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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-aremeasured — 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.
    Quantum fluctuations of the vacuum
    The science fiction writer Arthur C. Clarke famously formulated three laws of prediction. The third law, a guide for aficionados of ‘hard science fiction’ (characterized by its emphasis on scientific accuracy), declares that any sufficiently advanced technology is indistinguishable from magic. **
    So, here’s an interesting bit of quantum physics that at first glance looks indistinguishable from magic.
    Take two small metal plates and place them side by side a few millionths of a metre apart in a vacuum, insulated from any external electric and magnetic fields. There is no force between these plates, aside from an utterly insignificant gravitational attraction between them which, for the purposes of this experiment, can be safely ignored.
    Now here comes the magic. Although there can be no force between them, the plates are actually pushed very slightly together.
    This is an effect first identified by the Dutch physicist Hendrik Casimir in 1948.