Quantum Physics is not Weird. On the Contrary.. Paul J. van Leeuwen

for example, when you receive a payment reminder concerning your own obituary, the computer will often get the blame. In that case we apparently grant it some free will.

      Light as little hard balls or as waves

      Newton published also a theory about light. After experiments with refraction of light by prisms, he correctly concluded that white light was a composition of colored light. Colored light could not be split further by prisms. He concluded that light, like all other matter in the universe, consisted of small hard colored balls, corpuscles. That idea explained the observation that light traveled in straight lines. It explained the reflection of light in mirrors quite well by the collision laws of his own mechanics. However, with refraction there were some problems. First of all, how was it possible that those corpuscles could so easily travel through a solid medium like glass. In addition, a French scientist demonstrated with experiments and logic that the corpuscles should travel even faster in a solid medium than in a vacuum.

      Newton, however, had an illustrious contemporary, the Dutch scientist Christiaan Huygens ^[12] (1629-1695), who contested his model of light corpuscles and his idea of absolute movement, In 1678 Huygens proposed in his "Traité de la lumière" ^[13] that light should be viewed as a wave phenomenon. Huygens also convincingly demonstrated that space and all movement within it were only relative, an idea that Einstein later applied in his special theory of relativity. Nevertheless, Newton's scientific fame and status ensured that his corpuscle model and his absolute space both went uncontested in scientific circles throughout the next two hundred and fifty years.

      How did Huygens arrive at his idea about waves? He observed how light behaved in birefringent crystals. In these crystals, for instance calcite, an incident light beam will be split into two beams that each follow a different direction. To explain this effect, Huygens assumed that light was a wave phenomenon in which the wave vibrated perpendicular to the direction of the light beam. When that vibration takes place in only one direction, the light is called polarized. Sunlight is not polarized and therefore vibrates in all directions perpendicular to the light beam. So, polarization ^[14] is the direction in which a light wave vibrates. Polarization is not limited to light waves. You could call a surface wave, as on water, more or less vertically polarized because the water particles are mainly moving in the vertical direction.

      In birefringent crystals, the propagation speed of the light wave depends on the angle the polarization direction makes with the directions of the crystal lattice. This is because the properties of a birefringent crystal lattice are different when viewed from different angles. This phenomenon is called anisotropy. The two emerging bundles therefore obtain polarizations that are perpendicular to each other.

      Figure 2.5: Birefringence: splitting the incoming wave in two different polarized waves.

      Figure 2.5 shows a simplified image of birefringence in a calcite crystal. The parallelogram represents the crystal. The incoming non-polarized wave arrives from the left on the surface of the crystal. Inside the crystal, the portion that is vertically polarized more or less retains its speed and therefore continues in a straight line. The speed of the horizontally polarized portion of the wave is however reduced, causing it to break twice, on both incidence and exit. The result is two separate parallel and perpendicularly polarized beams of light.

      Figure 2.6 shows the explanation of the propagation of light as Huygens saw it. Huygens proposed that each part of a wave front became the source of a new circular expanding wave front extending forward, something he named an elementary wave source. The new wave front could be found by drawing the tangent line along those elementary waves.

      Figure 2.6: Huygens principle of light propagation.

      Imagine yourself looking from above at a swimming pool with a deep part C₁ and a shallow part C₂. See figure 2.7. Parallel running wave fronts enter from above left - crests light gray, troughs dark gray - arriving at an oblique angle at the border between C₁ and C₂, which is here a border between deep and shallow water. Waves slow down when rolling from deep into shallow water. The wave speed in C₂, which is the shallow part of the swimming pool, will therefore be less than in C₁. So the distance between the wave crests, which is the wavelength, will have to be smaller in C₂.

      Figure 2.7: Huygens principle of light refraction.

      Source: Wikimedia Commons.

      In order not to lose the continuity of the wavefronts across the boundary between the two media, the waves in C₂ have to change direction. According to Huygens principle, this angle can be found by supposing that each part of the wavefront passing the boundary between C₁ and C₂ becomes a new circular wave source - an elemental wave source - with the radius of the expanding circular front now corresponding to the slowed wave speed. The tangent line along the resulting circular wavefronts then represents the new wave front. So the direction of the movement of the wave - drawn here with the black arrows, pointing perpendicular to the wave fronts - bends away from the boundary. According to Huygens, this explains Snell's law ^[15] for the refraction of light waves.

      So it was Huygens' idea that each point of a wavefront can be considered as a new elemental wave source expanding in a circular fashion and that the resulting wave would be simply the sum of all those elemental waves. However, he could not explain with this elementary wave model why the waves expanding backwards from those elementary sources could not be treated in the same way.

      Figure 2.8: Wave refraction explained with contiguous wave fronts. φ₁ is the angle of incidence, φ₂ is the angle of refraction.

      It's not necessary for you to understand and follow Huygen's elemental waves, it is enough when you just think about the wavefronts having to be contiguous when crossing the boundary. The parallel lines figure 2.8 depict the parallel traveling wavefronts. The waves in C₂ do run slower than in C₁ while their frequency remains the same, which means that their wavelength λ₂ in C₂ has to be smaller than their wavelength λ₁ in C₁. This should be clear from figure 2.8.

      In order to remain both contiguous and parallel, the wave fronts entering C₂ must change their direction at the boundary. In C₂ they will have to run more parallel to the boundary. The dashed line in figure 2.8 drawn perpendicular to the boundary between C₁ and C₂ is called the normal. The angles with the normal, φ₁ and φ₂, are called the angle of incidence and of refraction. You should understand from this that the angle of refraction is smaller than the angle of incidence when the wave speed is slower in medium C₂.

      So, in general, when the wave speed is slower in the medium it enters, the wave fronts will tend to run more parallel to the boundary between the two media. This effect explains a phenomenon that you can easily observe standing on the beach. Perhaps you have noticed that incoming waves often will run almost parallel to the beach as they reach the shore. That is because the shallower the water, the slower the wave speed will be. Running slower and slower the closer they get to the shore, the wave fronts will, with each further slowing, change direction a little bit, finally running almost parallel to the shore. We can imagine Huygens, living near the