The Phenomenon of Waves

The phenomenon of waves is encountered in many different contexts throughout physics and can be described with the same mathematical formalism whenever it occurs. The same mathematical forms are used to describe a light wave, a vibrating guitar string, a sound wave, or a water wave.

In quantum theory, these forms are used again to describe the waves associated with particles. This time, however, the waves are much more abstract. They are closely related to the statistical nature of quantum theory, i.e. to the fact that atomic phenomena can only be described in terms of probabilities.

The information about the probabilities for a particle is contained in a quantity called the probability function, and the mathematical form of this quantity is that of a wave, that is to say, it is similar to the forms used for the description of other types of waves. The waves associated with particles, however, are not ‘real’ three-dimensional waves, like water waves or sound waves, but are ‘probability waves’; abstract mathematical quantities which are related to the probabilities of finding the particles in various places and with various properties.

The introduction of probability waves, in a sense, resolves the paradox of particles being waves by putting it in a totally new context; but at the same time it leads to another pair of opposite concepts which is even more fundamental, that of existence and non-existence. This pair of opposites, too, is transcended by the atomic reality. We can never say that an atomic particle exists at a certain place, nor can we say that it does not exist. Being a probability pattern, the particle has tendencies to exist in various places and thus manifests a strange kind of physical reality between existence and non-existence. We cannot, therefore, describe the state of the particle in terms of fixed opposite concepts. The particle is not present at a definite place, nor is it absent.

It does not change its position, nor does it remain at rest. What changes is the probability pattern, and thus the tendencies of the particle to exist in certain places.

In the words of Robert Oppenheimer, If we ask, for instance, whether the position of the electron remains the same, we must say ‘no’; if we ask whether the electron’s position changes with time, we must say ‘no’; if we ask whether the electron is at rest, we must say ‘no’; if we ask whether it is in motion, we must say ‘no’.’ The reality of the atomic physicist, like the reality of the Eastern mystic, transcends the narrow framework of opposite concepts. Oppenheimer’s words thus seem to echo the words of the Upanishads, It moves. It moves not. It is far, and It is near. It is within all this, And It is outside of all this.8 Force and matter, particles and waves, motion and rest, existence and non-existence-these are some of the opposite or contradictory concepts which are transcended in modern physics. Of all these opposite pairs, the last seems to be the most fundamental, and yet, in atomic physics we have to go even beyond the concepts of existence and non-existence. This is the feature of quantum theory which is most difficult to accept and which lies at the heart of the continuing dis- cussion about its interpretation. At the same time, the trans- cending of the concepts of existence and non-existence is also one of the most puzzling aspects of Eastern mysticism. Like the atomic physicists, the Eastern mystics deal with a reality which lies beyond existence and non-existence, and they frequently emphasize this important fact. Thus Ashvaghosha:

Suchness is neither that which is existence, nor that which is non-existence, nor that which is at once existence and non-existence, nor that which is not at once existence and non-existence.9 Faced with a reality which lies beyond opposite concepts, physicists and mystics have to adopt a special way of thinking, where the mind is not fixed in the rigid framework of classical logic, but keeps moving and changing its viewpoint. In atomic physics, for example, we are now used to applying both the particle and the wave concept in our description of matter. We have learned how to play with the two pictures, switching from one to the other and back, in order to cope with the atomic reality. This is precisely the way in which the Eastern mystics think when they try to interpret their experience of a reality beyond opposites. In the words of Lama Govinda, The Eastern way of thinking rather consists in a circling round the object of contemplation . . . a many-sided, i.e. multi-dimensional impression formed from the superimposition of single im- pressions from different points of view.‘lO To see how one can switch back and forth between the particle picture and the wave picture in atomic physics, let us examine the concepts of waves and particles in more detail. A wave is a vibrational pattern in space and time. We can look at it at a definite instant of time and will then see a periodic pattern in space, as in the following example. This pattern is characterized by an amplitude A, the extension of the vibration, and a wavelength L, the distance between two successive crests. a wave pattern Alternatively, we can look at the motion of a definite point of the wave and will then see an oscillation characterized by a certain frequency, the number of times the point oscillates

back and forth every second. Now let us turn to the particle picture. According to classical ideas, a particle has a well- defined position at any time, and its state of motion can be described in terms of its velocity and its energy of motion. Particles moving with a high velocity also have a high energy. Physicists, in fact, hardly use ‘velocity’ to describe the particle’s state of motion, but rather use a quantity called ‘momentum’ which is defined as the particle’s mass times its velocity. Quantum theory, now, associates the properties of a probability wave with the properties of the corresponding particle by relating the amplitude of the wave at a certain place to the probability of finding the particle at that place. Where the amplitude is large we are likely to find the particle if we look for it, where it is small, unlikely. The wave train pictured on p. 155, for example, has the same amplitude throughout its length, and the particle can therefore be found anywhere along the wave with the same likelihood.* 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.