By Anna Reeves
Many of you will probably have heard about quantum mechanics, and that it supposedly supports spirituality and mysticism, through films like “What the Bleep do we know” or Fritjof Capra’s book “The Tao of Physics”. I should warn you however that not all physicists agree with all of this, so you should be careful not to make uninformed claims about what quantum mechanics supposedly says, so as not to alienate those physicists further. I have for example heard spiritual people claiming things like “Quantum mechanics says that everything is possible.” It sounds grand and spiritual but is not true.
Having just attended a university lecture course on the foundations of quantum mechanics, I am now in a position to tell you exactly what quantum mechanics says and what it doesn’t. It actually goes much further than the wave mechanics that Capra explains in his book, and the implications of the full theory are much more profound than I have seen explained in such popular treatments. It not just takes apart our materialistic view of the world, but our entire picture of a “world”.
I will begin by quoting the five fundamental postulates of quantum mechanics. Everything else can be derived from these postulates, so they really are all that quantum mechanics says. Don’t worry if you don’t understand the equations and some of the terminology, I will explain what I think is important for you to understand.
I will then give my personal interpretation of the meaning of quantum mechanics and leave you to form your own. I just ask that, whenever you talk about this subject to others, you distinguish between what quantum mechanics says and what your own interpretation of it is.
The postulates of quantum mechanics
- The state of a system is a vector/ket in Hilbert space, denoted by |ψ>.
- Any quantity associated with the system that can be measured corresponds to a Hermitian operator on the Hilbert space and is called an observable.
The commutator of the position and momentum operators is ih/2π. All other operators bear the same functional relationship to these two operators as do the classical quantities.
- If a measurement of an observable is made on a system in state |ψ>, the possible outcomes of the measurement are the eigenvalues of the operator, and the probability of obtaining outcome λi is given by |i|ψ>|2, where |vi> is the eigenvector corresponding to the eigenvalue λi.
- Immediately after a measurement, the system will be in the state |vi>, the eigenvector corresponding to the measured eigenvalue.
- Between measurements, the state evolves according to the time-dependent Schroedinger equation:
H|ψ, t> = ih/2π(d/dt)|ψ, t>
I will begin with the first postulate, which is the most important one.
First of all, “system” can be anything – typically elementary particles, atoms, molecules, but in principle also macroscopic objects like footballs, cups, plates, planets, or even the whole universe.
Secondly, although quantum physicists talk about the “state” of a system, they do not mean what we usually think of being the state of a system – its position, mass, velocity, energy etc. What they mean is that the system has a so-called “Hilbert space” associated with it, which is nothing more than an abstract mathematical set in which the elements of the set (called “vectors” or “kets”) have certain mathematical properties, and that at every moment in time the system is one of the elements of the set. I’m running into problems with language here; I don’t want to say that the system is “described by” or “represented by” this element of the mathematical set, because that would make you think that the mathematics is just describing some underlying physical reality. But this is not so. There is no reality (my lecturer actually said this!), only the mathematics. This abstract mathematical vector is all there is to the system.
(To avoid confusion: By vector we don’t mean an arrow with a direction, but an abstract mathematical object with certain mathematical properties.)
But what about the “properties” of the system? Does it not have to have properties like position, velocity, energy, form, colour, etc? In classical mechanics, which has formed our picture of the world, every object exists and moves around in space and time and has intrinsic properties like these. Not so in quantum mechanics. Here, such properties are called “observables” and they only take on definite values if someone measures them. This is what the next two postulates are concerned with. They give the mathematical procedures for predicting the possible outcomes of measurements and their probabilities. The procedure is the same for every observable, you just have to use the operator which corresponds to the observable you are intending to measure. Here’s how it goes: (You can skip this paragraph if you want)
Every operator has certain vectors associated with it, the so-called “eigenvectors” of this operator. They have the defining property that if the operator operates on them, you get the same vector back, just multiplied by a constant value, called the eigenvalue of that eigenvector. So the first thing you do is to determine the eigenvectors and eigenvalues of the operator in question. Usually this is done by representing the operator as a matrix and applying the standard mathematical procedures for finding them. The eigenvalues found thus are the possible outcomes of a measurement of this observable. Note that they depend solely on the operator, not on the state the system is in. The state of the system only determines the probabilities with which you will get the possible outcomes. They are found thus: The state-vector |ψ> can be thought of as a linear combination of the eigenvectors of the operator, i.e. it consists of some percentage of the first eigenvector, some other percentage of the second eigenvector, and so on. This percentage, expressed as a fraction of 1, and then squared gives the probability of measuring the eigenvalue corresponding to that eigenvector. Sounds strange? It works, though. The predictions made in this way by quantum mechanics have been experimentally tested over and over again and have never been falsified.
The “collapse postulate”
Postulate number 4 deals with one of the strangest aspects of quantum mechanics, the fact that a measurement actually alters the state of the system being measured. At the precise instant of the measurement being made, the state of the system jumps from |ψ> to |vi>, the vector corresponding to the measurement result obtained. Before the measurement, |ψ> contained fractions of various other eigenvectors of the operator, but at the moment of the measurement, all other eigenvectors except for one of them simply vanish. Where did they go? I think this shows that you should not think of |ψ> as being anything real.
It also makes it clear that a “measurement” is not really measuring something that was there all along, it produces the property being measured.
But how do we know that |ψ> had not been |vi> all along, and we just didn’t know? First of all, we can do experiments where we prepare systems in any state that we want, so we can prepare it to be in a state which we know is not |vi>. Secondly, it is up to us and our free will which property of the system we measure. If we choose to measure another observable, this corresponds to a different operator, and this operator can have other eigenvectors associated with it, so the system has different choices of vectors to collapse into. (This btw is the basis of the uncertainty principle: Because the system cannot collapse into different eigenvectors at the same time, it is not possible to measure two properties simultaneously if they have different eigenvectors.) And thirdly, the eigenvectors which vanish at the moment of measurement can have had an effect on the time evolution of the state before the measurement.
The probabilities calculated in quantum mechanics are not expressions of our ignorance of the “true state of the system”. The “state” of a system while nobody is doing a measurement on it is |ψ>, nothing else, and it contains everything that can possibly be known about the system. This collapse of |ψ> at the moment of measurement does happen, as has been shown experimentally.
As Heisenberg, one of the founding fathers of quantum mechanics, says:
“The scientific method of analysing, explaining and classifying has become conscious of its limitations, which arise out of the fact that by its intervention science alters and refashions the object of investigation. In other words, method and object can no longer be separated. The scientific world-view has ceased to be a scientific view in the true sense of the word.”
“The atomic physicist has had to resign himself to the fact that his science is but a link in the infinite chain of man’s argument with nature, and that it cannot simply speak of nature ‘in itself’.”
The “Measurement Problem”
As you will probably already have noticed, the postulates talk of measurements and outcomes of measurements, but nowhere do they define what a measurement actually is. It’s quite embarrassing for quantum physicists to have to admit that their otherwise rigorous and exact theory contains such a vague and ill-defined concept. Does for example a photon hitting a detector constitute a measurement? If so, where exactly does the measurement take place? At the instant in time that the photon hits the surface of the detector material? Or when the detector registers it, for example by making a click or displaying a dot on a computer screen? Or does it happen when the person doing the experiment sees the dot on the screen or hears the click?
Now this is a question of the interpretation of quantum mechanics, and it is still an open question with many different opinions and theories about it. In practice, it makes virtually no difference to the predictions of what the observer will see at which point in the chain from the photon to the observer you put the measurement in order to do your calculations. But it makes a big difference if you wonder about what is “really going on”, because the collapse of the state-vector supposedly happens at the precise moment of the measurement taking place.
In principle, all interactions between particles can be treated entirely quantum-mechanically, i.e. their Hilbert spaces and state vectors can be combined in a certain way to give a new Hilbert space and state vector, which evolves in time according to the Schroedinger equation. There would be no inherent need to apply the collapse postulate at all, if there were not these “conscious observers”, i.e. us, observing nature.
The “Copenhagen Interpretation” of Quantum Mechanics
This is the standard interpretation which most physicists use. It is basically a way of allowing you to do your calculations without bothering much about the philosophical problem. It divides the world up into the “microscopic” and “macroscopic”, and treats microscopic objects (e.g. electrons, atoms, molecules) as vectors in Hilbert space and macroscopic objects, like apparatuses in the laboratory, as classical objects (i.e. objects with fixed properties). Whenever a microscopic object interacts with a macroscopic one, that constitutes a measurement of one of the properties of the microscopic object. This approach works well in practice, because the errors incurred in treating a macroscopic object as a classical instead of a quantum object are so small that they are practically unobservable.
But to anyone interested in a philosophical understanding of quantum mechanics, this is clearly a very unsatisfying picture. How could there be such a fundamental difference between microscopic and macroscopic objects? Don’t macroscopic objects consist of microscopic ones? And where exactly do you draw the line between microscopic and macroscopic? The fact remains that according to the fundamental postulates, quantum mechanics can be applied to the whole world, including macroscopic objects.
A short note on “wave mechanics”
Some of you may have heard about wavefunctions, and paradoxes like whether particles are waves or particles. Forget about all that. Particles are vectors in Hilbert space. The wavefunction is just the representation of this vector in the position-representation, i.e. in the basis of eigenvectors of the position operator. The vector can just as well be represented in bases of other observables, like momentum or energy or whatever, and then you would get different functions. So there is nothing real or fundamental about the wavefunction. You can’t think of it as existing in space and time. It is just a representation of the mathematics. So there is no paradox.
My personal interpretation of the meaning of quantum mechanics
Since every physical process can in theory (if not in practice) be described as vectors in Hilbert space evolving according to the Schroedinger equation, without the need for a “measurement” taking place until we enter the scene, I think it is obvious that it is our consciousness which “makes the measurements” that the postulates speak of. For a materialistic physicist who regards consciousness as a result of some strange processes in our braincells, this is of course difficult to accept, so that is why so many of them say that they don’t understand quantum mechanics. That is also why some of them have devised alternative interpretations or modifications which circumvent the need for an observer, for example the many-worlds interpretation, the de Broglie – Bohm pilot wave interpretation, or the Ghirardi-Rimini-Weber modification.
Now I don’t believe in any of those. I have no problem with quantum mechanics the way it is. I think it is telling us that consciousness is all there really is, that there is no “world out there” which exists objectively independent of whether we observe it. Because you can’t really regard this vector in Hilbert space as anything real, can you? It is just a mathematical abstraction. This becomes even more clear when you learn that there are other pictures of what is going on. The picture of a vector |ψ> evolving in time is just one of them; you can equally well keep |ψ> fixed at some point in time and then evolve the operators in time instead. Both pictures make exactly the same predictions for measurements you can make. So what is really going on? Which picture is the correct picture of what is happening? Obviously, these questions have no answers. There is nothing “really going on”. It is only our mind which thinks that there is a world that exists independently of what we are conscious of right now. But now quantum mechanics is telling us that this assumption that we make is wrong. I like this quote from Niels Bohr, one of the founding fathers of quantum mechanics: “There is no quantum world. There is only an abstract quantum mechanical description. It is wrong to think that the task of science is to find out how nature “is”. “
Loop Quantum Gravity
One of the leading contenders for a quantum mechanical treatment of general relativity (currently a purely “classical” theory) is loop quantum gravity. I find it very convincing because, unlike for example string theory, it makes no assumptions. It arises naturally if you keep the basic principles of general relativity and apply quantum mechanics to space-time. You can then treat volume, area and time as observables and calculate probabilities for measuring certain values of them. It turns out that these values are discrete and not continuous, btw. But the important thing to take from this is: Not even space and time exist when you are not measuring/observing them.
Beyond Quantum Mechanics?
As I have mentioned before, many physicists who are uncomfortable with the implications of quantum mechanics for the existence of the world and its inherent randomness have tried to devise new theories which reproduce the predictions of quantum mechanics. Such theories are called “hidden-variable theories” because they assume that particles have variables which determine the outcomes of measurements of its properties, but which can not themselves be measured. There has been an important development on this front which I want to briefly mention. Quantum mechanics is inherently non-local, essentially because the collapse of the state-vector happens instantaneously even if its position-representation (the wavefunction) is spread out in space. This was another thing that Einstein didn’t like about quantum mechanics, and he called it “spooky action at a distance”.
A physicist called John Bell derived certain inequalities which must hold if you assume locality and hidden variables. Quantum mechanics predicts that they are violated in certain cases, and it has been experimentally confirmed that they are violated in exactly this way. This is an important result because it is so general: No local hidden-variable theory can be consistent with these experimental results.
Put another way, either particles really have no properties and no variables associated with them when you are not making any measurement on them, or reality is non-local, or both.
There are hidden-variable theories like for example the de Broglie – Bohm pilot wave theory, but they are necessarily very non-local.