On the Use and Misuse of Quantum Mechanics
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00:00
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Transcript: English(auto-generated)
00:12
Ladies and gentlemen, this morning you will have heard lectures in three dialects of
00:20
English, one with a slight Swedish accent, one with a slight Japanese accent, and one with a strong American accent. Sometimes it helps for people for whom English isn't the native language to hear the material spoken with a slight accent.
00:44
I hope mine can be understood. Another thing, I believe that I was scheduled to stop this lecture promptly at 11.30 or as some people say half twelve.
01:00
I will try to stop sooner than that because I wish to go on the trip to my now. However, I also wish to eat lunch today so that, anyway, but I do have a lot of material
01:23
to cover and therefore I'm going to read the text reasonably quickly. Quantum mechanics is a very wonderful tool for dealing with problems in atomic, molecular, and condensed matter physics as well as with many parts of chemistry.
01:44
It can be based on a small number of axiomatic statements which are easy to apply but hard to understand. This same quantum mechanics can be extended in very plausible ways to apply to electromagnetic radiation in interaction with matter.
02:04
Many problems in nuclear physics have been treated with quantum mechanics and the reconciliation of quantum mechanics and special relativity theory has had considerable success. With general relativity, things are rather more difficult still.
02:22
Many problems of sub-nuclear physics and high energy phenomena have also been treated with quantum mechanics but the theory is pushed rather far from its roots when dealing with such problems. My main object in this lecture is to deal with the interpretation of quantum mechanics
02:41
on the atomic level and with the application of that theory to more macroscopic systems, which for example, among other things, might include measuring instruments. It is tempting to think that quantum mechanics might be applicable as well to much larger
03:01
parts of the universe, such as with life and beyond, but I will try to give you some reasons why such a temptation should be resisted. Experience and common sense teach us that to learn anything about a sub-microscopic system is a difficult task.
03:22
Our intuition is a most unreliable guide in this domain. We have ingrained concepts about the meaning of reality, which no doubt will be defined by Professor Wigner in the next lecture, and about causality, the relation between
03:40
cause and effect, and about our human role as observers of natural phenomena. All of these concepts are most inappropriate for dealing with the sub-microscopic world. The development of the mathematical structure of quantum mechanics proved easier than the
04:00
formulation of a satisfactory interpretation of the theory and description of the process of measurement. Quantum mechanics can do a lot for us if we regard it simply as a set of computational rules for dealing with simple dynamical systems. For instance, we can calculate the energies of stationary states of atoms and make an
04:22
immediate connection with high precision spectroscopic observations of the last century. For this, one does not need to worry very much about the interpretation of the wave function, which describes the state of the system, but only to calculate the energies of stationary states.
04:42
When we come to discuss scattering processes or radiative transitions between stationary states or the theory of measurement of dynamical variables, however, we have greater need for a better understanding of the meaning of the wave function. In the last few years, I have seen a number of papers dealing with interesting applications
05:04
of quantum mechanics to large-scale phenomena. These range from problems in physics, optical communication theory, molecular biology, all the way up to the theory of the whole universe. I have been quite suspicious about the validity of most of this research.
05:23
One example of this is found in the theory expounded by Kip Thorne and Carlton Caves of Caltech on quantum demolition measurements of gravity waves. Gravity waves probably have not been detected yet, perhaps they have, but not everybody
05:43
thinks so, but many people are working on much more sensitive detectors and probably someday gravity waves will be detected. However, exceeding the limitations imposed by quantum mechanics in such measurements
06:04
is the subject of the discipline called quantum non-demolition measurements. This work represents part of a major program at Caltech to build the most sensitive detector
06:22
possible for gravity waves. Estimates based on an intuitive application of the uncertainty principle to the experimental configuration set a quantum limit on the smallest signal which could be received. Thorne, Caves, and others have searched for methods to quote, beat the quantum limit
06:45
unquote. Their papers are very impressive and persuasive. Unfortunately for them, or more likely for me, they make use of a form of assumption
07:00
about quantum mechanics with which I do not agree. To put it briefly, I could say that they postulate the reduction of the wave packet hypothesis of von Neumann. There are very similar ideas expressed in the first chapter of Professor Durek's book on quantum mechanics.
07:21
I partly disagree with the statements expressed by those authors, but I disagree more strongly with the use of those hypotheses by people involved in gravity wave measurements.
07:43
A gravity wave detector typically consists of a very massive cylinder, one of its very high Q normal modes is put into vibration by a passing pulse of gravitational radiation. In simple terms, the system consists of a forced simple harmonic oscillator and should
08:05
be very familiar to all students of textbook quantum mechanics. At issue is the question how one can measure coordinate and momentum of such a large system. The method is to trace out the time-dependent shape of a gravitational wave pulse by following
08:25
the motion of a reference surface on the massive oscillator. There is general agreement that the gravitational fields which are to be studied in this research can be treated completely classically. So there is no need to worry about the quantum theory of gravitation in this study.
08:43
No need to talk of gravitons or other quantum aspects of gravitation. There will be plenty of non-quantum mechanical disturbances of the detector to cope with, but Thorne and Caves wish to go further than the quantum limits since this is required for the detection of some types of expected gravitational radiation.
09:05
My feeling is that they may probably succeed if the von Neumann hypothesis is correct and will fail if it is not. Perhaps in three or four years, one of those gentlemen will be here to tell you what they did.
09:26
Another field where quantum mechanics has been applied to macroscopic phenomena is in the theory of optical communication. Here we might consider a signal generator such as a laser, a transmission medium perhaps between the Earth and the Moon,
09:42
and a detector which might consist of a photoelectric device and associated electronic circuits. Since the von Neumann reduction hypothesis plays an essential role in this theory, the foundations of which were laid at Bell Telephone Labs but many other people work on the subject,
10:01
I believe that this theory is fatally flawed. Quantum mechanics is now over 50 years old, 55 years old. I taught graduate courses in that subject for over 35 years at Columbia, Stanford, Oxford, Yale, and the University of Arizona.
10:23
My lectures always began with an explanation that one must first learn the rules of calculation in quantum mechanics before one can understand the physical meaning of the subject. Somehow, the time always ran out before I could give a proper discussion of the interpretation of quantum mechanics.
10:43
I did give an hour's lecture on this subject here in Lindau in 1968. This was subsequently published in Physics Today. Two months ago, I gave a long series of Lee Page's lectures at Yale University on the theory of measurement in quantum mechanics,
11:07
and that will be eventually published in that university's press. In the little time available to me today, I will have to confine my discussion to a very simple form of quantum mechanics,
11:22
and I will have to keep hidden many elegant features of the more general theory. I will mostly be considering a dynamical system in which one particle is moving along a straight line. The dynamical variables to be measured will be limited to a coordinate such as X and an energy such as the Hamiltonian,
11:43
denoted by a symbol H for Hamiltonian. I will mostly use the wave mechanical formulation of quantum mechanics in which the state of a system is described by the Schrodinger wave function, which is a function of X and T, psi of X and T.
12:01
Suppose that we have a simple problem in classical mechanics. A particle of mass M moves along a line under the action of some specified conservative potential energy field, V of X. The system is described by the mass M and the form of the potential function V of X.
12:21
The state of the system can be specified by giving the particles coordinate X and velocity V. The object of the exercise is usually to predict the future state of the system at a time greater than zero, and that's done with the help of the Newtonian equations of motion given the initial state at time T equals zero.
12:43
It should be obvious that if at a certain time we want to change the mass of the particle or the force F of X which is acting on the particle, we will subsequently have a different dynamical system and a different set of differential equation or equations of motion to solve.
13:02
It is a good idea to know at all times what problem we are trying to solve. There's a great deal of profundity in that last sentence, and it is inadequately observed, I would say. With a little more sophistication, we can introduce concepts like momentum, which is the product of mass times velocity,
13:24
and we can introduce a Hamiltonian function of X and P. The Newtonian equations of motion are replaced by Hamilton's equations, which I believe are on the view graph near the bottom.
13:42
Well, I now want to deal with the corresponding problem in quantum mechanics. Starting simply, I look at the Schrodinger equation for a completely isolated system, and probably that's on the next view graph, which is back here.
14:08
That's the Schrodinger equation. Now, that equation may not be carved in marble, but I'm going to take it very literally.
14:22
I have seen students who wear this equation inscribed on T-shirts. As a bit of fancy, I would like to pretend that Moses found this equation on a tablet at Mount Sinai. While he understood thou shalt not kill and other commandments,
14:43
he did not know the meaning of the strange equation with its mysterious symbols, and did not wish to confuse his people by telling them about the extra tablet. As a result, we had to wait many thousands of years for another chance of enlightenment. Unlike Moses, we probably know today what h-bar is,
15:06
Planck's constant improved by Dirac by a factor of 1 over 2 pi, i, the square root of minus 1, and t, standing for time. The Hamiltonian operator h is derived from the classical Hamiltonian function, h of x and p,
15:23
by replacing the momentum p by a differential operator, h-bar over i, d by dx, sometimes a partial derivative. The hard thing to understand in this equation is the meaning of the wave function, psi of x and t.
15:41
The notion of a wave function is borrowed from classical field theories, but unlike those, there is no direct physical interpretation to be given of the Schrodinger wave function. Instead, certain rules are postulated for using the wave function to calculate quantities of physical interest.
16:00
Among those are the probability density, w of x and t, which is the absolute square of the wave function, and the expectation value of a dynamical quantity, f of x and p. There could be many dynamical quantities to be considered, but this stands for a general one.
16:20
And that is calculated by evaluating an integral consisting of a sandwich made of two wave functions and the operator f placed between them. The idea that the absolute square of a wave function is to be regarded as a probability density comes from the work of Max Born on collision theory
16:41
and from Dirac's more general formulation of quantum mechanics as a bridge between matrix and wave mechanics. I now consider a few simple cases. First, let the wave function be one of the eigenfunctions of the Hamiltonian operator, h of x and p.
17:05
And I think it's now time for the next view graph. That equation at the top, it represents an eigenvalue problem, and that is characterizing a stationary state called u sub n with an energy, stationary state energy, e sub n.
17:28
The probability density for that state is given by w, which would be called w sub n, and that's the absolute square of u n of x. And that is independent of the time, and hence the use of the word stationary state.
17:42
The wave function is taken to be normalized so that the total probability for finding the electron anyplace is unity. If one measures somehow the operator h for this state, if you measure the energy of this state,
18:01
you find the value e sub n with certainty. Now, I'm saying that, but the mere fact I say it doesn't explain how this is to be done. But a certain amount of that has to be absorbed in courses in quantum mechanics, as you well know. Each time one measures some other dynamical quantity, such as x, one may get a different value of the measurement.
18:26
And only when an ensemble of measurements has been studied, or when an ensemble of measurements has been made, does one obtain the probability density w sub n of x, which can be calculated by the equation given there.
18:45
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, and that has a subscript m on it, standing for the time of some measurement.
19:03
So, that wave function is a function of x and t for the time of measurement. And this is taken to be a linear combination of two of the stationary states of the atom. U1 and U2 are stationary state wave functions for two different energy eigenvalues, E1 and E2.
19:25
And they can be shown to be orthogonal, if you know what that means, but you don't need to. And if the wave functions are normalized, the wave function psi will also be normalized, if the complex coefficients C1 and C2 are normalized to unity,
19:43
in a way which you do not see there, but the sum of the squares of the C1 and C2 should be unity. If the dynamical variable h is repeatedly measured for a system with this wave function, one sometimes gets E1 and sometimes gets E2, and there is no way to predict in advance which result will be obtained.
20:09
The relative probabilities with which the two energy eigenvalues are obtained as the results of measurement will be given by the quantities C1 absolute squared and C2 absolute squared.
20:22
Well, I've now given you a little bit of measurement theory in quantum mechanics. If you find it vague, so do I. We talk about measurements, but we don't know how to make them. Talk is cheap, but you never get more than you pay for.
20:45
My attitude towards such problems has no doubt been influenced by contact with some research in experimental physics, in which single highly isolated atomic states are precisely manipulated by microwave or optical frequency fields.
21:01
In the discussion of the measurement of any dynamical variable of a physical system, I want to specify exactly, in the language of the quantum theory, what apparatus is necessary for the task and how to use it, at least in principle. I am not satisfied with hand waving or a black box approach or with a formal logical scheme.
21:26
My starting point is this Ferdinger equation for a completely isolated system, which we have already seen on a previous view graph. The manifest role of the wave equation is to allow us to calculate the future state of the system,
21:43
psi of t, from its initial state, psi of zero, for a system with the given Hamiltonian operator, which is usually of the form of a kinetic and a potential energy added together. The first problem is to get our system into the desired starting state, psi of zero.
22:00
This is called preparation of the initial state. We may then let the wave function evolve under the guidance of the Schrodinger's equation until a time t sub m, when a measurement is to be made. I gave a discussion of the problem of state preparation in my 1968 lecture, and although hardly anybody here heard that lecture, I will simply take the result for granted,
22:26
which is that we can start the system off pretty well in any state that we please. Not all quantum mechanical systems, even if isolated, are describable by a wave function. We simply may not know the starting wave function.
22:42
In that case, the best that we can do is to consider that the wave function might be one or another of several possible wave functions and assign a probability distribution for the various possibilities. The wave function would then be used to work out what should happen to each of these separate wave functions,
23:04
and predicted results would be obtained by averaging over the ensemble of wave functions. The theory would lose a great deal of its causality if we had to do this, but sometimes we would. In a case where a wave function description is possible, one speaks of a pure case, otherwise of a mixture.
23:24
The theory usually uses density matrices instead of wave functions for the necessary kind of bookkeeping, but it is possible to get along without using density matrices if one works with an ensemble of wave functions. Once converted into a mixture, a pure case can never be recovered without the use of some kind of filtering process,
23:45
which is equivalent to the preparation of a completely new state instead of making a measurement on the original system, which was the system we should have been concentrating on. The wave equation will apply only if the system has the Hamiltonian H equals T plus V,
24:08
but we do have to permit some disturbance of the system if we are to allow an observation of the system, for instance, a measurement of some quantity. Any disturbance whatsoever will represent a change of the dynamical problem,
24:22
and hence we will certainly have to use a different Schrodinger equation to describe the system during the time it's enjoying the process of measurement. Quantum mechanics allows, I would say, at most three general kinds of disturbances. The first, from the application of a classically describable external force
24:45
with a corresponding additional term added to the Hamiltonian. We might apply an external electric or a magnetic field and treat those fields classically. The second way would be from the dynamical coupling of another quantum mechanical system to the first system
25:04
to make a larger combined system, which from then on forevermore would be the system we should be studying. The third way would be from the intervention of an observer, putting the observer in quotation marks,
25:21
who attempts to learn something about the system by looking at it or looking at some associated measuring instrument which has for a time at least been part of an enlarged system. In the second case, the added system has to be defined in terms of new variables, not little x and p,
25:45
but let's say big X and p, and the discussion of this case is simplest when the appended system is in a known quantum state at the time of union. The enlarged but still isolated system is from then on regarded as a system of interest,
26:02
and its Schrodinger equation can be used to follow its time development. The third case will be discussed below, but perhaps I should allow you to anticipate that in my view a living observer is not a suitable object for a Hamiltonian treatment, whether in quantum mechanics or in classical.
26:24
The second and third cases play a central role in the theory of quantum mechanical measurements. In case three, an observer interacts with the system. von Neumann made a postulate often called the reduction of the wave pocket, wave packet hypothesis to deal simply with the change of the wave function in such a case.
26:46
This postulate states that when an observer gets a result of a measurement, perhaps we should say maximum measurement, the wave function of the system collapses into the eigenfunction appropriate for the variables being measured.
27:04
For reasons given below, I do not think that von Neumann's postulate is either helpful or necessary for the understanding of quantum mechanics and for the discussion of gravity wave detectors. Instead, one can try to give a quantum mechanical description of the combined system
27:23
consisting of the measuring apparatus in a known quantum state, brought into interaction with the original system of interest and proceed as in case two, as if we have a dynamical problem. As long as the two interacting systems are united into one isolated combined system,
27:41
the description is given by a wave function. However, to use the measuring instrument, we must separate it off from the original system and look at some property, such as a needle pointer position, from which we hope to infer something about the state of the original system. As a result of the separation of the united system into two parts,
28:04
neither of the separated systems will, from that time ever more, have a definite wave function. Each will be in an incoherent mixture of single system pure case states. One can interpret this as arising from the uncontrollable interaction
28:22
between the two parts of the system during the time that they were united. This is similar to what would happen in case one if a random perturbation were applied to a single system.
28:41
If you had a single system and a known perturbation, the system would remain in a pure case state. 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. A number of writers on this subject have assumed that after the separation, the measuring system would have a definite wave function
29:01
with a definite phase relationship between various components which were being added in the total wave function. This would leave them with the unwelcome situation of an essentially classical measuring instrument which might be in a state represented by a definite superposition of several needle pointer states.
29:25
The transfer of attention alluded to above from the system of interest to the combined system and then to the measuring instrument only postpones the need for understanding the measurement process which surely becomes more difficult as the system becomes larger and larger.
29:45
Ultimately, one would be led to consider still larger systems such as the electromagnetic field of optical radiation, the retina and optic nerve of the eye, the brain, the mechanism of consciousness and eventually the whole universe.
30:02
For the measurement of position of an electron within an atom, I adopted for my 1968 lecture a method which is the quantum mechanical transcription of a classical one that might be used to determine the probability distribution for a fly in a room.
30:22
One would quickly clasp one's fingers around a small region at point X sub M and find out by some subsequent but non-quantum mechanical operation whether one had caught an electron or a fly or not. Then the process would be repeated many times for similarly prepared atoms
30:42
to build up a probability distribution. This represents a rather destructive procedure as the electron's wave function is disturbed even if the electron is not found. When an electron is found or when an electron is caught, one will have prepared the state of a very well localized particle
31:04
but that is of no help for solution of the original problem. On most of the occasions, an electron will not be caught but its wave function in the room will nevertheless be seriously affected by the effort.
31:21
It is here that the reduction hypothesis runs into trouble. If the electron is caught, its state is pretty well known. That might be thought to match the reduction hypothesis. Maybe so, but the state is that of a different problem than the one we were supposed to be considering. We have engaged in preparation, not measurement.
31:45
If the electron is not caught, the future development of the wave function is disturbed and if its gravity waves were trying to detect or something like that, there will be serious problems. When similar considerations are applied to gravity wave detection,
32:01
the ensuing complications are highly undesirable. I should mention that in his 1933 book, Mathematical Foundations of Quantum Mechanics, von Neumann gave a quite different method for measuring a position coordinate which had the advantage over the one I just described to you of more faithfully modeling a conventional measuring apparatus.
32:23
The system of interest had canonical variables little x and little p. The measurement system had variables capital X and capital P. The interaction Hamiltonian was taken to be proportional to the product little x capital P and this would be very hard to realize in practice, but I wouldn't quibble about that.
32:44
But what I would complain about is that except in an absurd limiting case, the method does not avoid the conversion of the wave function into a statistical mixture and therefore the hypothesis fails, but this was not recognized.
33:03
Let us now consider a system which at the time of measurement T sub m is described by the two state superposition wave function which we have there. At later times, the wave function will evolve according to the Schrodinger equation
33:24
into a wave function like this which differs from the form above by the presence of two exponential factors which have an absolute magnitude of one, but a phase that depends on the time elapsed after the time of measurement.
33:40
T minus T sub m would be the time elapsed. And notice that the probabilities for finding the electron in state one or state two are given by the same expressions as before and they are independent of the time because the exponential factors have unit moduli.
34:02
We will have to repeat this whole operation many times in order to determine the values of the probabilities P sub m. The different members of the ensemble could easily have different waiting times. No doubt this would happen quite naturally
34:27
while we were thinking about how we could determine their energy values. That would of course convert the pure case wave function into a statistical mixture of randomly phased wave functions which would be described by a density matrix instead of a pure case wave function.
34:44
Well, I have to discuss the way that we would tell whether the atom was in state one or two and that is done with a Stern-Gerlach apparatus which we can think of as being a kind of coupling of the system to a system for measurement and that discussion is pretty well known.
35:07
So I think that I shouldn't take the opportunity to go on until the appointed time for termination of the lecture
35:20
but the conclusion with which I would like to leave you is that when one is studying the motion of a massive cylinder of thoroughly macroscopic size, it is going to be awfully hard to know the Hamiltonian acting on the system
35:44
and if you do make a measurement, you will have to take into account with exquisite precision the introduction of any additional terms in the Hamiltonian and no matter what you do, you will lose any knowledge of the wave function of the oscillator
36:02
that you might have had to begin with so that as the measurement procedure goes on, looking again and again to see how the gravity wave is evolving, it will be necessary to take into account the fact that the wave function of the system of interest
36:22
is becoming even less a pure case wave function than it might have been and the result will be that the calculation will have to be done all over again. Caves and Thorne have made the calculation on the basis
36:40
that after every little measurement that they made, they could say that the wave function had collapsed to the value that it would have had if they had the system in an eigenstate, but the procedures for getting a system in an eigenstate of energy are sufficiently elaborate that they would not ever be able to do that.
37:04
Furthermore, in order to conduct the procedures that quantum mechanics requires, they would have to have an ensemble of gravity wave detectors and they would be well off if they had an ensemble of gravity waves falling on the system and a gravity wave is something that you take when you get it
37:21
and you can't be sure the next one will be the same kind of wave. So the final conclusion is that they will not beat the quantum limit, but let them come and tell you about how they did it.