The information about the particle’s state of motion is contained in the wavelength and frequency of the wave. The wavelength is inversely proportional to the momentum of the particle, which means that a wave with a small wavelength corresponds to a particle moving with a high momentum (and thus with a high velocity). The frequency of the wave is pro- portional to the particle’s energy; a wave with a high frequency means that the particle has a high energy. In the case of light, for example, violet light has a high frequency and a short wavelength and consists therefore of photons of high energy and high momentum, whereas red light has a low frequency and a long wavelength corresponding to photons of low energy and momentum.

A wave which is spread out like the one in our example does not tell us much about the position of the corresponding particle. It can be found anywhere along the wave with the same likelihood. Very often, however, we deal with situations where the particle’s position is known to some extent, as for example in the description of an electron in an atom. In such

a case, the probabilities of finding the particle in various places must be confined to a certain region. Outside this region they must be zero. This can be achieved by a wave pattern like the one in the following diagram which corresponds to a particle confined to the region X. Such a pattern is called a wave packet.* It is composed of several wave trains with various 157 Beyond the World of Opposites a wave packet corresponding to a particle located somewhere in the region X wavelengths which interfere with each other destructively** outside the region X, so that the total amplitude-and thus the probability of finding the particle there-is zero, whereas they build up the pattern inside X. This pattern shows that the particle is located somewhere inside the region X, but it does not allow us to localize it any further. For points inside the region we can only give the probabilities for the presence of the particle. (The particle is most likely to be present in the centre where the probability amplitudes are large, and less likely near the ends of the wave packet where the amplitudes are small.) The length of the wave packet represents therefore the uncertainty in the location of the particle. The important property of such a wave packet now is that it has no definite wavelength, i.e. the distances between two successive crests are not equal throughout the pattern. There

is a spread in wavelength the amount of which depends on the length of the wave packet: the shorter the wave packet, the larger the spread in wavelength. This has nothing to do with quantum theory, but simply follows from the properties of waves. Wave packets do not have a definite wavelength. Quantum theory comes into play when we associate the wave- length with the momentum of the corresponding particle. If the wave packet does not have a well-defined wavelength, the particle does not have a well-defined momentum.

This means that there is not only an uncertainty in the particle’s position, corresponding to the length of the wave packet, but also an uncertainty in its momentum, caused by the spread in wave- length. The two uncertainties are interrelated, because the spread in wavelength (i.e. the uncertainty of momentum) depends on the length of the wave packet (i.e. on the un- certainty of position). tf we want to localize the particle more precisely, that is, if we want to confine its wave packet to a smaller region, this will result in an increase in the spread in wavelength and thus in an increase in the uncertainty of the particle’s momentum.

The precise mathematical form of this relation between the uncertainties of position and momentum of a particle is known as Heisenberg’s uncertainty relation, or uncertainty principle.

It means that, in the subatomic world, we can never know both the position and momentum of a particle with great accuracy. The better we know the position, the hazier will its momentum be and vice versa. We can decide to undertake a precise measurement of either of the two quantities; but then we will have to remain completely ignorant about the other one. It is important to realize, as was pointed out in the previous chapter, that this limitation is not caused by the imperfection of our measuring techniques, but is a limitation of principle. If we decide to measure the particle’s position precisely, the particle simply does not have a well-defined momentum, and vice versa.

The relation between the uncertainties of a particle’s position and momentum is not the only form of the uncertainty principle. Similar relations hold between other quantities, for example between the time an atomic event takes and the energy it involves. This can be seen quite easily by picturing our wave

hat there are pairs of concepts which are interrelated and cannot be defined simultaneously in a precise way. The more we impose one concept on the physical ‘object’, the more the other concept becomes uncertain, and the precise relation between the two is given by the uncertainty principle.

For a better understanding of this relation between pairs of classical concepts, Niels Bohr has introduced the notion of complementarity. He considered the particle picture and the wave picture as two complementary descriptions of the same reality, each of them being only partly correct and having a limited range of application. Each picture is needed to give a full description of the atomic reality, and both are to be applied within the limitations given by the uncertainty principle.

This notion of complementarity has become an essential part of the way physicists think about nature and Bohr has often suggested that it might be a useful concept also outside the field of physics; in fact, the notion of complementarity proved to be extremely useful 2,500 years ago. It played an essential role in ancient Chinese thought which was based on the insight that opposite concepts stand in a polar-or com- plementary-relationship to each other. The Chinese sages represented this complementarity of opposites by the archetypal poles yin and yang and saw their dynamic interplay as the essence of all natural phenomena and all human situations. Niels Bohr was well aware of the parallel between his concept of complementarity and Chinese thought.

When he visited China in 1937, at a time when his interpretation of quantum theory had already been fully elaborated, he was deeply impressed by the ancient Chinese notion of polar opposites, and from that time he maintained an interest in Eastern culture. Ten years later, Bohr was knighted as an acknowledg- ment of his outstanding achievements in science and important contributions to Danish cultural life; and when he had to choose a suitable motif for his coat-of-arms his choice fell on the Chinese symbol of t’ai-chi representing the complementary relationship of the archetypal opposites yin and yang. In choosing this symbol for his coat-of-arms together with the inscription Contraria sunt complementa (Opposites are com- plementary), Niels Bohr acknowledged the profound harmony between ancient Eastern wisdom and modern Western science.