Farewell to Reality (7 page)

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.

Heisenberg eventually bowed to the pressure. He accepted Bohr's view and the Copenhagen interpretation was born.

The uncertainty principle is not limited to position and momentum. It applies to other pairs of physical properties, called
conjugate
properties, such as energy and time. It also applies to the different spin orientations of quantum particles.

For example, photon polarization can be ‘vertical' or ‘horizontal', which implies some kind of reference frame against which we judge these orientations. ‘Vertical' must mean vertical with regard to some co-ordinate axis. When applied to polarization, the uncertainty principle tells us that certainty in one co-ordinate axis means complete uncertainty in another. If in the laboratory I fix a piece of Polaroid film so that its transmission axis lies along the
z
axis (say), and I measure photons passing through this film, then I have determined that these photons have vertical polarization measured along the
z
axis, with a high degree of certainty.
^*
This implies a high degree of uncertainty for polarizations oriented along either the
x
or
y
axis.