by Steve Bryson
Last time, we learned that waves have a lot to do with particles. We learned that the wavelength of a wave determines the particle's state of motion and the strength of the wave determines the particle's possible position. However, a wave having a well defined wavelength is somehow incompatible with that wave being localized in space. If the wave is localized, it is not clear what you mean by wavelength, as the wavelength of a wave is the distance between exactly repeating points on the wave and if a wave is localized it cannot repeat over a great distance. Thus the state of motion and the position of a particle corresponding to that wave are somehow incompatible. This is known as the Heisenberg uncertainty principle, and is now a well established principle in physics.
Adding waves to make new waves
In figuring out what this means it now becomes very important for us to understand the relationship between a particle and it's associated wave. The first thought (due to Einstein and DeBroglie) was that the wave somehow "guided" the particle in a way that was not understood. To understand how this idea did not work out we need to know some more things about waves.
To motivate the following discussion, consider the following wave:
(This is actually a part of the wave which repeats forever.)
This is a rather complicated wave. I know, however, that it is really the sum of two simple sine waves (because this is how I drew it):
The two simple sine waves on top are of the same strength and slightly different wavelengths. The complicated wave at the bottom is the result of adding the two simple sine waves. Thus, in some sense, the complicated wave is made out of the two simple sine waves. Where the two sine waves both have wiggles in the same direction, the sum of the two waves is a wiggle twice as large in the same direction. Where the two sine waves have wiggles in opposite directions, they add up to no wiggle at all. Stare at the above figure for a while and see if you can see this.
We call the process of adding two waves to make a new wave interference. We say that the two waves interfere with each other to make a new wave when they add together as in the above example. The really fun thing about waves is that any wave, no matter how complicated, may be expressed as a (possibly infinite) sum of simple sine waves.
Now how is one to interpret interference in the context of the quantum rules? If the complicated wave were the quantum mechanical wave associated to a particle, what could we say about the particle's state of motion? If we take the quantum rules literally, then we must say that the particle somehow has both states of motion corresponding to the two simple sine waves that make up the complicated wave. This is an extremely radical statement, which anticipates our discussion of measurement.
To say that a particle can have two states of motion (in this case two velocities) at once seems completely against our intuition. (We will find later that according to the quantum theory of measurement, we only see one state of motion when we measure even though quantum mechanics says that there may be many.) Yet this is what we must believe if we assume the quantum rules associating waves to particles.
So now we are really up against the wall of either trusting our intuition or believing quantum mechanics. In this position, the physicist always turns to experiment. In 1927 Davisson and Germer performed an experiment in which a beam of electrons showed interference. In fact, the interference observed exactly matched that expected from the quantum mechanical wave associated with the electrons! They did not, however, observe interference in the wave. They observed interference in the particles!
This result could not possibly be stranger. It was not that an individual electron showed interference, yet when you looked at where many electrons went in this experiment, you found that they fell exactly where the interfered quantum wave for the electrons had high strength.
To Summarize: What we observe to be particles in nature seem to behave in a way determined by these mysterious waves. The wave determine the behavior of the particles in the following ways:
-If the particle's wave is a simple sine wave, then the particle has a velocity (actually momentum) determined by the waves wavelength.
-If the particle's wave is just a spike at a point then the particle will be found at that point.
If the particle's wave has a form other than these two, then the particle will be observed to have one of several possible velocities (actually momenta) and may be found wherever the wave has a non-zero strength. The reason that there are several possible velocities for the particle is that no matter what the shape of the wave, that wave may be considered as the sum of many different simple sine waves each giving a possible exact velocity. The stronger the constituent sine wave, the more likely that the velocity given by that sine wave will be found.
This puts us in a position to discuss just how one views the act of measurement in quantum mechanics. Consider the wave on the first page of this handout which was the sum of two simple sine waves:
Let's say that this wave is the quantum wave associated to some particle. Then we can ask "Where will I find the particle and how fast will it be going?". We already know that the particle can be found wherever the strength of the particle is not zero, that is anywhere but the places where there is no wiggle:
probably here------- not here-------------probably here------------------------
Now what can one say about the possible velocities of the particle that one can expect to observe? The particle's wave is exactly the sum of these two simple sine waves:
The two sine waves that make up this wave are of equal strength and slightly different wavelengths. This means that when we measure the velocity of the particle with the above complicated wave we are equally likely to find the particle with one of the two velocities given by the two constituent simple sine waves. Further, these two velocities are the only two possible velocities.
Now let's say that we have measured the particle's velocity and found it to be the velocity determined by the top of the two possible sine waves. Now we know the particle's velocity exactly and therefore the particle's quantum wave is a simple sine wave:
By the act of observation we changed the wave associated to the particle! The change of the wave due to the act of observation is called the collapse of the wave function. The wave, which before the observation was a sum of simple sine waves, collapsed due to the observation into one of the simple sine waves in that sum.
Now let's say that after we measured the particle's velocity exactly (which, by the way, made the particle's position totally indeterminate) we decide to measure the particle's position. Then after the measurement we know the particle's position exactly and so it's wave looks like this:
Now the wave has collapsed from a simple sine wave to a single pulse wave. Now this pulse wave is the sum of many more than two simple sine waves, so if we again measure the velocity of the particle we will collapse the wave back to some simple sine wave.
It is the collapse of the wave function that is at the heart of the controversies surrounding the interpretation of quantum mechanics. The central question in the interpretation of quantum mechanics is: What is the quantum wave and how does it collapse?
When phrased as above, it sounds like the quantum wave associated with a particle is somehow a measure of our knowledge of the particle and nothing more. But in last week's class we found that the quantum waves are directly observed in interference experiments and furthermore these waves explain the quantization of electron orbits in atoms. In other words we need to give some reality to the waves besides our knowledge in order to understand the physics we see.
What is that reality? The most popular views are summarized in the next section of this handout.
One thing I want to make clear at this point. The mathematics of quantum mechanics tells you what kind of quantum waves you will find in given situations. The mathematics does not obviously tell you about the collapse of the wave function or now it comes about. The collapse of the wave function has to be inserted into quantum mechanics as an interpretation (except possibly for the many worlds interpretation).
Interpretations of the Wave Aspects of Matter
As we have been learning, matter in the world appears to have wave aspects. This is a short description of how different physicists view these wave aspects as parts of their physical world. There are several questions that come up:
Is the wave function a fundamental aspect of reality (as opposed to arising from some more fundamental particle system)?
Why is the probability of finding a particle proportional to the strength of the wave?
What is the relationship between particles and their associated waves?
Why does the wave function seem to collapse in the act of observation?
The waveguide theory (DeBroglie, Einstein, Bohm): The wave is a fundamental aspect of reality which is separate from particles which are also fundamental. Thus there two kinds of objects in the world, waves and particles. The particles are in some yet to be understood way 'guided' by the waves and that is why the probability of finding a particle is given by the strength of the wave. The wave is also controlled by the particles, so when we observe a particle we actually change that particle's associated wave. Thus the wave does not actually collapse.
A vocal minority of physicist takes this point of view these days, and it was one of the first interpretations.
The Copenhagen Interpretation (Bohr, Heisenberg and a host of others): We just do not know what the wave associated to a particle is. We do, however, know how to predict observations from the wave function. Thus let's just make those predictions in terms of things that we do know (classical observables) and ask the above questions only during happy hour, not in the labs. This amounts to saying that observable quantities only make sense in the context of the observation. Put more strongly, it is improper to speak of, say, a particle's velocity unless you describe how you measure it. Thus as velocity is measured in a completely different way than position, it is no surprise that you cannot know the velocity and the position of a particle at the same time. In fact, it is not clear that it is meaningful to say that a particle has a velocity and a position at the same time, as you cannot even in principle measure them in the same experiment.
This is the popular point of view, in that it allows physics to proceed.
Hidden variables (Bohm): The wave function is not fundamental. It arises from some averaging process over more classical particle properties that we just do not yet know about. All the above questions are therefore misplaced and we should be worrying about what these hidden more fundamental particle properties are.
Due to the work of Bell and the observations by Aspect and others, we now know that in order to correctly describe the world hidden variable theories must postulate influences that travel infinitely fast. According to relativity theory, these signals would perhaps then enable one to send messages backwards in time thus playing hell with the notion of cause and effect. On the other hand, orthodox quantum theory correctly predicts all observations made so far without such problems, so hidden variable theories become somewhat less palatable.
Consciousness theories (Wigner): The world is built exactly as quantum theory says it is, and the wave function is the fundamental object which when is itself quantized gives rise to all particle phenomena. The collapse of the wave function is caused by the interaction of consciousness with the physical system described by the wave. Thus the world is not as consciousness sees it until consciousness sees it. The probability interpretation somehow arises out of the interaction of the world with consciousness (in a way that has yet to be understood).
While this point of view has it's charms as a philosophical view of consciousness observing the world, it's construction as a physical theory is highly problematical. Not a widely held view.
Many Worlds (Everett, Dewitt, Grahm, originally Wheeler but Wheeler has moved away from this lately): The world is built exactly as quantum theory says it is, and the wave function is the fundamental object which when is itself quantized gives rise to all particle phenomena. There is in fact no mystery generated by the above questions. All you have to do is take the mathematical structures of quantum theory as literal representations of how the world is structured. Then you discover that the wave function never collapses and the world continually splits into many worlds in which all the quantum allowed possibilities occur. However, no one branch of the split can ever know of any others and so it feels like the wave function collapsed. Also discover that the probability interpretation falls in your lap from the mathematics of quantum theory.
It is accepted that this point of view is technically correct and consistent, but due to the stated splitting of universes not many physicists actually believe it. Also, as the predictions of the many worlds interpretation exactly matches the predictions of quantum mechanics many physicists do not feel a need to choose it so they stay Copenhagen. I feel a need to stress, however, that the many worlds point of view is not strictly an interpretation--it is merely an observation of what the mathematics of quantum mechanics says, in particular if you some part of a quantum wave the world would appear to you exactly as it appears to us.
My own personal view (embedded as it is in the many worlds approach) on the general topic of the physical implications of quantum theory is less technically expressed in the following, which is an excerpt from an essay that I wrote some time ago:
Let us look at what quantum mechanics itself says reality is.
The matter of the universe is not made of particles. Spacetime, the setting of the universe, is not populated by tiny baseballs that are moving around hitting one another. Rather spacetime is the setting for waves in a very different space that is completely outside of our personal experience, though our experience is made of the interactions of these waves. What we experience as particles is, in the reality of quantum mechanics, an intricate and subtle tapestry woven by the interfering of these waves with one another. (It is important to understand that the quantum mechanical waves are not the particles.) Particles arise from the these waves in much the same way that shapes arise in a painting viewed from afar. Even more strikingly, when these waves interfere with each other, many new strands of the tapestry are created. These new strands will never weave with one another, but go off their own ways creating whole new separate realms of experience. This is known to the physicist as the 'many worlds interpretation of quantum mechanics'. It follows directly by taking quantum mechanics literally.
Let us now dip lightly into the main controversies surrounding quantum mechanics.
The quantum mechanical view of reality has direct implications for our experiential reality. Inherent in the way that particles arise out of the interactions of the quantum waves is the fact that the particles do not behave in a deterministic way. Rather than making precise predictions about the properties of particles, quantum mechanics only gives probabilities for certain particle processes to occur. This has been very hard for many physicists to accept. Many physicists have felt that as quantum mechanics cannot make detailed predictions about particles it must be, in some way, an incomplete description of reality. Then the probabilities that arise out of quantum mechanics reflect our ignorance of what is happening in nature. These people have proposed other theories of nature that would be more complete and underlay quantum mechanics. The most common versions of these theories are called hidden variable theories.
John Bell thought hard about these hidden variable theories. He realized that if some (actually any) hidden variable theory is the underlying reality (and if interactions in spacetime only depend on what is happening at the place of the interaction), then experiments should come out one way, and if quantum mechanics is the underlying reality the experiments should come out another way. This is the content of Bell's theorem. The experiments have been done and they support quantum mechanics as the underlying reality. While the final Judgement is not yet in, it seems that there can be no nice hidden variable theory underlying quantum mechanics. This means that the properties of particles are inherently probabilistic, leading some physicists to conclude that there is no such thing as an objective, determined reality. These physicists are insisting on a reality composed particles behaving in a deterministic way. Some even hope that something like a deterministic hidden variable theory will yet come along.
Yet quantum mechanics is a completely deterministic theory. It determines the behavior of the abstract waves in the abstract space that quantum mechanics says is the reality. Since particles are not in the underlying reality of quantum mechanics it is no surprise that the properties of particles do not act like the elements of an objective reality.