Because the Schrödinger equation has the vital property of linearity mentioned earlier, we can always add two or more solutions and get another. In particular, we can add plane waves. Although each separate solution is a regular wave throughout space, when the solutions are added the interference between them can create surprising patterns. This makes possible the beautiful construction of Schrödinger’s wave packets (Box 15).
BOX 15 Static Wave Packets
A wave with its latent classical histories perpendicular to the wave crests is shown at the top of Figure 47. Using the linearity, we add an identical wave with crests inclined by 5° to the original wave. The lower part of the computer-generated diagram shows the resulting probability density (blue mist). The superposition of the inclined waves has a dramatic effect. Ridges parallel to the bisector of the angle between them (i.e. nearly perpendicular to the original wave fields) appear, and start to ‘highlight’ the latent histories. In fact, these emergent ridges are the interference fringes that show up in the two-slit experiment (Box 11), in which two nearly plane waves are superimposed at a small angle, and also in Young’s illustration of interference (Figure 22).
Much more dramatic things happen if we add many waves, especially if they all have a crest (are in phase) at the same point. At that point all the waves add constructively, and a ‘spike’ of probability density begins to form. At other points the waves sometimes add constructively, though to a lesser extent, and sometimes destructively. Wave patterns like those shown in Figure 48 are obtained.
Figure 47 If two inclined but otherwise identical plane waves like the one at the top are added, the figure at the bottom is obtained. The ridges run along the direction of the light rays’ in the original plane waves. (The top figure shows the amplitude, the bottom the square of the added waves, since in quantum mechanics that measures the probability density.)
Figure 48 brings to mind a passage in A Midsummer Night’s Dream that has haunted poets for centures:
And, as imagination bodies forth
The forms of things unknown, the poet’s pen
Turns them to shapes, and gives to airy nothing
A local habitation, and a name.
The intersection of two wave fields does not result in any distinguished point, just a field of parallel ridges. There is no ‘local habitation’. But if the crests of three or more waves intersect at a common point – so that the waves are in phase there – and their amplitudes are varied appropriately, then a point becomes distinguished. A localized ‘blob’ is formed. As Schrödinger realized with growing excitement in the winter of 1925/6, this begins to look like a particle.
Figure 48 These wave patterns are obtained (from the bottom upward) by adding increasing numbers of plane waves oriented within a small range of directions. All waves have a crest where the ‘spike’ rises from the ‘choppy’ pattern. Their amplitudes also vary in a range, since otherwise ‘ridges’ like those in Figure 47 are obtained.
The pièce de résistance is finally achieved if the waves of different wavelengths move and do so with different speeds. This often happens in nature. In most media – above all in vacuum – light waves all propagate with the same speed. However, in some media the waves of different wavelengths travel at different speeds. Since waves of different wavelengths have different colours, this can give rise to beautiful chromatic effects. In quantum mechanics, the waves associated with ordinary matter particles like electrons, protons and neutrons always propagate at different speeds, depending on their wavelengths. The relationship between the wavelength and the speed of propagation is called their dispersion relation.
Figure 49 has been constructed using such a dispersion relation. The initial ‘spike’ (wave packet) at the bottom is the superposition of waves of different angles in a small range of wavelengths. The dispersion relation makes each wave in the superposition move at a different speed. At the initial time, the waves are all in phase at the position of the ‘spike’, but the position at which all the waves are in phase moves as the waves move. The ‘spike’ moves! Its positions are shown at three times (earliest at the bottom, last at the top). This wave packet disperses quite rapidly because relatively few waves have been used in its construction. In theoretical quantum mechanics, one often constructs so-called Gaussian wave packets, which contain infinitely many waves all perfectly matched to produce a concentrated wave packet. These persist for longer.
It is a remarkable fact about waves in general and quantum mechanics in particular that the wave packet moves with a definite speed, which is known as the group velocity and is determined by the dispersion law. It is quite different from the velocities of any of the individual waves that form the packet. Only when there is no dispersion and all the waves travel at the same speed is the velocity of the packet the same as the speed of propagation of the waves. These remarkable purely mathematical facts about superposition of waves were well known to Schrödinger at the time he made his great discoveries – one of which was that this beautiful mathematics seemed to be manifested in nature.
SCHRÖDINGER’S HEROIC FAILURE
This led him to propose the wave-packet interpretation of quantum mechanics. His main concern was to show how a theory based on waves could nevertheless create particle-like formations. A potential strength of his proposal was that particle-like behaviour could be expected only above a certain scale. Over short enough distances, within atoms or in colliding wave packets, the full wave theory would have to be used, but in many circumstances it seemed that particles should be present. With total clarity, which shines through his marvellous second paper on wave mechanics, he saw that if particles are associated with waves, then in atomic physics we must expect an exact parallel with geometrical optics. There will be many circumstances in which ordinary Newtonian particles seem to be present, but in the interior of atoms, for example, where the potential changes rapidly, we shall have to use the full wave theory. Schrödinger’s second paper contains wonderful insights.
Figure 49 A moving wave packet obtained by adding plane waves having slightly different orientations, wavelengths and propagation velocities. The initially sharply peaked packet disperses quite quickly, as shown in the two upper figures.
Unfortunately, his idea soon ran into difficulties. He had been aware of one from the start. For a single particle, the configuration space is ordinary space, and the idea that the wave function represents charge density makes sense. But he was well aware that his wave function was really defined for a system of particles and therefore had a different value for each configuration of them. I highlighted this earlier by imagining ‘wave-function meters’ in a room which showed the effect of moving individual atoms in models of molecules. It is difficult to see how the wave function can be associated with the charge density of a single particle in space.