equation with its mysterious symbols.
127
00:14:48.189 --> 00:14:53.942
And did not wish to confuse his people by telling them about the extra tablet.
128
00:14:53.942 --> 00:14:59.546
As a result we had to wait many thousands of years for another chance of enlightenment.
129
00:14:59.546 --> 00:15:11.628
Unlike Moses we probably know today what H bar is, Planck constant improved by Dirac by a factor of 1 over 2 pi.
130
00:15:12.805 --> 00:15:17.005
I, the square root of minus 1 and T standing for time.
131
00:15:17.005 --> 00:15:22.622
The Hamiltonian operator H is derived from the classical Hamiltonian function,
132
00:15:22.622 --> 00:15:32.514
H of X and P and by replacing the momentum P by a differential operator, H bar over I, D by DX.
133
00:15:32.514 --> 00:15:35.669
Sometimes a partial derivative.
134
00:15:35.669 --> 00:15:41.478
The hard thing to understand in this equation is the meaning of the wave function, psi of X and T.
135
00:15:41.478 --> 00:15:45.533
The notion of a wave function is borrowed from classical field theories
136
00:15:45.533 --> 00:15:52.480
but unlike those there is no direct physical interpretation to be given of the Schrödinger wave function.
137
00:15:52.685 --> 00:15:59.480
Instead certain rules are postulated for using the wave function to calculate quantities of physical interest.
138
00:15:59.743 --> 00:16:08.890
Among those are the probability density, W of X and T which is the absolute square of the wave function.
139
00:16:08.890 --> 00:16:14.154
And the expectation value of a dynamical quantity F of X and P,
140
00:16:14.154 --> 00:16:21.503
there could be many dynamical quantities to be considered but this stands for a general one.
141
00:16:21.503 --> 00:16:27.381
And that is calculated by evaluating an … (inaudible 16.25) consisting of a sandwich
142
00:16:27.381 --> 00:16:33.635
made of 2 wave functions and the operator F placed between them.
143
00:16:33.635 --> 00:16:38.899
The idea that the absolute square of a wave function is to be regarded as a probability density
144
00:16:38.899 --> 00:16:41.762
comes from the work of Max Born on collision theory
145
00:16:41.762 --> 00:16:50.370
and from Dirac’s more general formulation of quantum mechanics as a bridge between matrix and wave mechanics.
146
00:16:50.370 --> 00:16:53.038
I now consider a few simple cases.
147
00:16:53.038 --> 00:17:05.292
First let the wave function be one of the eigen functions of the Hamiltonian operator H of X and P.
148
00:17:05.292 --> 00:17:09.418
And I think its now time for the next view graph.
149
00:17:12.852 --> 00:17:17.318
That equation at the top represents an eigen value problem
150
00:17:17.318 --> 00:17:27.316
and that is characterising a stationary state called U sub N with an energy, stationary state energy E sub N.
151
00:17:27.316 --> 00:17:37.182
The probability density for that state is given by W, which would be called W sub N and that’s the absolute square of UN of X.
152
00:17:37.182 --> 00:17:40.274
And that is independent of the time.
153
00:17:40.274 --> 00:17:43.314
And hence the use of the word stationary state.
154
00:17:43.314 --> 00:17:52.100
The wave function is taken to be normalised so that the total probability for finding the electron any place is unity.
155
00:17:52.359 --> 00:17:58.789
If one measures somehow the operator H for this state,
156
00:17:59.010 --> 00:18:05.683
if you measure the energy of this state you find the value E sub N with certainty.
157
00:18:05.683 --> 00:18:10.820
Now I’m saying that but the mere fact I say it doesn’t explain how this is to be done.
158
00:18:10.820 --> 00:18:17.937
But a certain amount of that has to be absorbed in courses in quantum mechanics as you well know.
159
00:18:17.937 --> 00:18:25.801
Each time one measures some other dynamical quantity such as X, one may get a different value of the measurement
160
00:18:25.801 --> 00:18:34.143
and only when an ensemble of measurements has been studied or when an ensemble of measurements has been made
161
00:18:34.143 --> 00:18:45.651
does one obtain the probability density W sub N of X which can be calculated by the equation given there.
162
00:18:45.651 --> 00:18:57.721
As a second case let us consider a wave function psi of X and what you can’t read there stands for the variable T
163
00:18:57.721 --> 00:19:03.879
and that has a subscript M on it standing for the time of some measurement.
164
00:19:03.879 --> 00:19:09.529
So that wave function is a function of X and T for the time of measurement.
165
00:19:09.529 --> 00:19:16.360
And this is taken to be a linear combination of 2 of the stationary states of the atom,
166
00:19:16.360 --> 00:19:23.049
U1 and U2 are stationary state wave functions for 2 different energy eigen values,
167
00:19:23.049 --> 00:19:30.674
E1 and E2 and they can be shown to be orthogonal, if you know what that means, but you don’t need to.
168
00:19:31.075 --> 00:19:38.721
And if the wave functions are normalised the wave function of psi will also be normalised if the complex coefficients,
169
00:19:38.721 --> 00:19:47.462
C1 and C2 are normalised to unity in a way which you do not see there
170
00:19:47.462 --> 00:19:52.515
but the sum of the squares of the C’s, 1 or C2 should be unity.
171
00:19:52.726 --> 00:20:01.236
If the dynamical variable H is repeatedly measured for a system with this wave function one sometimes gets E1
172
00:20:01.236 --> 00:20:03.825
and sometimes gets E2.
173
00:20:04.036 --> 00:20:08.599
And there is no way to predict in advance which result will be obtained.
174
00:20:08.599 --> 00:20:14.102
The relative probabilities with which the 2 energy eigen values are obtained
175
00:20:14.102 --> 00:20:22.679
as the results of measurement will be given by the quantity C1 absolute squared and C2 absolute squared.
176
00:20:22.924 --> 00:20:31.780
Well I’ve now given you a little bit of measurement theory in quantum mechanics, if you find it vague so do I.
177
00:20:31.780 --> 00:20:36.449
We talk about measurements but we don’t know how to make them.
178
00:20:36.449 --> 00:20:44.306
Talk is cheap but you never get more than you pay for.
179
00:20:44.306 --> 00:20:51.685
My attitude towards such problems has no doubt been influenced by contact with some research and experimental physics
180
00:20:51.685 --> 00:21:01.254
in which single highly isolated atomic states are precisely manipulated by microwave or optical frequency fields.
181
00:21:01.254 --> 00:21:06.495
In the discussion of the measurement of any dynamical variable of a physical system
182
00:21:06.495 --> 00:21:16.685
I want to specify exactly in the language of the quantum theory what apparatus is necessary for the task and how to use it,
183
00:21:16.933 --> 00:21:18.685
at least in principle.
184
00:21:18.897 --> 00:21:27.013
I am not satisfied with hand waving or a black box approach or with a formulogical scheme.
185
00:21:27.013 --> 00:21:31.296
My starting point is the Schrödinger equation for a completely isolated system
186
00:21:31.296 --> 00:21:37.477
which we have already seen on a previous view graph.
187
00:21:37.477 --> 00:21:42.693
The manifest role of the wave equation is to allow us to calculate the future state of the system,
188
00:21:42.693 --> 00:21:49.863
psi of T from its initial state psi of zero for a system with a given Hamiltonian operator
189
00:21:49.863 --> 00:21:54.542
which is usually of the form of a kinetic and a potential energy added together.
190
00:21:54.542 --> 00:21:59.944
The first problem is to get our system in to the desired starting state psi of zero,
191
00:21:59.944 --> 00:22:03.009
this is called preparation of the initial state.
192
00:22:03.259 --> 00:22:08.163
We may then let the wave function evolve under the guidance of the Schrödinger equation
193
00:22:08.163 --> 00:22:13.032
until a time T sub M when a measurement is to be made.
194
00:22:13.277 --> 00:22:23.165
I gave a discussion of the problem of state preparation in my 1968 lecture and although hardly anybody here heard that lecture,
195
00:22:23.165 --> 00:22:32.765
I will simply take the result for granted which is that we can start the system off pretty well in any state that we please.
196
00:22:32.765 --> 00:22:39.358
Not all quantum mechanical systems, even if isolated are describable by a wave function.
197
00:22:39.358 --> 00:22:42.617
We simply may not know the starting wave function.
198
00:22:42.617 --> 00:22:46.322
In that case the best that we can do is to consider that the wave function
199
00:22:46.322 --> 00:22:55.774
might be one or another of several possible wave functions and the sign of probability distribution for the various possibilities.
200
00:22:55.774 --> 00:23:04.223
The wave function would then be used to work out what should happen to each of these separate wave functions
201
00:23:04.223 --> 00:23:09.873
and predicted results would be obtained by averaging over the ensemble of wave functions.
202
00:23:09.873 --> 00:23:16.542
The theory would lose a great deal of its causality if we had to do this but sometimes we would.
203
00:23:16.542 --> 00:23:23.912
In a case where a wave function description is possible one speaks of a pure case, otherwise of a mixture.
204
00:23:23.912 --> 00:23:30.790
The theory usually uses density matrices instead of wave functions for the necessary kind of book keeping.
205
00:23:30.790 --> 00:23:37.322
But it is possible to get along without using density matrixes if one works with an ensemble of wave functions.
206
00:23:37.322 --> 00:23:45.430
Once converted into a mixture a pure case can never be recovered without the use of some kind of filtering process
207
00:23:45.430 --> 00:23:54.088
which is equivalent to the preparation of a completely new state instead of making a measurement on the original system
208
00:23:54.088 --> 00:23:59.100
which was the system we should have been concentrating on.
209
00:23:59.100 --> 00:24:08.066
The wave equation will apply only if the system has the Hamiltonian H equals T plus V
210
00:24:08.066 --> 00:24:15.093
but we do have to permit some disturbance of the system if we are to allow an observation of the system,
211
00:24:15.093 --> 00:24:18.299
for instance a measurement of some quantity.
212
00:24:18.299 --> 00:24:22.545
Any disturbance whatsoever will represent a change of the dynamical problem
213
00:24:22.545 --> 00:24:26.698
and hence we will certainly have to use a different Schrödinger equation
214
00:24:26.698 --> 00:24:31.960
to describe the system during the time its enjoying the process of measurement.
215
00:24:32.162 --> 00:24:38.481
Quantum mechanics allows I would say at most 3 general kinds of disturbances.
216
00:24:38.705 --> 00:24:47.778
The first, from the application of a classically describable external force with a corresponding additional term
217
00:24:47.778 --> 00:24:50.646
added to the Hamiltonian.
218
00:24:50.646 --> 00:24:56.375
We might apply an external electric or a magnetic field and treat those fields classically.
219
00:24:56.375 --> 00:25:03.114
The second way would be from the dynamical coupling of another quantum mechanical system to the first system
220
00:25:03.114 --> 00:25:13.297
to make a larger combined system which from then on, forever more would be the system we should be studying.
221
00:25:13.297 --> 00:25:21.000
The third way would be from the intervention of an observer, putting the observer in quotation marks,
222
00:25:21.000 --> 00:25:24.802
who attempts to learn something about the system by looking at it
223
00:25:24.802 --> 00:25:36.712
or looking at some associated measuring instrument which has for a time at least been part of an enlarged system.
224
00:25:36.963 --> 00:25:46.969
In the second case the added system has to be defined in terms of new variables, not little X and P but lets say big X and P
225
00:25:46.969 --> 00:25:54.764
and the discussion of this case is simplest when the appendage system is in a known quantum state at the time of union.
226
00:25:54.764 --> 00:26:02.002
The enlarged but still isolated system is from then on regarded as the system of interest
227
00:26:02.002 --> 00:26:06.282
and its Schrödinger equation can be used to follow its time development.
228
00:26:06.538 --> 00:26:13.227
The third case will be discussed below but perhaps I should allow you to anticipate
229
00:26:13.488 --> 00:26:19.401
that in my view a living observer is not a suitable object for a Hamiltonian treatment,
230
00:26:19.627 --> 00:26:24.103
whether in quantum mechanics or in classical.
231
00:26:24.415 --> 00:26:29.770
The second and third cases play a central role in the theory of quantum mechanical measurements.
232
00:26:29.770 --> 00:26:33.715
In case 3 an observer interacts with the system.
233
00:26:33.715 --> 00:26:42.619
Von Neumann made a postulate often called the reduction of the wave packet hypothesis to deal simply
234
00:26:42.619 --> 00:26:46.240
with the change of the wave function in such a case.
235
00:26:46.240 --> 00:26:54.676
This postulate states that when an observer gets a result of a measurement, perhaps we should say maximum measurement,
236
00:26:54.912 --> 00:27:03.421
the wave function of the system collapses into the eigen function appropriate for the variables being measured.
237
00:27:03.421 --> 00:27:10.357
For reasons given below I do not think that Von Neumann’s postulate is either helpful or necessary
238
00:27:10.357 --> 00:27:17.878
for the understanding of quantum mechanics and for the discussion of gravity wave detectors.
239
00:27:18.196 --> 00:27:23.469
Instead one can try to give a quantum mechanical description of the combined system,
240
00:27:23.469 --> 00:27:29.667
consisting of the measuring apparatus in a known quantum state brought into interaction
241
00:27:29.667 --> 00:27:36.409
with the original system of interest and proceed as in case 2, as if we have a dynamical problem.
242
00:27:36.409 --> 00:27:43.468
As long as the 2 interacting systems are united in the one isolated combined system the description is given by a wave function.
243
00:27:43.668 --> 00:27:48.933
However to use the measuring instrument we must separate it off from the original system
244
00:27:48.933 --> 00:27:53.523
and look at some property such as a needle pointer position
245
00:27:53.741 --> 00:27:58.558
from which we hope to infer something about the state of the original system.
246
00:27:58.558 --> 00:28:05.938
As a result of the separation of the united system into 2 parts neither of the separated systems
247
00:28:06.140 --> 00:28:11.189
will from that time ever more have a definite wave function.
248
00:28:11.189 --> 00:28:17.046
Each will be in an incoherent mixture of single system pure case states.
249
00:28:17.046 --> 00:28:24.007
One can interpret this as arising from the uncontrollable interaction between the 2 parts of the system
250
00:28:24.007 --> 00:28:32.520
during the time that they were united.
251
00:28:32.771 --> 00:28:40.015
This is similar to what would happen in case 1 if a random perturbation were applied to a single system.
252
00:28:40.261 --> 00:28:46.980
If you had a single system in the known perturbation the system would remain in a pure case state,
253
00:28:46.980 --> 00:28:52.953
but if you had a random perturbation and didn’t know it you would have to make an ensemble and then you would have a mixture.
254
00:28:53.266 --> 00:28:57.890
A number of writers on this subject have assumed that after the separation,
255
00:28:57.890 --> 00:29:03.954
the measuring system would have a definite wave function with a definite phase relationship
256
00:29:03.954 --> 00:29:11.078
between various components which were being added in the total wave function.
257
00:29:11.078 --> 00:29:16.708
This would leave them with the unwelcome situation of an essentially classical measuring instrument
258
00:29:16.708 --> 00:29:23.169
which might be in a state represented by a definite super position of several needle pointer states.
259
00:29:23.169 --> 00:29:31.834
The transfer of attention alluded to above from the system of interest to the combined system
260
00:29:31.834 --> 00:29:38.785
and then to the measuring instrument only postpones the need for understanding the measuring process
261
00:29:38.998 --> 00:29:45.715
which surely becomes more difficult as the system becomes larger and larger.
262
00:29:45.970 --> 00:29:52.444
Ultimately one would be led to consider still larger systems such as the electromagnetic field of optical radiation,
263
00:29:52.648 --> 00:30:00.881
the retina and optic nerve of the eye, the brain, the mechanism of consciousness and eventually the whole universe.
264
00:30:01.191 --> 00:30:08.876
For the measurement of position of an electron within an atom I adopted for my 1968 lecture a method
265
00:30:08.876 --> 00:30:13.059
which is the quantum mechanical transcription of a classical one
266
00:30:13.296 --> 00:30:19.195
that might be used to determine the probability distribution for a fly in a room.
267
00:30:19.539 --> 00:30:30.428
One would quickly clasp one’s fingers around a small region of point X of M and find out by some subsequent
268
00:30:30.675 --> 00:30:37.547
but non quantum mechanical operation whether one had caught an electron or a fly or not.
269
00:30:37.547 --> 00:30:44.919
Then the process would be repeated many times for similarly prepared atoms to build up a probability distribution.
270
00:30:45.166 --> 00:30:54.120
This represents a rather destructive procedure as the electrons wave function is disturbed even if the electron is not found.
271
00:30:54.120 --> 00:31:02.852
When an electron is found or when an electron is caught one will have prepared the state of a very well localised particle.
272
00:31:03.154 --> 00:31:07.947
But that is of no help for solution of the original problem.
273
00:31:07.947 --> 00:31:12.111
On most of the occasions an electron will not be caught
274
00:31:12.321 --> 00:31:19.772
but its wave function in the room will nevertheless be seriously affected by the effort.
275
00:31:20.037 --> 00:31:24.956
It is here that the reduction hypothesis runs into trouble.
276
00:31:24.956 --> 00:31:28.372
If the electron is caught its state is pretty well known.
277
00:31:28.577 --> 00:31:32.633
That might be thought to match the reduction hypothesis, maybe so.
278
00:31:32.633 --> 00:31:37.166
But the state is that of a different problem than the one we were supposed to be considering.
279
00:31:37.411 --> 00:31:45.252
We have engaged in preparation, not measurement.
280
00:31:45.252 --> 00:31:50.126
If the electron is not caught the future development of the wave function is disturbed.
281
00:31:50.126 --> 00:31:56.364
And if its gravity waves we’re trying to detect for something like that there would be serious problems.
282
00:31:56.614 --> 00:32:03.673
When similar considerations are applied to gravity wave detection the ensuing complications are highly undesirable.
283
00:32:04.902 --> 00:32:09.936
I should mention that in his 1933 book, Mathematical Foundations of Quantum Mechanics,
284
00:32:09.936 --> 00:32:15.045
Von Neumann gave quite different method for measuring a position coordinate
285
00:32:15.255 --> 00:32:23.004
which had the advantage over the one I’ve just described to you of more faithfully modelling a conventional measuring apparatus.
286
00:32:23.004 --> 00:32:27.724
The system of interest had canonical variables, little x and little p.
287
00:32:27.724 --> 00:32:31.800
The measurement system had variables capital X and capital P.
288
00:32:32.602 --> 00:32:37.909
The interaction Hamiltonian was taken to be proportional to the product little X capital P
289
00:32:37.909 --> 00:32:43.894
and this would be very hard to realise in practice but I wouldn’t quibble about that.
290
00:32:43.894 --> 00:32:49.342
But what I would complain about is that except in an absurd limiting case the method
291
00:32:49.342 --> 00:32:55.046
does not avoid the conversion of the wave function into a statistical mixture.
292
00:32:56.440 --> 00:33:03.706
And therefore the hypothesis fails but this was not recognised.
293
00:33:03.706 --> 00:33:09.542
Let us now consider a system which at the time of measurement T sub M
294
00:33:09.542 --> 00:33:16.742
is described by the 2 state super position wave function of which we have there.
295
00:33:16.742 --> 00:33:26.675
At later times the wave function will evolve according to the Schrödinger equation into a wave function like this,
296
00:33:26.675 --> 00:33:34.649
which differs from the form above by the presence of 2 exponential factors which have an absolute magnitude of 1.
297
00:33:34.649 --> 00:33:43.533
But a phase that depends on the time elapsed after the time of measurement, T minus T sub M would be the time elapsed.
298
00:33:43.784 --> 00:33:55.744
And notice that the probabilities for finding the electron in state 1 or state 2 are given by the same expressions as before
299
00:33:55.744 --> 00:34:02.742
and they are independent of the time because the exponential factors have unit modularly.
300
00:34:02.742 --> 00:34:08.799
We will have to repeat this whole operation many times in order to determine the values of the probabilities piece have been.
301
00:34:09.471 --> 00:34:13.902
The different members of the ensemble could easily have different waiting times,
302
00:34:25.288 --> 00:34:31.043
no doubt this would happen quite naturally while we were thinking about how we could determine their energy values.
303
00:34:31.043 --> 00:34:36.626
That would of course convert the pure case wave function into a statistical mixture of randomly phased wave functions
304
00:34:36.626 --> 00:34:44.222
which would be described by a density matrix instead of a pure case wave function.
305
00:34:44.222 --> 00:34:50.152
Well I have to discuss the way that we would tell whether the atom was in state 1 or 2
306
00:34:50.152 --> 00:34:56.378
and that’s done with a Stern-Gerlach apparatus which we can think of as being a kind of coupling of the system
307
00:34:56.378 --> 00:35:06.007
to a system for measurement and that discussion is pretty well known.
308
00:35:07.473 --> 00:35:17.919
So that I think that I shouldn’t take the opportunity to go on until the appointed time for the termination of the lecture.
309
00:35:17.919 --> 00:35:27.398
But the conclusion with which I would like to leave you is that
310
00:35:27.398 --> 00:35:36.217
when one is studying the motion of a massive cylinder of thoroughly macroscopic size,
311
00:35:36.217 --> 00:35:44.566
it is going to be awfully hard to know the Hamiltonian acting on the system.
312
00:35:44.566 --> 00:35:53.990
And if you do make a measurement you will have to take into account with exquisite precision the introduction
313
00:35:53.990 --> 00:35:57.392
of any additional terms in the Hamiltonian.
314
00:35:57.392 --> 00:36:02.685
And no matter what you do you will lose any knowledge of the wave function of the oscillator
315
00:36:02.685 --> 00:36:05.212
that you might have had to begin with.
316
00:36:05.212 --> 00:36:11.227
So that as the measurement procedure goes on looking again and again
317
00:36:11.227 --> 00:36:18.566
to see how the gravity wave is evolving it would be necessary to take into account the fact
318
00:36:18.566 --> 00:36:26.572
that the wave function of the system of interest is becoming even less a pure case wave function than it might have been.
319
00:36:26.572 --> 00:36:35.835
And the result would be that the calculation will have to be done all over again.
320
00:36:36.083 --> 00:36:43.144
Caves and Thorne have made the calculation on the basis that after every little measurement that they made
321
00:36:43.144 --> 00:36:51.011
they could say that the wave function had collapsed to the value that it would have had if they had the system in an eigen state.
322
00:36:51.011 --> 00:36:58.290
But the procedures for getting a system in an eigen state of energy are sufficiently elaborate
323
00:36:58.290 --> 00:37:02.157
that they would not ever be able to do that.
324
00:37:02.157 --> 00:37:09.479
Furthermore in order to conduct the procedures that quantum mechanics requires they would have to have
325
00:37:09.479 --> 00:37:17.701
an ensemble of gravity wave detectors and they would be well off if they had an ensemble of gravity waves falling on the system.
326
00:37:17.701 --> 00:37:25.348
And a gravity wave is something that you take when you get it and you can’t be sure the next one will be the same kind of wave.
327
00:37:25.348 --> 00:37:36.715
So the final conclusion is that they will not meet the quantum limit but let them come and tell you about how they did it, so.
328
00:37:41.466 --> 00:37:43.449
(Applause).