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.*