by Jim Baggott
Wave particle duality and the Copenhagen interpretation
But this couldn’t simply be a case of reverting to a purely particulate description of light. Nobody was denying all the evidence of wave-like behaviour, such as diffraction and interference. Besides, Einstein had retained the central property of ‘frequency’ in his description of the light quanta, and frequency is a property of waves.
So, how could particles of light also be waves? Particles are by definition localized bits of stuff — they are ‘here’ or ‘there’. Waves are delocalized disturbances in a medium; they are ‘everywhere’, spreading out beyond the point where the disturbance is caused. How could photons be here, there and everywhere?
Einstein believed that photons are first and foremost particles, following predetermined trajectories through space. In this scheme, the trajectories are determined by some other, unguessed property that leads to the appearance of wave behaviour as a result of statistical averaging. According to this interpretation, in the two-slit interference experiment each photon follows a precise and predetermined trajectory. It is only when we have observed a large number of photons that we see that the trajectories bunch together in some places and avoid other places, and we interpret this bunching as interference.
There was another view, however. Danish physicist Niels Bohr and German Werner Heisenberg argued that particles and waves are merely the shadowy projections of an unfathomable reality into our empirical world of measurement and perception. They claimed that it made no sense to speculate about what photons really are. Better to focus on how they appear — in this kind of experiment they appear as waves, in that kind of experiment they appear as particles.
Bohr is credited with the statement:
There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.6
This approach to quantum theory became known as the Copenhagen interpretation, named for the city in which Bohr had established a physics institute where much of the debate about the interpretation of quantum theory took place. At the heart of this interpretation lies Bohr’s notion of complementarity, a fundamental duality of wave and particle behaviour.
Photon spin and polarization
Before we continue with this exploration of the historical development of quantum theory, and what it tells us about the ways in which we try to understand light, it’s important to take a short diversion to look more closely at the properties of photons.
Photons are massless particles which in a vacuum travel at the speed of light. The Particle Data Group, an international collaboration of some 170 physicists that reviews data on elementary particles and produces an annual ‘bible’ of recommended data for practitioners, suggests that photons cannot possess a so-called ‘rest mass’ greater than about two millionths of a billionth of a billionth (2 × 10-24) of the rest mass of the electron.* If the photon does have a rest mass, it is indeed very, very small.
Photons are also electrically neutral — they carry no electrical charge. They are instead characterized by their energies, which are directly related to their frequencies or inversely to their wavelengths. This reference to wave-like behaviour, with properties of frequency and wavelength, belies a property called phase that all photons (and in fact all types of quantum particle) possess.
In one sense, the phase of a wave is simply related to the position it has in its peak and trough cycle. However, in practical terms, the phase of a quantum particle can never be observed directly. Measurements reveal properties that are affected by the phase, but not the phase itself. According to the Copenhagen interpretation, we should deal only with what we can measure. Whatever the origin of phase, it results in the observation of behaviour that we interpret in terms of waves, and we must leave it at that.
Photons are also characterized by their intrinsic angular momentum, or what physicists call spin. In classical Newtonian mechanics, we associate angular momentum with objects that spin around some central axis. For a solid body spinning on its axis, such as a spinning top or the earth, the angular momentum is calculated from its moment of inertia (a measure of the body’s resistance to rotational motion) multiplied by the speed of the rotation.
This obviously can’t be applicable for photons. For one thing, photons are massless: they have no central axis they can spin around. They can’t be made to spin faster or slower. The term ‘spin’ is actually quite misleading. It is a hangover from an early stage in the development of quantum theory, when it was thought that this property could be traced to an intrinsic ‘self-rotation’, literally quantum particles spinning like tops. This was quickly dismissed as impossible, but the term ‘spin’ was retained.
As with phase, it doesn’t help to look too closely at the property of spin and ask what a photon is really doing. We do know that the property of spin is manifested as angular momentum. The interactions between photons and matter are governed by the conservation of angular momentum, and experiments performed over many years have demonstrated this. When we create an intense beam of photons (such as a laser beam), selected so that the spins of the photons are aligned, the angular momentum of all the individual photons adds up and the beam imparts a measurable torque. A target placed in the path of the beam will be visibly twisted in the direction of the beam’s rotation.
Physicists characterize the spin properties of quantum particles according to a spin quantum number, which provides a measure of the particles’ intrinsic angular momentum.7 This quantum number can take half-integral or integral values. Quantum particles with half-integral spins are called fermions, named for Italian physicist Enrico Fermi. If we persist in pushing the spinning top analogy, we find that fermions would have to spin twice around their axes in order to get back to where they started (which shows that persisting with classical analogies in the quantum world usually leads only to headaches).
The most important thing to know about fermions is that they are forbidden from occupying the same quantum ‘state’. In this context the state of a quantum particle is defined in terms of the various properties the particle possesses, such as energy and angular momentum. These properties are characterized by their quantum numbers, so a quantum state is defined in terms of its particular set of quantum numbers. Thus when we say that fermions are forbidden from occupying the same quantum state, we mean that no two fermions can have the same set of quantum numbers.
This is called the Pauli exclusion principle, first devised by Austrian physicist Wolfgang Pauli in 1925, and it explains the existence of all material substance and the structure of the periodic table of the elements.
The photon has a spin quantum number of 1, which classifies it as a boson, named for Indian physicist Satyendra Nath Bose. Unlike fermions, photons can ‘condense’ into a single quantum state in which many photons possess the same quantum numbers. A laser is just one striking example of such ‘Bose condensation’. It is not possible to create a ‘laser’ using electrons instead of photons, since electrons are not bosons and so cannot be formed in a single quantum state in this way.
Although the spin of a quantum particle is a rather mysterious property, we do know that it can ‘point’ in different directions. Generally speaking, the spin of a quantum particle with spin quantum numbers can point in a number of directions given by twice the value of s plus 1. For electrons, s is equal to ½, so there are two times ½ plus 1, or two different spin directions. We tend to call these ‘spin up’ and ‘spin down’.
This would suggest that the spin of the photon (spin quantum number 1) can point in three different directions (two times 1 plus 1), but there’s a caveat. The spins of particles that travel at the speed of light cannot point in the direction they’re travelling.
The reasons for this are both subtle and complex. But let’s suppose for a minute that we observe a spinning top moving in a straight line across a table. Suppose further that from
our vantage point above the spinning top we observe it to be spinning clockwise as it moves to the right (in the direction of three o’clock). Now, the total speed of a single point on the leading edge of the top is the vector sum of the rotation speed at that point plus the speed of linear motion across the table. So, as the top rotates from three o’clock to nine o’clock, the total speed is a little less than the speed of linear motion. But as the top rotates from nine o’clock back to three o’clock, the total speed is a little greater than the speed of linear motion. And there’s the rub. If the object is a photon moving at the speed of light, then, as we will discover in Chapter 4, Einstein’s special theory of relativity insists that this is a speed that cannot be exceeded. A photon ‘rotating’ such as to give a total speed slightly greater than the speed of light is physically unacceptable and is indeed forbidden by the theory. This reduces the number of possible spin directions for the photon to two.
These two spin ‘orientations’ of the photon correspond to the two known types of circular polarization. In right-circular polarization the amplitude of the photon wave can be thought to rotate clockwise as seen from the perspective of the source of the photon. In left-circular polarization the photon wave rotates counterclockwise. When viewed in terms of rotating waves, it’s perhaps not hard to appreciate the connection between spin and angular momentum.
If circular polarization is unfamiliar to you, don’t worry. Waves are very malleable things. They can be combined in ways that particles cannot. If we don’t like one kind of wave, we can add other kinds to it to form something called a superposition. Add left- and right-circular polarized waves together in just the right ways and you get linear polarization — vertical and horizontal — which is much more familiar.* Photons in different polarization states have been used since the 1970s in some of the most profound tests of the interpretation of quantum theory ever performed.
Quantum probability and the collapse of the wavefunction
Newtonian physics is characterized by a determinism founded on a strong connection between cause and effect. In the old version of empirical reality described by Newton’s theories, if I do this, then that will happen. No question. One hundred per cent.
In quantum theory this kind of predictability is lost. Think about what happens in a two-slit interference experiment when the intensity of the light is reduced so low that, on average, only one photon passes through the apparatus at a time. What we see is that each photon is detected on the other side of the slits, perhaps as a tiny white dot formed on a piece of photographic film. If we wait patiently for lots and lots of photons to pass through the apparatus, one at a time, then we will observe that the white dots form an interference pattern (see Figure 1).
How does this work? The photon is single quantum of light. It is an indivisible particle, unable to split in two and pass through both slits simultaneously. But waves can do just this, and the photon is also a wave. A wave is described mathematically by something called a wavefunction. To explain what happens in this case, we assume that the wavefunction corresponding to a single photon passes through both slits. The secondary wavefunctions emerging from the two slits then diffract and interfere. By the time they reach the photographic film, the wavefunctions have combined to produce bright fringes (constructive interference) and dark fringes (destructive interference).
We now make a further important assertion. The amplitude of the wavefunction at a particular point in space and time provides a measure of the probability that a photon is present.* When the wavefunction corresponding to a single photon interacts with chemicals in the photographic emulsion, it ‘collapses’. At this point the photon mysteriously appears, as an indivisible bundle of energy, and a white dot is formed on the film. Such dots are more likely to be formed in regions of the photographic plate where the amplitude of the wavefunction is high, less likely where it is low. After many photons have been recorded, the end result is a set of interference fringes.
We need to be very careful here. Quantum probability is not like ‘ordinary’ probability, of the kind we associate with tossing a coin. When I toss a coin, I know that there’s a 50 per cent probability that it will land ‘heads’ and a 50 per cent probability that it will land ‘tails’. I don’t know what result I’m going to get for any specific toss because I’m ignorant of all the variables involved — the weight of the coin, speed of the toss, air currents, the force of the coin’s impact on the ground, and so on. If I could somehow acquire knowledge of some of these variables and eliminate others completely, then I might actually be able to use Newton’s laws of motion to compute in advance what result I’m going to get.
Quantum probability is quite different. The Copenhagen interpretation insists that in a quantum system like the two-slit interference experiment with single photons, there are no other variables of which we are ignorant. There is nothing in quantum theory that tells us how an indivisible photon particle navigates its way past the two slits. This doesn’t necessarily mean that we’re missing something; that the theory is somehow incomplete. What it does mean is that the particle picture is not relevant here and we can’t use it to understand what’s going on. We use the wave picture instead and revert to the particle picture only when the wavefunction collapses and the photon is detected.
Figure 1 We can observe quantum particles as they pass, one at a time, through a two-slit apparatus by recording where they strike a piece of photographic film. Each white dot indicates that ‘a quantum particle struck here’. Photographs (a)—(e) show the resulting images when, respectively, 10, 100, 3,000, 20,000 and 70,000 particles have been detected. The interference pattern becomes more and more visible as the number of particles increases. From A. Tonomura et al., American Journal of Physics, 57 (1989), pp. 117—20.
‘We have to remember that what we observe is not nature in itself but nature exposed to our method of questioning,’ Heisenberg said.
This notion of the collapse of the wavefunction doesn’t completely break the connection between cause and effect, but it does weaken it considerably. In the quantum domain if I do this, then that will happen with a certain probability. No certainty. Some doubt. Quantum events like the detection of a photon appear to be left entirely to chance.
Einstein didn’t like it at all:
Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.8
The collapse of the wavefunction continues to plague the current authorized version of reality.
The uncertainty principle
According to Newton’s mechanics, although I might not have the means at my disposal, I can have some confidence that I can measure with arbitrary precision the position and momentum of an object moving through space.
Imagine we fire a cannonball. The cannonball shoots out of the cannon with a certain speed and traces a parabolic path through the air before hitting the ground. It moves through space, passing instantaneously through a series of positions with specific speeds. Although here again I would need to figure out all the different variables at play (wind speed, the precise pull of earth’s gravity), there is nothing in principle preventing me from finding out what these are. After some computation, I produce a map of position and speed throughout the cannonball’s trajectory.
If I know the mass of the cannonball (which I can measure separately), I can calculate the momentum at each point in the trajectory, by multiplying together mass and speed at that point.
Once again, however, in quantum mechanics things are rather different. Quantum particles are also waves. Suppose we were somehow able to localize a quantum wave particle in a specific region of space so that we could measure its position with arbitrary precision. In the wave description, this is in principle possible by combining a large number of wave forms of different frequencies in a superposition (called a ‘wavepacket’),
such that they add up to produce a resultant wave which is large in one location in space and small everywhere else. Great. This gives us the position.
What about the momentum? That’s a bit of a problem. We localized the wave by combining lots of waves with different frequencies. This means that we have a spread of frequencies in the superposition and hence a spread of wavelengths.9 According to French physicist Louis de Broglie, the wavelength of a quantum wave is inversely proportional to the quantum particle’s momentum.10 The spread of wavelengths therefore means there’s a spread of momenta.
We can measure the position of a quantum wave particle with arbitrary precision, but only at the cost of uncertainty in the particle’s momentum.
The converse is also true. If we have a quantum wave particle described by a single wave with a single frequency, this implies a single wavelength which we can measure with arbitrary precision. From de Broglie’s relation we determine the momentum. But then we can’t localize the particle. It remains spread out in space. We can measure the momentum of a quantum wave particle with arbitrary precision, but only at the cost of uncertainty in the particle’s position.
This is Heisenberg’s famous uncertainty principle, which he discovered in 1927.11
Heisenberg initially interpreted his principle in terms of what he thought of as the unavoidable ‘clumsiness’ with which we try to probe the quantum domain with our essentially classical measuring instruments. Bohr had come to a different conclusion, however, and they argued bitterly. The clumsiness argument implied that quantum wave particles actually possess precise properties of position and momentum, and we could in principle measure these if only we had the wit to devise experiments of greater subtlety.
Bohr was adamant that these properties simply do not exist in our empirical reality. This is a reality that consists of things-as-they-are measured — the wave shadows or the particle shadows, as appropriate. Bohr insisted that it is this fundamental duality, this complementarity of wave and particle behaviour, that lies at the root of the uncertainty principle, much as the explanation given above suggests. It is not possible for us to conceive experiments of greater subtlety, because such experiments are inconceivable.